A Novel Regression Model for Fractiles: Formulation, Computational Aspects, and Applications to Medical Data

Abstract

Covariate-related response variables that are measured on the unit interval frequently arise in diverse studies when index and proportion data are of interest. A regression on the mean is commonly used to model this relationship. Instead of relying on the mean, which is sensitive to atypical data and less general, we can estimate such a relation using fractile regression. A fractile is a point on a probability density curve such that the area under the curve between that point and the origin is equal to a specified fraction. Fractile or quantile regression modeling has been considered for some statistical distributions. Our objective in the present article is to formulate a novel quantile regression model which is based on a parametric distribution. Our fractile regression is developed reparameterizing the initial distribution. Then, we introduce a functional form based on regression through a link function. The main features of the new distribution, as well as the density, distribution, and quantile functions, are obtained. We consider a brand-new distribution to model the fractiles of a continuous dependent variable (response) bounded to the interval (0, 1). We discuss an R package with random number generators and functions for probability density, cumulative distribution, and quantile, in addition to estimation and model checking. Instead of the original distribution-free quantile regression, parametric fractile regression has lately been employed in several investigations. We use the R package to fit the model and apply it to two case studies using COVID-19 and medical data from Brazil and the United States for illustration.

1. Introduction, State-of-the-Art, and Motivation

We provide in this section an introduction, the bibliographical review, our objectives, and the plan of this article, as well as a motivating example of the proposed methodology from medicine.

1.1. Bibliographical Review

When information about indices, percentages, proportions, and rates is relevant in an area of research, data derived from continuous random variables constrained to a range of possible values frequently appear. This type of information has become crucial during the COVID-19 pandemic to investigate issues regarding infected and recovered cases, as well as deaths worldwide [1,2]. However, this pandemic also has inducted indirect aspects related to morbidity. For example, the mentioned pandemic produced serious problems of a sedentary lifestyle because of physical inactivity. The medical applications presented in our investigation are associated with these two aspects, that is, COVID-19 pandemic and sedentary lifestyle.

To analyze data constrained to a range of possible values, the regression technique is a statistical method which is popularly used and largely utilized for parameter estimation and hypothesis testing [3]. One estimates the expectation (conditional) of the dependent or response variable employing this technique, while the independent variables (covariates or explanatory variables) are assumed to take a specific value. The response conditional mean is often described by the covariates utilizing a linear or non-linear equation. However, if the response includes extreme atypical values or outliers, common in different types of data, the estimated conditional mean is affected by these outliers, and consequently, the predictions will be inaccurate.

Under the unit mean response modeling, the beta regression [4] comes to mind first when connecting the conditional expectation of the response established over the unit interval with covariates. The beta regression has been proposed by reparameterizing its probability density function (PDF) according to its expected value. This parameterization can be applied to probability distributions with a closed-form expectation, and any model parameter can be transformed by using this expected value.

A fractile is a point on a PDF curve such that the area under the curve between that point and the origin is equal to a specified fraction. The fractile or quantile regression (QR) technique was introduced in [5,6], which can be presented alternatively to a mean regression [7,8,9]. The QR originally proposed has no assumptions about the distribution of the response and tends to be robust to outlying observations. This property is an advantage of the QR in relation to the mean regression. Similarly to this regression, a QR based on a probability distribution can be postulated via a reparameterized form of any distribution parameter according to its quantile function (QF). The QR models relate a set of covariates to specific quantiles of a response variable, with the median being often considered; see [10] for a study and implementation of this class of models and their applications, including QR for continuous, discrete, and unit responses.

Several parametric QR structures were postulated based on diverse statistical distributions, including the arc-secant hyperbolic Weibull (ASHW), Johnson SB (JOSB), Kumaraswamy (KUMA), log-extended exponential-geometric (LEEG), unit-Birnbaum-Saunders (UBSA), unit-Burr-XII (UBUR), unit-Chen (UCHE), unit-Gompertz (UGOM), unit-Gumbel (UGUM), unit-logistic (ULOG), unit-generalized half-normal-E (UGHE), unit-generalized half-normal-X (UGHX), unit-Weibull-E (UWEE), unit-Weibull-X (UWEX), and Vasicek (VASI) models [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. Some of these distributions are based on classical models [27,28,29]. In [10], the authors provided a number of parametric QRs, as well as examples and software packages, which can be employed to model rates and indices.

Nevertheless, to our best knowledge, QR structures under the Weibull–Marshall–Olkin distribution have not been established until now. Note that the Weibull–Marshall–Olkin distribution is widely flexible in statistical terms to model diverse types of data, such as shown in the theoretical shape analysis and empirical applications considered in the present study. This flexibility allows the Weibull–Marshall-Olkin distribution to provide a better performance in different classes of statistical modeling in relation to its competing distributions, as confirmed in the mentioned empirical applications here stated.

Therefore, our objective in the present article is to formulate a novel Weibull–Marshall–Olkin QR model. First, we introduce a reparameterization of the Weibull–Marshall–Olkin distribution by adding a quantile parameter and so we formulate the novel regression model, providing high flexibility and allowing us to state a framework such as in GLM—generalized linear models—[30]. We demonstrate that our new model outperforms recently proposed QR models in terms of model fitting.

1.2. An Motivation Case Study from Medicine

Researchers are interested in statistical models for describing data with quantiles that incorporate the effect of covariates. However, the statistical methods for this type of data analysis remain a matter of discussion and are not yet fully disseminated.

A motivation for our investigation comes from a real-world medical study associated with a data set related to the body fat percentage of people who received assistance in a government hospital of the state of Paraná, in the city of Curitiba, Brazil.

The inclusion criteria to recruit people were the following: (i) healthy people (men and women) between 18 and 90 years old having a body mass index (BMI) of 18.5 and 29.9 kg/m${}^2$; (ii) those who do not take supplements or hormone replacement therapy that might change their appearance; and (iii) those who are physically active and are able to walk without the aid of orthoses or prostheses.

The following persons were excluded from the investigation: (i) people with chronic illnesses; (ii) those taking prescription, illicit drugs, or drugs that affect body composition, such as corticosteroids, thyroid hormones in suppressive doses, or insulin-dependent diabetes; and (iii) those who were thin with a BMI under 18.5 kg/m${}^2$ or with a BMI of around 30 kg/m${}^2$.

For the recruited people (participants), all subjects’ anthropometric measurements (weight and height) were collected the same day after the total body densitometry of the body fat, lean, and bone masses’ total.

All participants filled out the validated Portuguese form of the International Physical Activity Questionnaire (IAPQ) [31], which frequently measures physical activity. The IPAQ-short is made out of eight questions. This questionnaire permits the assessment of week-after-week time spent on proactive tasks of moderate areas of strength for, and in a variety of contexts associated with day-to-day life. These contexts include work, leisure, transportation, and housework, in addition to time spent engaging in straightforward sitting exercises.

Most of the volunteers answered the IPAQ through self-administration or an individual interview. The volunteers were then split into three groups based on how much exercise they had performed. Sedentary individuals do not engage in any form of physical activity for more than ten consecutive minutes each week. At least ten continuous minutes of physical activity at least five days per week or 150 min per week is required to be considered insufficiently active. Assets spend at least 150 min a week doing at least 20 min of vigorous or moderate exercise three times a week, 30 min of walking five times a week, or any additional activity five days a week or more.

The data set was considered in [32], which consists of 298 records about the fat percentage of people who received assistance in a government hospital in Curitiba, Paraná, Brazil. The data set includes two categorical and two continuous covariates, and the five responses correspond to the fat percentages at the android, arms, body, gynaecoid, and legs. The person’s BMI (in kg/m${}^2$) and age (in years) are continuous covariates. In contrast, gender (female or male) and IPAQ (sedentary patient, insufficiently active, or active) are the categorical covariates.

We consider the fat percentage at the legs as the response variable mainly, but the other responses are also considered later. We perform a descriptive statistical analysis [33] for continuous variables of a data set related to body fat. Univariate descriptive statistics and correlation analysis for each variable are presented in

Table 1. Univariate descriptive statistics of data related to the fat percentage at the legs.

Variable Grouped by Level	Mean	SD	Minimum	Maximum	1st Quartile	2nd Quartile	3rd Quartile	CS	CK
legs	0.324	0.11	0.068	0.574	0.235	0.319	0.416	0.021	−0.843
legs (by F)	0.407	0.074	0.203	0.574	0.355	0.406	0.459	−0.205	−0.350
legs (by M)	0.241	0.072	0.068	0.444	0.195	0.238	0.289	0.174	0.021
legs (by A)	0.303	0.117	0.068	0.574	0.213	0.290	0.397	0.223	−0.900
legs (by I)	0.342	0.097	0.133	0.563	0.259	0.350	0.423	−0.098	−0.732
legs (by S)	0.359	0.095	0.174	0.565	0.289	0.357	0.442	0.050	−0.988
age	46.000	19.879	18.000	87.000	25.000	47.000	65.000	0.158	−1.359
bmi	24.716	3.151	18.500	29.900	22.300	24.900	27.200	−0.101	−0.943

Where legs is fat (in %) at legs, male (M), female (F), sedentary (S), insufficiently active (I), active (A); and SD is the standard deviation.

. Histograms and scatterplots represent the variables: age, legs, and BMI in Figure 1. The variables age and BMI are substantially correlated. Note that, at 1%, each correlation is statistically significant, encouraging the development of a regression model. When these models are formulated, a collinearity problem may be detected between BMI and age.

Our correlation analysis indicates a Spearman coefficient ($\rho$) that is significant at 1% when testing ${\mathrm{H}}_0:\rho =0$. Regarding the mentioned variables, we report: legs versus BMI with $r=0.230$ and $p\text{-}\mathrm{value}<0.001$; legs versus age with $r=0.284$ and $p\text{-}\mathrm{value}<0.001$; and BMI versus age with $r=0.470$ and $p\text{-}\mathrm{value}<0.001$, where r is the sample correlation coefficient.

Before statistically formulating a QR model, we must evaluate any potential collinearity issues in the exploratory data analysis. The variance inflation factor (VIF) is a more formal and often applied method to determine the collinearity between covariates. If $\mathrm{VIF}>10$, then collinearity can be present. For the covariates under analysis, we report the corresponding VIF: age with $\mathrm{VIF}=1.662$; BMI with $\mathrm{VIF}=1.411$; gender/sex with $\mathrm{VIF}=1.083$; and IPAQ with $\mathrm{VIF}=1.280$. Thus, we can continue with our QR analysis because every VIF value is less than 10. In summary, derived from our exploratory and VIF analyses, we state that a QR structure with high flexibility may describe these data well and better than its competing models. We recall that the Weibull–Marshall–Olkin distribution is widely flexible in statistical terms to model diverse types of data. Then, the new QR model derived in the present study can provide a better performance in relation to its competitors. Thus, the motivation example inspires our research from an empirical perspective.

1.3. Objectives and Plan of Sections

As mentioned, our objective in the present study is to formulate a new alternative QR model based on a Weibull–Marshall–Olkin distribution to relate covariates and specific quantiles of a response variable. Moreover, we demonstrate how our QR model can be used with real data sets.

Our article is organized into sections. We establish the new model and its methodology in Section 2, whereas Section 3 introduces two case studies as illustration. Section 4 details more information on our study’s findings and discusses them. We provide some conclusions, limitations, and ideas for further investigation in Section 4.

2. Methodology

In this section, the new QR model and its statistical methodology is presented.

2.1. A New Weibull–Marshall–Olkin PDF

Let X follow a Weibull model based on $F(x;\varsigma ,\nu )=1-exp\left[-\left(x/\varsigma \right){}^{\nu }\right]$, that is, on the cumulative distribution function (CDF). Also, consider the Marshall–Olkin method to add a new parameter into a distribution proposed in [34]. PDF, CDF, and QF expressions for the extended Weibull Marshall-Olkin (EWMO) model are, respectively, given by

$$\begin{array}{ccc}\hfill f\left(x;\alpha ,\varsigma ,\nu \right)& =& \frac{\alpha \nu x^{\nu -1}exp\left[-\left(x/\varsigma \right){}^{\nu }\right]}{\varsigma {}^{\nu }{\left\{1-\overline{\alpha }exp\left[-\left(x/\varsigma \right){}^{\nu }\right]\right\}}^2},x>0,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill F\left(x;\alpha ,\varsigma ,\nu \right)& =& \frac{1-exp\left[-\left(x/\varsigma \right){}^{\nu }\right]}{1-\overline{\alpha }exp\left[-\left(x/\varsigma \right){}^{\nu }\right]},Q\left(\tau ;\alpha ,\varsigma ,\nu \right)=\varsigma {\left[-log\left(\frac{1-\tau }{1-\tau \overline{\alpha }}\right)\right]}^{\frac{1}{\nu }},\hfill \end{array}$$

(2)

where $\varsigma >0$ is a scale parameter, $\nu >0$ and $\alpha >0$ are shape parameters, $\overline{\alpha }=(1-\alpha )$ is a tilt parameter, and $0<\tau <1$ is a fractile parameter. As a particular case of the EWMO model, we obtain the extended Marshall–Olkin exponential distribution if $\alpha =1$. The PDF, CDF, and QF stated in (1) and (2) correspond to the Weibull distribution when $\nu =1$. Note that starting with a baseline CDF for a random variable X, $F_0(x;\mathbf{\theta })$, namely, we can obtain the extended Marshall–Olkin distribution with a CDF defined as

$$F(x;\mathbf{\theta },\alpha )=\left(1-\alpha {\overline{F}}_0(x;\mathbf{\theta })\right)/\left(1-\overline{\alpha }{\overline{F}}_0(x;\mathbf{\theta })\right),$$

(3)

where ${\overline{F}}_0(x;\mathbf{\theta })$ defined in (3) is the survival function of X, that is, ${\overline{F}}_0(x;\mathbf{\theta })=1-F_0(x;\mathbf{\theta }),$ and $\mathbf{\theta }={({\theta }_1,\dots ,{\theta }_k)}^{\top }$ is a vector of parameters.

Considering the transformation $X=exp(-Y)$, we define the unit Weibull–Marshall–Olkin (UWMO) model, with PDF, CDF, and QF being specified as

$$\begin{array}{ccc}\hfill f\left(y;\alpha ,\varsigma ,\nu \right)& =& \frac{\alpha \nu {\left[-log\left(y\right)\right]}^{\nu -1}exp\left\{-{\left[-log\left(y\right)/\varsigma \right]}^{\nu }\right\}}{\varsigma {}^{\nu }y{\left(1-\overline{\alpha }exp\left\{-{\left[-log\left(y\right)/\varsigma \right]}^{\nu }\right\}\right)}^2},0<y<1,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill F\left(y;\alpha ,\varsigma ,\nu \right)& =& \frac{\alpha }{exp\left[{\left(-log\left(y\right)/\varsigma \right)}^{\nu }\right]-\overline{\alpha }},Q\left(\tau ;\alpha ,\varsigma ,\nu \right)=exp\left\{-\varsigma log{\left[\left(\alpha +\tau \overline{\alpha }\right)/\tau \right]}^{\frac{1}{\nu }}\right\},\hfill \end{array}$$

(5)

with $\alpha >0$ and $\nu >0$ being parameters associated with the shape of the distribution, $\overline{\alpha }=(1-\alpha )$ being a tilt parameter, and $0<\tau <1$ being a fractile parameter. Note that now $\varsigma >0$ is not a parameter related to the scale of the distribution, since $F\left(y;\alpha ,\varsigma ,\nu \right)$ is not equal to $F(y/\varsigma ;\alpha ,\nu )$. For $\alpha =1$, we have the unit-Weibull model. In applications, the parameters $\alpha$, $\varsigma$, and $\nu$ do not have direct interpretations regarding the observed data. However, we can reparameterize the parameter $\alpha$, taking a one-to-one change into consideration as: $(\alpha ,\varsigma ,\nu )\to (\mu ,\varsigma ,\nu )$, where $\mu =Q$ is a quantile defined in (5). Hence, we have that

$$\alpha =h^{-1}\left(\mu \right)=\frac{\tau }{(1-\tau )}\left(exp\left\{{\left[-\frac{log\left(\mu \right)}{\varsigma }\right]}^{\nu }\right\}-1\right).$$

(6)

From (6), we generate the unit Weibull–Marshall–Olkin quantile (UWMOQ) distribution of parameters $\mu >0$, $\varsigma >0$, $\nu >0$, and $0<\tau <1$. Figure 2, Figure 3 and Figure 4 show plots of potential shapes for the PDF of the UWMOQ distribution for particular values of $\mu$, $\varsigma$, $\nu$, and $\tau$. The graphical plots display skewness to the left and right as well as bathtub shapes of the UWMOQ distribution, showing the wide shape flexibility of the UWMOQ distribution.

2.2. A Novel Quantile Regression Model

Contrasted with regression models that determine the conditional mean of the response variable, we observe that the QR models are a solid substitute. Since they estimate the conditional quantiles of the covariates, these models are also robust regressions. QR models based on probability distributions are generally proposed via a reparameterized form of a distribution parameter according to their QF. Since the QF of the UWMOQ distribution is transformable by its parameters, it is possible to suggest an alternative QR model derived from the UWMOQ PDF.

Now that we have the reparameterized PDF based on (6), we can concentrate on the QR model employing the UWMOQ distribution. Assume observations obtained from the reparameterized UWMOQ PDF with parameters ${\mu }_i$, $\mathbf{\beta }$, and known quantile level $\tau$. Hence, a UWMOQ model for the quantile ${\mu }_i$ is established by

$$g\left({\mu }_i\right)={\mathit{x}}_i{\mathbf{\beta }}^{\top },i\in \{1,\dots ,n\},$$

(7)

where $\mathbf{\beta }={({\beta }_0,{\beta }_1,\dots ,{\beta }_p)}^{\top }$ and ${\mathit{x}}_i=(1,x_{i1},\dots ,x_{ip})$ represent the QR coefficients vector (unknown) and i-th vector of values (known) of the covariates, respectively. Similarly to (7), we can model ${\varsigma }_i$ and ${\nu }_i$. Note that g stated in (7) is the link that connects the covariates to the quantile of the response variable. In addition, the link is determined by the distribution domain. Since the domain of the UWMOQ distribution is bounded on the interval (0, 1), logit or probit links can be used to relate the response variable quantile to covariates. Thus, we utilize a link related to the logit function, which is displayed by

$$g\left({\mu }_i\right)=\mathrm{logit}\left({\mu }_i\right)=log\left(\frac{{\mu }_i}{1-{\mu }_i}\right)={\mathit{x}}_i{\mathbf{\beta }}^{\top },i\in \{1,\dots ,n\}.$$

2.3. Estimating the Model Parameters

Presently, the estimation of model parameters is carried out with the maximum likelihood (ML) method, considering the parameterizations $(\alpha ,\varsigma ,\nu )$ described. Here, we also propose a diagnostic checking for regression [35] based on a residual analysis.

Assume a sample of size n, $\mathit{Y}={(Y_1,\dots ,Y_n)}^{\top }$, namely, extracted from a UWMO distribution, which has parameters $\alpha$, $\varsigma$, $\nu$, and let its observations be stated as $\mathit{y}={(y_1,\dots ,y_n)}^{\top }$. Using the PDF defined in (4), the proportional log-likelihood function for ${(\alpha ,\varsigma ,\nu )}^{\top }$ is associated with

$$\begin{array}{ccc}\hfill \ell (\alpha ,\varsigma ,\nu ;\mathit{y})& \propto & nlog\left(\alpha \right)+nlog\left(\nu \right)-n\nu log\left(\varsigma \right)-\nu \sum _{i=1}^{n}log\left(y_i\right)+\sum _{i=1}^{n}log\left\{\frac{{\left[-log\left(y_i\right)\right]}^{\nu -1}exp\left\{-{\left[-\frac{log\left(y_i\right)}{\varsigma }\right]}^{\nu }\right\}}{1-\left(1-\alpha \right)exp\left\{-{\left[-\frac{log\left(y_i\right)}{\varsigma }\right]}^{\nu }\right\}}\right\}.\hfill \end{array}$$

(8)

ML estimates of ${(\alpha ,\varsigma ,\nu )}^{\top }$ are generated by maximizing the logarithmic likelihood function established in (8) and resolving the score elements $U_{\alpha }=\partial \ell \left(\alpha ,\varsigma ,\nu ;\mathit{y}\right)/\partial \alpha =0$, $U_{\varsigma }=\partial \ell \left(\alpha ,\varsigma ,\nu ;\mathit{y}\right)/\partial \varsigma =0$, and $U_{\nu }=\partial \ell \left(\alpha ,\varsigma ,\nu ;\mathit{y}\right)/\partial \nu =0$ accordingly, expressed as

$$\begin{array}{ccc}\hfill U_{\alpha }& =& \frac{n}{\alpha }+\sum _{i=1}^n\frac{exp\left\{-{\left[-\frac{log\left(y_i\right)}{\varsigma }\right]}^{\nu }\right\}}{1-\left(1-\alpha \right)exp\left\{{\left[-\frac{log\left(y_i\right)}{\varsigma }\right]}^{\nu }\right\}},\hfill \\ \hfill U_{\varsigma }& =& \frac{1}{\varsigma }\left\{\nu \sum _{i=1}^n\frac{{\left[-\frac{log\left(y_i\right)}{\varsigma }\right]}^{\nu }}{1-\left(1-\alpha \right)exp\left\{-{\left[-\frac{log\left(y_i\right)}{\varsigma }\right]}^{\nu }\right\}}-n\nu \right\},\hfill \\ \hfill U_{\nu }& =& \frac{n}{\nu }-nlog\left(\varsigma \right)-\sum _{i=1}^{n}log\left(y_i\right)\hfill \\ & & -\sum _{i=1}^n\frac{log\left[-log\left(y_i\right)\right]\left(1-\alpha \right)exp\left\{-{\left[-\frac{log\left(y_i\right)}{\varsigma }\right]}^{\nu }\right\}+{\left[-\frac{log\left(y_i\right)}{\varsigma }\right]}^{\nu }log\left[-\frac{log\left(y_i\right)}{\varsigma }\right]}{.}\hfill \\ & & +\sum _{i=1}^n\frac{log\left[-log\left(y_i\right)\right]}{1-\left(1-\alpha \right)exp\left\{-{\left[-\frac{log\left(y_i\right)}{\varsigma }\right]}^{\nu }\right\}}.\hfill \end{array}$$

The associated inverse Hessian matrix calculates the asymptotic standard errors (SEs) of $\widehat{\alpha }$, $\widehat{\varsigma }$, and $\widehat{\nu }$. The quantile reparameterization of the UWMO distribution requires considering that

$$\alpha =\frac{\tau }{(1-\tau )}\left\{exp\left({\left[-\frac{log\left(\mu \right)}{\varsigma }\right]}^{\nu }\right)-1\right\}.$$

Thus, we want to evaluate how the covariates affect ${(\alpha ,\varsigma ,\nu )}^{\top }$ by means of

$$\begin{array}{ccc}\hfill g_1\left({\mu }_i\right)& =& {\eta }_i={\beta }_0+{\beta }_1{x}_{i1}+\cdots +{\beta }_p{x}_{ip},\hfill \\ \hfill g_2\left({\varsigma }_i\right)& =& {\zeta }_i={\delta }_0+{\delta }_1{z}_{i1}+\cdots +{\delta }_q{z}_{iq},\hfill \\ \hfill g_3\left({\nu }_i\right)& =& {\rho }_i={\gamma }_0+{\gamma }_1{v}_{i1}+\cdots +{\gamma }_r{v}_{ir},\hfill \end{array}$$

using functions of the explanatory variables ${\mathit{x}}_i={(1,x_{i1},\dots ,x_{ip})}^{\top }$, ${\mathit{z}}_i={(1,z_{i1},\dots ,z_{iq})}^{\top }$ and ${\mathit{v}}_i={(1,v_{i1},\dots ,v_{ir})}^{\top }$, respectively. We assume that $g_1$, $g_2$, and $g_3$ consist of functions which are strictly monotonic, twice-differentiable, and that map the $\tau$-th quantile ${\mu }_i$, ${\varsigma }_i$, and ${\nu }_i$ to the real line. Suitable choices of $g_1$ are the Cauchy: Cauchit link; maximum extreme value: log–log link; minimum extreme value: complementary log–log link; the inverse CDF of the logistic: logit link; normal: probit link; distributions. We consider the log link for $g_2$ and $g_3$. Observe that $\mathit{x}$, $\mathit{z}$, and $\mathit{v}$ may be identical repetitions or subsets one of another.

We use 1st-order and 2nd-order derivatives of the logarithm of the likelihood function to estimate the model parameters. This log-likelihood function is stated as ${\ell }_i={\ell }_i\left(\mathbf{\Theta }\right)=log\left[f\left(y_i;\mathbf{\Theta },{\mathit{x}}_i,{\mathit{z}}_i,{\mathit{v}}_i,\tau \right)\right]$, and then the score vector is constructed with elements defined as

$$\frac{\partial {\ell }_i}{\partial {\beta }_j}=\frac{\partial {\ell }_i}{\partial {\mu }_i}\frac{\partial {\mu }_i}{\partial {\eta }_i}\frac{\partial {\eta }_i}{\partial {\beta }_j},\frac{\partial {\ell }_i}{\partial {\delta }_j}=\frac{\partial {\ell }_i}{\partial {\varsigma }_i}\frac{\partial {\varsigma }_i}{\partial {\zeta }_i}\frac{\partial {\zeta }_i}{\partial {\delta }_j},\frac{\partial {\ell }_i}{\partial {\gamma }_j}=\frac{\partial {\ell }_i}{\partial {\nu }_i}\frac{\partial {\nu }_i}{\partial {\rho }_i}\frac{\partial {\rho }_i}{\partial {\gamma }_j},$$

where $\mathbf{\Theta }={({\mathbf{\beta }}^{\top },{\mathbf{\delta }}^{\top },{\mathbf{\gamma }}^{\top })}^{\top }$; $\mathbf{\beta }={({\beta }_0,\dots ,{\beta }_p)}^{\top }$ is a $(p+1)\times 1$ vector of parameters related to an $n\times (p+1)$ covariates matrix $\mathit{X}$; $\mathbf{\delta }={({\delta }_0,\dots ,{\delta }_q)}^{\top }$ is a $(q+1)\times 1$ vector of parameters related to an $n\times (q+1)$ matrix of covariates $\mathit{Z}$; and $\mathbf{\gamma }={({\gamma }_0,\dots ,{\gamma }_r)}^{\top }$ is an $(r+1)\times 1$ vector of parameters linked to an $n\times (r+1)$ matrix of covariates $\mathit{V}$. We have the score of the log-likelihood function expressed as $\partial \ell /\partial \mathbf{\beta }={\mathit{X}}^{\top }\mathrm{diag}\left({\mathit{W}}_{\mu }\right){\dot{\ell }}_{\mu }$, $\partial \ell /\partial \mathbf{\delta }={\mathit{Z}}^{\top }\mathrm{diag}\left({\mathit{W}}_{\varsigma }\right){\dot{\ell }}_{\varsigma }$, and $\partial \ell \partial \mathbf{\gamma }={\mathit{V}}^{\top }\mathrm{diag}\left({\mathit{W}}_{\nu }\right){\dot{\ell }}_{\nu },$ with “diag” referring to a diagonal matrix of size n and

$${\mathit{W}}_{\mu }=\left[\frac{\partial {\mu }_1}{\partial {\eta }_1},\dots ,\frac{\partial {\mu }_n}{\partial {\eta }_n}\right],{\mathit{W}}_{\varsigma }=\left[\frac{\partial {\varsigma }_1}{\partial {\zeta }_1},\dots ,\frac{\partial {\varsigma }_n}{\partial {\zeta }_n}\right],{\mathit{W}}_{\nu }=\left[\frac{\partial {\nu }_1}{\partial {\rho }_1},\dots ,\frac{\partial {\nu }_n}{\partial {\rho }_n}\right],$$

$${\dot{\ell }}_{\mu }={\left[\frac{\partial \ell }{\partial {\mu }_1},\dots ,\frac{\partial \ell }{\partial {\mu }_n}\right]}^{\top },{\dot{\ell }}_{\varsigma }={\left[\frac{\partial \ell }{\partial {\varsigma }_1},\dots ,\frac{\partial \ell }{\partial {\varsigma }_n}\right]}^{\top },{\dot{\ell }}_{\nu }={\left[\frac{\partial \ell }{\partial {\nu }_1},\dots ,\frac{\partial \ell }{\partial {\nu }_n}\right]}^{\top }.$$

Then, the corresponding Hessian matrix has elements stated as

$$\begin{array}{ccc}\hfill \frac{{\partial }^2\ell }{\partial \mathbf{\beta }\partial {\mathbf{\beta }}^{\top }}& =& {\mathit{X}}^{\top }\mathrm{diag}\left({\ddot{\ell }}_{\mu \mu }\right)\mathrm{diag}\left({\mathit{W}}_{\mu }^2\right)\mathit{X},\hfill \\ \hfill \frac{{\partial }^2\ell }{\partial \mathbf{\delta }\partial {\mathbf{\delta }}^{\top }}& =& {\mathit{Z}}^{\top }\mathrm{diag}\left({\ddot{\ell }}_{\varsigma \varsigma }\right)\mathrm{diag}\left({\mathit{W}}_{\varsigma }^2\right)\mathit{Z},\hfill \\ \hfill \frac{{\partial }^2\ell }{\partial \mathbf{\gamma }\partial {\mathbf{\gamma }}^{\top }}& =& {\mathit{V}}^{\top }\mathrm{diag}\left({\ddot{\ell }}_{\nu \nu }\right)\mathrm{diag}\left({\mathit{W}}_{\nu }^2\right)\mathit{V},\hfill \\ \hfill \frac{{\partial }^2\ell }{\partial \mathbf{\beta }\partial {\mathbf{\delta }}^{\top }}& =& {\mathit{X}}^{\top }\mathrm{diag}\left({\ddot{\ell }}_{\mu \varsigma }\right)\mathrm{diag}\left({\mathit{W}}_{\mu }\right)\mathrm{diag}\left({\mathit{W}}_{\varsigma }\right)\mathit{Z},\hfill \\ \hfill \frac{{\partial }^2\ell }{\partial \mathbf{\beta }\partial {\mathbf{\gamma }}^{\top }}& =& {\mathit{X}}^{\top }\mathrm{diag}\left({\ddot{\ell }}_{\mu \nu }\right)\mathrm{diag}\left({\mathit{W}}_{\mu }\right)\mathrm{diag}\left({\mathit{W}}_{\nu }\right)\mathit{V},\hfill \\ \hfill \frac{{\partial }^2\ell }{\partial \mathbf{\delta }\partial {\mathbf{\gamma }}^{\top }}& =& {\mathit{Z}}^{\top }\mathrm{diag}\left({\ddot{\ell }}_{\varsigma \nu }\right)\mathrm{diag}\left({\mathit{W}}_{\varsigma }\right)\mathrm{diag}\left({\mathit{W}}_{\nu }\right)\mathit{V},\hfill \end{array}$$

where

$${\ddot{\ell }}_{\mu \mu }=\left[\frac{{\partial }^2\ell }{\partial {\mu }_1\partial {\mu }_1},\dots ,\frac{{\partial }^2\ell }{\partial {\mu }_n\partial {\mu }_n}\right],{\ddot{\ell }}_{\varsigma \varsigma }=\left[\frac{{\partial }^2\ell }{\partial {\varsigma }_1\partial {\varsigma }_1},\dots ,\frac{{\partial }^2\ell }{\partial {\varsigma }_n\partial {\varsigma }_n}\right],{\ddot{\ell }}_{\nu \nu }=\left[\frac{{\partial }^2\ell }{\partial {\nu }_1\partial {\nu }_1},\dots ,\frac{{\partial }^2\ell }{\partial {\nu }_n\partial {\nu }_n}\right],$$

$${\ddot{\ell }}_{\mu \varsigma }=\left[\frac{{\partial }^2\ell }{\partial {\mu }_1\partial {\varsigma }_1},\dots ,\frac{{\partial }^2\ell }{\partial {\mu }_n\partial {\varsigma }_n}\right],{\ddot{\ell }}_{\mu \nu }=\left[\frac{{\partial }^2\ell }{\partial {\mu }_1\partial {\nu }_1},\dots ,\frac{{\partial }^2\ell }{\partial {\mu }_n\partial {\nu }_n}\right],{\ddot{\ell }}_{\varsigma \nu }=\left[\frac{{\partial }^2\ell }{\partial {\varsigma }_1\partial {\nu }_1},\dots ,\frac{{\partial }^2\ell }{\partial {\varsigma }_n\partial {\nu }_n}\right].$$

The ML estimates ${({\widehat{\beta }}_0,\dots ,{\widehat{\beta }}_p)}^{\top }$, ${({\widehat{\delta }}_0,\dots ,{\widehat{\delta }}_q)}^{\top }$, and ${({\widehat{\gamma }}_0,\dots ,{\widehat{\gamma }}_r)}^{\top }$ can be obtained utilizing all-purpose optimization techniques such as conjugate-gradient, quasi-Newton, and Nelder-Mead. These techniques are available in the function optim of R [36] or in SAS proc NLMIXED [37], and they allow us to attain at least a local maximum of the logarithm of the likelihood function. By definition, an ML estimate is the global maximum point of the log-likelihood function, but this estimate can also be a local maximum point or a saddle point. Directions for identifying such points may be found in [38].

Taking into account certain regularity conditions, we have that

$$\widehat{\mathbf{\Theta }}\dot{\sim }{\mathrm{N}}_{p+q+r+3}(\mathbf{\Theta },{\left(\mathcal{I}(\mathbf{\Theta })\right)}^{-1}),$$

(9)

where $\mathcal{I}(\mathbf{\Theta })$ represents the expected information matrix that is written as

$$\mathcal{I}(\mathbf{\Theta })=\mathrm{E}\left[-\frac{{\partial }^2\ell (\mathbf{\Theta })}{\partial \mathbf{\Theta }\partial {\mathbf{\Theta }}^{\top }}\right].$$

(10)

The mentioned regularity conditions include that the true value of the parameter $\mathbf{\Theta }$ must be interior to the parameter space, the log-likelihood function must be thrice differentiable, and its third derivatives must be bounded. The interested reader can consult a standard textbook for more technical details and proofs about such conditions, which are general and well-known in statistical inference; see, for example, [39]. By using the expression given in (9), one can arrive at approximate confidence intervals (CIs). However, to obtain the matrix described in (10), we may use the observed information matrix described by $\mathcal{J}(\mathbf{\Theta })=-{\partial }^2\ell (\mathbf{\Theta })\partial \mathbf{\Theta }\partial {\mathbf{\Theta }}^{\top },$ which must be evaluated at $\mathbf{\Theta }=\widehat{\mathbf{\Theta }}$.

3. Empirical and Computational Aspects

We provide in this section some details of an R package developed by the authors for conducting QR with 15 existing distributions. The new UWMO model introduced in the present article is being added to this package. As an alternative to the new in-progress version of such a package, we developed a SAS macro to perform our numerical examples [37,40]. In addition to the description of the functionality of the R package, in this section, we also present the numerical and empirical applications of our work. First, a Monte Carlo simulation study is conducted. Then, two medical illustrations involving COVID-19 and sports medicine data sets are introduced.

3.1. Computational Package

We employ an R package named unitquantreg, which was developed in the setting stats::lm package. The unitquantreg() function of this package is flexible and permits us, through the ns() function, to utilize regression splines. This function is available in the splines package. Also, we use the gam() function of the mgcv package. The unitquantreg package can be secured from https://github.com/AndrMenezes/unitquantreg (accessed on 30 January 2023) and installed using devtools::install_github(‘‘AndrMenezes/unitquantreg’’). Next, we describe the models of the unitquantreg package and its commands. QR structures implemented in the unitquantreg package use: (i) ASHW; (ii) JOSB; (iii) KUMA; (iv) LEEG; (v) UBSA; (vi) UBUR; (vii) UCHE; (viii) UGOM; (ix) UGUM; (x) ULOG; (xi) UGHE; (xii) UGHX; (xiii) UWEE; (xiv) UWEX; and (xv) VASI distributions. As mentioned, the addition of the UWMO distribution into the unitquantreg package is in progress. The 15 models stated in the R package named unitquantreg follow the standard nomenclature of the R software, employing the prefixes r, p, d, and q for the random generation, CDF, PDF, and QF, respectively. In the case of a distribution with the unit-Weibull structure, we utilize:

• duweibull(x, mu, theta, tau = 0.5, log = FALSE)
• puweibull(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
• quweibull(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
• ruweibull(n, mu, theta, tau = 0.5)

The functions d, p, q, and r employ C++ under the Rcpp package [41,42]. The unitquantreg() function of the unitquatreg package operates such as the frameworks simplexreg(), lm(), cdfquantreg(), betareg(), gamlss(), and glm(), following:

• unitquantreg(formula, data, subset, na.action, tau, family,
•        link = c(‘‘logit’’, ‘‘probit’’, ‘‘cloglog’’, ‘‘cauchit’’),
•        link.theta = c(‘‘log’’, ‘‘sqrt’’, ‘‘identity’’), start = NULL,
•        control = unitquantreg.control(),
•        model = TRUE, x = FALSE, y = TRUE)

Similarly, for example, in the R package named simplexreg [43], a regression structure may be specified with the formula package of R [44]. Hence, to state formulations for both quantile and shape structures, one may define formula = y ~ x1 + x2 | z1 + z2, with y ~ x1 + x2 establishing the QR, where the explanatory variables z1 and z2 are associated with the shape parameter. Information of the generic functions coef, print, plot and summary can be extracted from the function unitquantreg, that is, from

• > methods(class = ‘‘unitquantreg’’)
•  [1] coef confint fitted hnp logLik model.frame model.matrix
•  [8] plot predict print residuals summary terms update vcov

The QR parameters may be estimated by the ML method employing the R package named optimx [45], an optimization function that calls other tools for optimizing in R. Note that optimx unifies such tools, permitting some optimization techniques and sanity checks to be considered. The command unitquantreg.control enables us to model the process, with its default values being stated as:

• method        hessian    gradient  maxit    factr     trace  dowarn    starttests fnscale
• ‘‘L-BFGS-B’’  ‘‘FALSE’’  ‘‘TRUE’’  ‘‘5000’’ ‘‘1e+07’’ ‘‘0’’  ‘‘FALSE’’ ‘‘FALSE’’  ‘‘1’’

with the two most relevant elements being: hessian and gradient, that allow for controlling whether the optimx must employ the analytical Hessian and score elements, respectively, or not. For the 15 distributions that are available in the unitquantreg package, the Hessian and score are programmed in C++ to generate accurate estimates and obtain a good performance computationally. Initial values for $\mathbf{\beta }$, the vector of regression parameters, may be provided by the user. Otherwise, these values for $\mathbf{\beta }$ can be generated from the QR, with the response variable being stated by $y_i^{*}=g_1\left(y_i\right)$, with $g_1$ being the structure that links with $\mu$, that is, with the quantile parameter. The quantreg package [46] has a function named rq that is used to generate the initial values. In the case of ${\delta }_j$, for $j\in \{1,\dots ,q\}$, these values are fixed at $0.1$ when assuming a logarithmic link, and at the value $1.1$ for links related to the inverse and square root functions. We obtain the SEs by means of the observed version of the Fisher matrix. We employ a Cholesky decomposition to reach the numerical stability of the inverse Hessian matrix.

Note that uniquantreg verifies whether the optimization algorithm converges or not. In this last case, a message as “optimization failed to converge” warns the user and then she/he should be cautious about the obtained results. Moreover, conditions as Karush–Kuhn–Tucker must be verified by optimx::optimx function. If these conditions are not met, a message is similarly generated and then the user must once gain be cautious about the reached results. The quantreg package also gives the Moore-Penrose inverse when the Hessian matrix is full rank, and as it is well-known, such a matrix has one negative eigenvalue, at least. We assume a distributional structure for the dependent (response) variable by means of an argument named family, and the following labels are utilized for the members of this family:

• families <- c(‘‘arc-secant hyperbolic Weibull’’ = ‘‘ashw’’,
•            ‘‘Johnson-SB’’ = ‘‘johnsonsb’’,
•            ‘‘Kumaraswamy’’ = ‘‘kum’’,
•            ‘‘log-extended exponential-geometric’’ = ‘‘leeg’’,
•            ‘‘unit-Birnbaum-Saunders’’ = ‘‘ubs’’,
•            ‘‘unit-Burr-XII’’ = ‘‘uburrxii’’,
•            ‘‘unit-Chen’’ = ‘‘uchen’’,
•            ‘‘unit-generalized half-normal-E’’ = ‘‘ughne’’,
•            ‘‘unit-generalized half-normal-X’’ = ‘‘ughnx’’,
•            ‘‘unit-Gompertz’’ = ‘‘ugompertz’’,
•            ‘‘unit-Gumbel’’ = ‘‘ugumbel’’,
•            ‘‘unit-logistic’’ = ‘‘ulogistic’’,
•            ‘‘unit-Weibull’’ = ‘‘uweibull’’)

3.2. Simulation Analyses

To detect performance, our Monte-Carlo study uses empirical values of the root-mean-square error (RMSE) and bias of the ML estimators for the UWMO QR coefficients, $\mathbf{\beta }$, and for the parameters $\varsigma$ and $\nu$ of the UWMOQ distribution. Also, we calculate the empirical coverage probability (CP) of the $95\%$ CI (CP${}_{95\%}$) following the asymptotic normality of the ML estimators.

We consider sample sizes $n\in \{50,70,90,110,130,150\}$; $\tau \in \{0.25,0.50,0.75\}$; $\varsigma \in \{0.50,1.50,2.00\}$, and $\nu \in \{0.50,1.00,2.00\}$ inserted on a regression framework stated as $\mathrm{logit}\left({\mu }_i\right)={\beta }_0+{\beta }_1{z}_{i1}$, for ${\beta }_0=0.50$, ${\beta }_1=0.50$, and $z_{i1}\sim \mathrm{Bernoulli}(p=0.5)$.

We report that the time spent in the simulation study is less than 30 min.

Table 2. Empirical RMSE, bias, and $95\%$ CP (true values: ${\beta }_0=0.50$, ${\beta }_1=0.50$, $\varsigma =0.50$, and $\nu =0.50$) of the ML estimator for the indicated parameter with simulated data.

		Bias				RMSE				CP${}_{95\%}$
$\mathbf{\tau }$	$\mathit{n}$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$
0.25	50	$0.0638$	$-0.0043$	$0.0903$	$0.0081$	$0.6356$	$0.7791$	$0.5878$	$0.1281$	$93.87$	$95.00$	$80.65$	$91.42$
	70	$0.0478$	$0.0060$	$0.1068$	$0.0039$	$0.5499$	$0.6759$	$0.5684$	$0.1104$	$94.30$	$94.49$	$83.18$	$92.54$
	90	$0.0396$	$0.0118$	$0.1131$	$0.0028$	$0.5010$	$0.6077$	$0.5505$	$0.0990$	$94.14$	$94.34$	$85.07$	$93.08$
	110	$0.0357$	$0.0105$	$0.1066$	$0.0020$	$0.4427$	$0.5492$	$0.5203$	$0.0898$	$94.38$	$94.18$	$86.17$	$93.54$
	130	$0.0305$	$0.0089$	$0.1010$	$0.0017$	$0.3957$	$0.4966$	$0.4920$	$0.0834$	$94.42$	$94.46$	$87.29$	$93.46$
	150	$0.0249$	$0.0078$	$0.0978$	$0.0006$	$0.3685$	$0.4619$	$0.4744$	$0.0781$	$94.62$	$94.94$	$88.11$	$94.10$
0.50	50	$-0.0020$	$-0.0025$	$0.1589$	$0.0333$	$0.6048$	$0.7770$	$0.6468$	$0.1702$	$94.16$	$94.28$	$81.11$	$87.76$
	70	$-0.0006$	$0.0069$	$0.1607$	$0.0201$	$0.5259$	$0.6752$	$0.6222$	$0.1489$	$94.08$	$94.01$	$82.87$	$88.64$
	90	$0.0006$	$0.0104$	$0.1643$	$0.0138$	$0.4798$	$0.6037$	$0.6095$	$0.1353$	$94.02$	$94.12$	$83.90$	$89.06$
	110	$0.0028$	$0.0099$	$0.1587$	$0.0107$	$0.4249$	$0.5469$	$0.5868$	$0.1243$	$94.20$	$93.94$	$85.47$	$89.56$
	130	$0.0022$	$0.0085$	$0.1496$	$0.0090$	$0.3806$	$0.4945$	$0.5544$	$0.1161$	$94.42$	$94.14$	$86.22$	$90.28$
	150	$0.0008$	$0.0070$	$0.1401$	$0.0061$	$0.3549$	$0.4596$	$0.5327$	$0.1086$	$94.50$	$94.70$	$87.44$	$91.42$
0.75	50	$-0.0872$	$-0.0173$	$0.2397$	$0.1019$	$0.6291$	$0.7549$	$0.7244$	$0.2487$	$94.12$	$94.67$	$82.90$	$85.40$
	70	$-0.0655$	$-0.0074$	$0.2266$	$0.0738$	$0.5453$	$0.6581$	$0.7018$	$0.2114$	$93.76$	$94.43$	$82.38$	$85.90$
	90	$-0.0528$	$-0.0013$	$0.2236$	$0.0574$	$0.4950$	$0.5902$	$0.6955$	$0.1887$	$93.98$	$94.28$	$82.82$	$86.64$
	110	$-0.0444$	$-0.0002$	$0.2174$	$0.0449$	$0.4399$	$0.5360$	$0.6791$	$0.1730$	$94.24$	$94.02$	$82.95$	$86.00$
	130	$-0.0382$	$-0.0010$	$0.2247$	$0.0403$	$0.3964$	$0.4853$	$0.6680$	$0.1614$	$94.34$	$94.34$	$84.52$	$87.00$
	150	$-0.0348$	$-0.0018$	$0.2190$	$0.0317$	$0.3717$	$0.4520$	$0.6615$	$0.1508$	$93.78$	$94.72$	$84.27$	$87.40$

,

Table 3. Empirical RMSE, bias, and $95\%$ CP (true values: ${\beta }_0=0.50$, ${\beta }_1=0.50$, $\varsigma =0.50$, and $\nu =1.00$) of the ML estimator for the indicated parameter with simulated data.

		Bias				RMSE				CP${}_{95\%}$
$\mathbf{\tau }$	$\mathit{n}$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$
0.25	50	$0.0420$	$0.0012$	$0.0552$	$0.0113$	$0.3212$	$0.4135$	$0.3934$	$0.2461$	$92.82$	$93.98$	$89.02$	$92.04$
	70	$0.0305$	$0.0005$	$0.0447$	$0.0058$	$0.2777$	$0.3527$	$0.3429$	$0.2083$	$93.72$	$93.98$	$90.84$	$93.22$
	90	$0.0250$	$0.0020$	$0.0382$	$0.0041$	$0.2522$	$0.3144$	$0.3079$	$0.1864$	$93.68$	$94.12$	$91.85$	$93.64$
	110	$0.0217$	$0.0022$	$0.0328$	$0.0041$	$0.2231$	$0.2850$	$0.2797$	$0.1660$	$93.96$	$94.06$	$92.84$	$94.00$
	130	$0.0183$	$0.0017$	$0.0293$	$0.0033$	$0.1995$	$0.2576$	$0.2594$	$0.1553$	$94.48$	$94.42$	$93.46$	$94.22$
	150	$0.0150$	$0.0012$	$0.0233$	$0.0023$	$0.1858$	$0.2399$	$0.2350$	$0.1420$	$94.32$	$94.88$	$93.60$	$94.98$
0.50	50	$0.0061$	$-0.0005$	$0.0977$	$0.0522$	$0.3031$	$0.4097$	$0.4264$	$0.3212$	$93.38$	$93.46$	$90.11$	$88.68$
	70	$0.0045$	$-0.0010$	$0.0724$	$0.0309$	$0.2642$	$0.3500$	$0.3703$	$0.2778$	$93.54$	$93.66$	$91.17$	$89.56$
	90	$0.0041$	$0.0004$	$0.0557$	$0.0197$	$0.2409$	$0.3113$	$0.3318$	$0.2546$	$93.56$	$93.86$	$91.57$	$89.90$
	110	$0.0042$	$0.0013$	$0.0460$	$0.0147$	$0.2138$	$0.2828$	$0.3027$	$0.2338$	$93.70$	$93.72$	$92.12$	$90.50$
	130	$0.0033$	$0.0010$	$0.0401$	$0.0132$	$0.1916$	$0.2559$	$0.2790$	$0.2179$	$94.12$	$94.16$	$92.92$	$90.96$
	150	$0.0022$	$0.0006$	$0.0311$	$0.0089$	$0.1788$	$0.2380$	$0.2549$	$0.2028$	$94.06$	$94.60$	$93.16$	$92.14$
0.75	50	$-0.0327$	$-0.0177$	$0.2026$	$0.1701$	$0.3126$	$0.3983$	$0.5293$	$0.4552$	$93.02$	$93.52$	$89.28$	$86.08$
	70	$-0.0249$	$-0.0158$	$0.1542$	$0.1181$	$0.2726$	$0.3422$	$0.4644$	$0.3885$	$93.04$	$94.00$	$87.74$	$86.12$
	90	$-0.0199$	$-0.0124$	$0.1250$	$0.0906$	$0.2479$	$0.3049$	$0.4187$	$0.3477$	$93.34$	$94.20$	$87.03$	$87.10$
	110	$-0.0169$	$-0.0099$	$0.1012$	$0.0698$	$0.2213$	$0.2780$	$0.3837$	$0.3190$	$93.94$	$93.56$	$86.85$	$86.92$
	130	$-0.0149$	$-0.0092$	$0.0904$	$0.0621$	$0.1997$	$0.2525$	$0.3568$	$0.2992$	$93.74$	$94.10$	$86.83$	$87.34$
	150	$-0.0136$	$-0.0086$	$0.0726$	$0.0484$	$0.1870$	$0.2350$	$0.3295$	$0.2795$	$93.30$	$94.56$	$87.25$	$87.90$

,

Table 4. Empirical RMSE, bias, and $95\%$ CP (true values: ${\beta }_0=0.50$, ${\beta }_1=0.50$, $\varsigma =0.50$, and $\nu =2.00$) of the ML estimator for the indicated parameter with simulated data.

		Bias				RMSE				CP${}_{95\%}$
$\mathbf{\tau }$	$\mathit{n}$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$
0.25	50	$0.0236$	$-0.0052$	$-0.0046$	$0.0182$	$0.1646$	$0.2210$	$0.1904$	$0.4499$	$92.26$	$93.74$	$93.46$	$93.50$
	70	$0.0172$	$-0.0046$	$-0.0037$	$0.0105$	$0.1423$	$0.1889$	$0.1620$	$0.3730$	$93.28$	$93.96$	$94.72$	$94.30$
	90	$0.0141$	$-0.0030$	$-0.0023$	$0.0092$	$0.1291$	$0.1681$	$0.1444$	$0.3287$	$93.60$	$93.88$	$95.44$	$94.58$
	110	$0.0121$	$-0.0021$	$-0.0014$	$0.0088$	$0.1142$	$0.1527$	$0.1307$	$0.2928$	$93.74$	$94.26$	$95.78$	$94.90$
	130	$0.0101$	$-0.0019$	$-0.0005$	$0.0086$	$0.1020$	$0.1383$	$0.1208$	$0.2713$	$94.54$	$94.32$	$96.06$	$94.96$
	150	$0.0085$	$-0.0020$	$-0.0012$	$0.0062$	$0.0951$	$0.1289$	$0.1108$	$0.2485$	$94.10$	$94.78$	$95.94$	$95.34$
0.50	50	$0.0035$	$-0.0046$	$0.0047$	$0.0629$	$0.1545$	$0.2180$	$0.1970$	$0.5780$	$93.02$	$93.56$	$94.34$	$89.83$
	70	$0.0024$	$-0.0041$	$0.0000$	$0.0350$	$0.1348$	$0.1861$	$0.1716$	$0.4984$	$93.18$	$93.76$	$95.12$	$90.66$
	90	$0.0021$	$-0.0026$	$-0.0019$	$0.0222$	$0.1229$	$0.1651$	$0.1552$	$0.4493$	$93.68$	$94.08$	$95.36$	$91.60$
	110	$0.0020$	$-0.0016$	$-0.0016$	$0.0186$	$0.1092$	$0.1504$	$0.1418$	$0.4098$	$93.68$	$93.80$	$95.48$	$92.38$
	130	$0.0016$	$-0.0014$	$-0.0003$	$0.0191$	$0.0979$	$0.1365$	$0.1307$	$0.3785$	$93.78$	$94.18$	$95.42$	$92.66$
	150	$0.0010$	$-0.0014$	$-0.0018$	$0.0124$	$0.0913$	$0.1268$	$0.1215$	$0.3535$	$94.04$	$94.54$	$95.58$	$93.50$
0.75	50	$-0.0139$	$-0.0186$	$0.0368$	$0.1896$	$0.1604$	$0.2160$	$0.2182$	$0.7219$	$92.30$	$93.38$	$89.30$	$88.70$
	70	$-0.0106$	$-0.0156$	$0.0199$	$0.1352$	$0.1401$	$0.1854$	$0.1954$	$0.6536$	$92.84$	$93.72$	$88.64$	$87.92$
	90	$-0.0085$	$-0.0125$	$0.0115$	$0.1006$	$0.1276$	$0.1648$	$0.1796$	$0.5981$	$93.16$	$94.18$	$89.18$	$87.89$
	110	$-0.0071$	$-0.0102$	$0.0074$	$0.0808$	$0.1139$	$0.1508$	$0.1668$	$0.5533$	$93.70$	$93.68$	$88.98$	$88.46$
	130	$-0.0065$	$-0.0095$	$0.0048$	$0.0678$	$0.1028$	$0.1373$	$0.1599$	$0.5273$	$93.76$	$94.02$	$88.68$	$88.08$
	150	$-0.0060$	$-0.0084$	$0.0015$	$0.0508$	$0.0963$	$0.1276$	$0.1498$	$0.4917$	$93.22$	$94.40$	$89.74$	$88.84$

,

Table 5. Empirical RMSE, bias, and $95\%$ CP (true values: ${\beta }_0=0.50$, ${\beta }_1=0.50$, $\varsigma =1.50$, and $\nu =0.50$) of the ML estimator for the indicated parameter with simulated data.

		Bias				RMSE				CP${}_{95\%}$
$\mathbf{\tau }$	$\mathit{n}$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$
0.25	50	$0.0380$	$-0.0356$	$-0.4592$	$0.0023$	$0.7550$	$0.8812$	$0.9959$	$0.1077$	$94.46$	$96.02$	$72.59$	$93.82$
	70	$0.0288$	$-0.0075$	$-0.3521$	$0.0007$	$0.6545$	$0.7760$	$0.9550$	$0.0910$	$94.35$	$95.32$	$76.19$	$94.62$
	90	$0.0243$	$0.0062$	$-0.2902$	$0.0006$	$0.5970$	$0.7030$	$0.9189$	$0.0807$	$94.62$	$94.80$	$78.18$	$94.68$
	110	$0.0232$	$0.0091$	$-0.2407$	$0.0005$	$0.5275$	$0.6416$	$0.8944$	$0.0722$	$94.38$	$94.31$	$80.00$	$94.90$
	130	$0.0198$	$0.0077$	$-0.1993$	$0.0010$	$0.4727$	$0.5806$	$0.8651$	$0.0648$	$94.42$	$94.60$	$82.10$	$95.28$
	150	$0.0149$	$0.0078$	$-0.1592$	$0.0006$	$0.4404$	$0.5427$	$0.8517$	$0.0596$	$94.58$	$94.98$	$82.90$	$95.38$
0.50	50	$-0.0486$	$-0.0287$	$-0.3356$	$0.0133$	$0.7303$	$0.8918$	$1.0055$	$0.1390$	$94.77$	$95.85$	$77.73$	$90.32$
	70	$-0.0350$	$-0.0017$	$-0.2630$	$0.0059$	$0.6350$	$0.7837$	$0.9731$	$0.1224$	$94.52$	$95.00$	$80.15$	$91.08$
	90	$-0.0283$	$0.0123$	$-0.1943$	$0.0046$	$0.5792$	$0.7095$	$0.9485$	$0.1096$	$94.40$	$94.56$	$82.56$	$91.98$
	110	$-0.0214$	$0.0147$	$-0.1578$	$0.0035$	$0.5121$	$0.6463$	$0.9123$	$0.0993$	$94.36$	$94.11$	$84.45$	$92.60$
	130	$-0.0185$	$0.0130$	$-0.1156$	$0.0027$	$0.4582$	$0.5854$	$0.8938$	$0.0928$	$94.56$	$94.44$	$85.38$	$92.74$
	150	$-0.0178$	$0.0119$	$-0.0791$	$0.0013$	$0.4273$	$0.5457$	$0.8837$	$0.0865$	$94.68$	$94.82$	$86.98$	$93.68$
0.75	50	$-0.1576$	$-0.0327$	$-0.3069$	$0.0500$	$0.7760$	$0.8854$	$1.0537$	$0.1889$	$95.25$	$96.08$	$76.61$	$87.18$
	70	$-0.1191$	$-0.0066$	$-0.2609$	$0.0323$	$0.6698$	$0.7783$	$1.0367$	$0.1640$	$94.42$	$95.38$	$77.88$	$87.58$
	90	$-0.0976$	$0.0073$	$-0.2285$	$0.0229$	$0.6072$	$0.7026$	$1.0191$	$0.1485$	$94.34$	$94.88$	$79.49$	$88.02$
	110	$-0.0810$	$0.0118$	$-0.1730$	$0.0179$	$0.5371$	$0.6428$	$1.0023$	$0.1370$	$94.38$	$94.19$	$80.98$	$88.24$
	130	$-0.0712$	$0.0102$	$-0.1314$	$0.0147$	$0.4821$	$0.5823$	$0.9917$	$0.1287$	$94.66$	$94.54$	$82.26$	$88.70$
	150	$-0.0636$	$0.0097$	$-0.1057$	$0.0103$	$0.4511$	$0.5438$	$0.9790$	$0.1207$	$94.18$	$94.80$	$83.77$	$89.68$

,

Table 6. Empirical RMSE, bias, and $95\%$ CP (true values: ${\beta }_0=0.50$, ${\beta }_1=0.50$, $\varsigma =1.50$, and $\nu =1.00$) of the ML estimator for the indicated parameter with simulated data.

		Bias				RMSE				CP${}_{95\%}$
$\mathbf{\tau }$	$\mathit{n}$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$
0.25	50	$0.0239$	$-0.0054$	$-0.2572$	$-0.0012$	$0.4183$	$0.5146$	$0.8332$	$0.1835$	$93.10$	$93.96$	$77.75$	$95.36$
	70	$0.0153$	$-0.0036$	$-0.1758$	$-0.0022$	$0.3631$	$0.4422$	$0.7859$	$0.1512$	$93.42$	$93.94$	$80.87$	$96.06$
	90	$0.0120$	$-0.0003$	$-0.1236$	$0.0000$	$0.3303$	$0.3959$	$0.7510$	$0.1287$	$93.90$	$93.98$	$82.17$	$95.60$
	110	$0.0110$	$0.0002$	$-0.0921$	$-0.0002$	$0.2934$	$0.3589$	$0.7225$	$0.1155$	$93.84$	$93.92$	$83.62$	$95.70$
	130	$0.0091$	$-0.0004$	$-0.0656$	$-0.0002$	$0.2633$	$0.3243$	$0.6940$	$0.1076$	$94.16$	$94.40$	$85.41$	$95.66$
	150	$0.0067$	$-0.0006$	$-0.0510$	$-0.0008$	$0.2457$	$0.3022$	$0.6726$	$0.0988$	$94.10$	$94.76$	$85.77$	$95.44$
0.50	50	$-0.0285$	$0.0084$	$-0.0992$	$0.0070$	$0.4057$	$0.5268$	$0.8121$	$0.2305$	$93.64$	$93.56$	$87.09$	$92.90$
	70	$-0.0233$	$0.0073$	$-0.0428$	$0.0033$	$0.3529$	$0.4508$	$0.7612$	$0.1944$	$93.72$	$93.78$	$89.54$	$93.94$
	90	$-0.0193$	$0.0088$	$-0.0169$	$0.0023$	$0.3217$	$0.4019$	$0.7193$	$0.1736$	$93.76$	$93.88$	$90.60$	$94.10$
	110	$-0.0153$	$0.0083$	$0.0033$	$0.0026$	$0.2852$	$0.3639$	$0.6865$	$0.1537$	$93.78$	$93.68$	$91.70$	$94.64$
	130	$-0.0133$	$0.0067$	$0.0082$	$0.0021$	$0.2554$	$0.3285$	$0.6454$	$0.1437$	$94.14$	$94.12$	$92.01$	$94.66$
	150	$-0.0126$	$0.0058$	$0.0215$	$0.0021$	$0.2384$	$0.3056$	$0.6245$	$0.1285$	$94.36$	$94.68$	$92.28$	$95.12$
0.75	50	$-0.0803$	$0.0102$	$-0.0126$	$0.0367$	$0.4320$	$0.5312$	$0.8604$	$0.2994$	$93.38$	$93.66$	$89.44$	$89.34$
	70	$-0.0625$	$0.0089$	$0.0196$	$0.0198$	$0.3736$	$0.4548$	$0.8164$	$0.2604$	$93.46$	$94.02$	$90.50$	$90.32$
	90	$-0.0517$	$0.0104$	$0.0343$	$0.0124$	$0.3383$	$0.4049$	$0.7821$	$0.2376$	$93.72$	$94.12$	$91.62$	$90.74$
	110	$-0.0426$	$0.0101$	$0.0381$	$0.0102$	$0.2994$	$0.3672$	$0.7340$	$0.2151$	$93.92$	$93.84$	$92.62$	$91.52$
	130	$-0.0374$	$0.0085$	$0.0436$	$0.0075$	$0.2694$	$0.3318$	$0.7047$	$0.2026$	$94.28$	$94.14$	$93.45$	$91.82$
	150	$-0.0326$	$0.0074$	$0.0415$	$0.0051$	$0.2514$	$0.3082$	$0.6658$	$0.1874$	$93.70$	$94.62$	$94.02$	$92.90$

,

Table 7. Empirical RMSE, bias, and $95\%$ CP (true values: ${\beta }_0=0.50$, ${\beta }_1=0.50$, $\varsigma =1.50$, and $\nu =2.00$) of the ML estimator for the indicated parameter with simulated data.

		Bias				RMSE				CP${}_{95\%}$
$\mathbf{\tau }$	$\mathit{n}$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$
0.25	50	$0.0073$	$-0.0096$	$-0.2878$	$-0.0130$	$0.2325$	$0.2838$	$0.7168$	$0.3056$	$92.62$	$93.70$	$69.63$	$95.84$
	70	$0.0036$	$-0.0069$	$-0.2310$	$-0.0131$	$0.2018$	$0.2437$	$0.6651$	$0.2517$	$93.40$	$93.76$	$72.48$	$96.14$
	90	$0.0027$	$-0.0041$	$-0.1836$	$-0.0092$	$0.1835$	$0.2182$	$0.6327$	$0.2179$	$93.62$	$93.82$	$74.76$	$95.36$
	110	$0.0024$	$-0.0029$	$-0.1515$	$-0.0077$	$0.1630$	$0.1977$	$0.6024$	$0.1944$	$93.46$	$93.94$	$76.40$	$95.56$
	130	$0.0016$	$-0.0026$	$-0.1236$	$-0.0063$	$0.1463$	$0.1785$	$0.5773$	$0.1761$	$94.22$	$94.34$	$78.69$	$95.40$
	150	$0.0006$	$-0.0024$	$-0.1089$	$-0.0069$	$0.1367$	$0.1664$	$0.5564$	$0.1632$	$93.90$	$94.74$	$78.94$	$95.24$
0.50	50	$-0.0227$	$0.0099$	$-0.1173$	$-0.0021$	$0.2268$	$0.2931$	$0.6122$	$0.3425$	$93.56$	$93.62$	$82.63$	$95.74$
	70	$-0.0182$	$0.0083$	$-0.0875$	$-0.0040$	$0.1969$	$0.2500$	$0.5490$	$0.2800$	$93.74$	$93.86$	$84.97$	$96.16$
	90	$-0.0149$	$0.0084$	$-0.0594$	$-0.0008$	$0.1792$	$0.2229$	$0.5139$	$0.2408$	$93.90$	$93.98$	$85.52$	$95.50$
	110	$-0.0119$	$0.0076$	$-0.0439$	$-0.0006$	$0.1584$	$0.2015$	$0.4828$	$0.2151$	$94.04$	$93.80$	$86.84$	$95.64$
	130	$-0.0103$	$0.0063$	$-0.0355$	$-0.0006$	$0.1417$	$0.1817$	$0.4498$	$0.1999$	$94.20$	$94.26$	$88.11$	$95.60$
	150	$-0.0094$	$0.0056$	$-0.0309$	$-0.0021$	$0.1323$	$0.1689$	$0.4287$	$0.1848$	$94.16$	$94.72$	$88.17$	$95.32$
0.75	50	$-0.0423$	$0.0203$	$-0.0396$	$0.0083$	$0.2439$	$0.3020$	$0.5576$	$0.4262$	$93.24$	$93.64$	$91.95$	$94.06$
	70	$-0.0317$	$0.0163$	$-0.0234$	$0.0043$	$0.2099$	$0.2566$	$0.4894$	$0.3511$	$93.60$	$93.90$	$93.06$	$94.82$
	90	$-0.0256$	$0.0150$	$-0.0137$	$0.0051$	$0.1898$	$0.2281$	$0.4429$	$0.3067$	$93.86$	$93.98$	$94.07$	$94.86$
	110	$-0.0207$	$0.0132$	$-0.0110$	$0.0038$	$0.1675$	$0.2060$	$0.4051$	$0.2771$	$93.74$	$93.90$	$94.35$	$95.00$
	130	$-0.0176$	$0.0111$	$-0.0053$	$0.0053$	$0.1501$	$0.1858$	$0.3747$	$0.2522$	$94.40$	$94.20$	$94.64$	$95.46$
	150	$-0.0153$	$0.0098$	$-0.0062$	$0.0035$	$0.1402$	$0.1722$	$0.3466$	$0.2308$	$94.02$	$94.74$	$94.78$	$95.44$

,

Table 8. Empirical RMSE, bias, and $95\%$ CP (true values: ${\beta }_0=0.50$, ${\beta }_1=0.50$, $\varsigma =2.00$, and $\nu =0.50$) of the ML estimator for the indicated parameter with simulated data.

		Bias				RMSE				CP${}_{95\%}$
$\mathbf{\tau }$	$\mathit{n}$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$
0.25	50	$0.0289$	$-0.0453$	$-0.7745$	$0.0012$	$0.7831$	$0.9007$	$1.2837$	$0.1043$	$94.66$	$96.14$	$69.26$	$94.16$
	70	$0.0216$	$-0.0120$	$-0.6466$	$0.0001$	$0.6784$	$0.7955$	$1.2059$	$0.0873$	$94.39$	$95.57$	$73.20$	$94.84$
	90	$0.0185$	$0.0055$	$-0.5457$	$0.0005$	$0.6187$	$0.7236$	$1.1584$	$0.0764$	$94.72$	$94.88$	$75.44$	$95.08$
	110	$0.0185$	$0.0087$	$-0.4864$	$0.0003$	$0.5465$	$0.6605$	$1.1138$	$0.0685$	$94.34$	$94.39$	$77.55$	$95.08$
	130	$0.0160$	$0.0083$	$-0.4257$	$0.0008$	$0.4898$	$0.5992$	$1.0700$	$0.0616$	$94.46$	$94.62$	$80.02$	$95.46$
	150	$0.0116$	$0.0084$	$-0.3798$	$0.0003$	$0.4564$	$0.5601$	$1.0418$	$0.0572$	$94.60$	$95.10$	$80.43$	$95.42$
0.50	50	$-0.0616$	$-0.0355$	$-0.6168$	$0.0100$	$0.7606$	$0.9165$	$1.2559$	$0.1333$	$95.05$	$96.18$	$76.07$	$90.82$
	70	$-0.0444$	$-0.0062$	$-0.5178$	$0.0044$	$0.6605$	$0.8047$	$1.1971$	$0.1165$	$94.62$	$95.32$	$78.79$	$91.94$
	90	$-0.0361$	$0.0129$	$-0.4164$	$0.0036$	$0.6020$	$0.7324$	$1.1593$	$0.1040$	$94.50$	$94.72$	$81.55$	$92.58$
	110	$-0.0278$	$0.0154$	$-0.3561$	$0.0028$	$0.5318$	$0.6668$	$1.1086$	$0.0938$	$94.50$	$94.23$	$83.58$	$93.10$
	130	$-0.0239$	$0.0144$	$-0.3036$	$0.0020$	$0.4755$	$0.6051$	$1.0754$	$0.0879$	$94.70$	$94.48$	$84.79$	$93.16$
	150	$-0.0225$	$0.0135$	$-0.2668$	$0.0009$	$0.4433$	$0.5645$	$1.0484$	$0.0820$	$94.74$	$94.84$	$85.82$	$93.94$
0.75	50	$-0.1752$	$-0.0398$	$-0.6059$	$0.0407$	$0.8120$	$0.9129$	$1.3070$	$0.1786$	$95.69$	$96.28$	$75.07$	$87.68$
	70	$-0.1311$	$-0.0077$	$-0.5413$	$0.0261$	$0.6985$	$0.8042$	$1.2682$	$0.1555$	$94.72$	$95.79$	$77.11$	$88.30$
	90	$-0.1070$	$0.0098$	$-0.4848$	$0.0182$	$0.6337$	$0.7281$	$1.2365$	$0.1409$	$94.52$	$95.09$	$78.79$	$88.74$
	110	$-0.0884$	$0.0136$	$-0.3955$	$0.0142$	$0.5595$	$0.6646$	$1.2127$	$0.1297$	$94.62$	$94.47$	$80.64$	$89.08$
	130	$-0.0778$	$0.0135$	$-0.3578$	$0.0110$	$0.5015$	$0.6042$	$1.1859$	$0.1227$	$94.80$	$94.66$	$81.67$	$89.42$
	150	$-0.0688$	$0.0127$	$-0.3159$	$0.0079$	$0.4685$	$0.5640$	$1.1576$	$0.1144$	$94.52$	$94.88$	$83.39$	$90.52$

,

Table 9. Empirical RMSE, bias, and $95\%$ CP (true values: ${\beta }_0=0.50$, ${\beta }_1=0.50$, $\varsigma =2.00$, and $\nu =1.00$) of the ML estimator for the indicated parameter with simulated data.

		Bias				RMSE				CP${}_{95\%}$
$\mathbf{\tau }$	$\mathit{n}$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$
0.25	50	$0.0169$	$-0.0049$	$-0.5311$	$-0.0028$	$0.4356$	$0.5311$	$1.0717$	$0.1741$	$93.22$	$93.98$	$72.97$	$95.62$
	70	$0.0096$	$-0.0026$	$-0.4125$	$-0.0034$	$0.3776$	$0.4569$	$1.0000$	$0.1429$	$93.46$	$93.98$	$76.42$	$96.16$
	90	$0.0074$	$0.0008$	$-0.3300$	$-0.0013$	$0.3436$	$0.4093$	$0.9493$	$0.1224$	$94.02$	$94.02$	$78.52$	$95.62$
	110	$0.0071$	$0.0011$	$-0.2836$	$-0.0012$	$0.3051$	$0.3709$	$0.9045$	$0.1096$	$93.68$	$93.98$	$80.08$	$95.62$
	130	$0.0057$	$0.0004$	$-0.2272$	$-0.0009$	$0.2739$	$0.3350$	$0.8730$	$0.1010$	$94.24$	$94.40$	$82.10$	$95.52$
	150	$0.0036$	$0.0001$	$-0.2041$	$-0.0014$	$0.2556$	$0.3123$	$0.8397$	$0.0928$	$94.20$	$94.80$	$82.56$	$95.40$
0.50	50	$-0.0364$	$0.0119$	$-0.2897$	$0.0041$	$0.4242$	$0.5473$	$0.9877$	$0.2140$	$93.86$	$93.64$	$84.65$	$94.04$
	70	$-0.0294$	$0.0103$	$-0.1996$	$0.0013$	$0.3683$	$0.4679$	$0.9169$	$0.1795$	$93.84$	$93.88$	$87.43$	$94.76$
	90	$-0.0243$	$0.0116$	$-0.1399$	$0.0017$	$0.3354$	$0.4172$	$0.8697$	$0.1572$	$93.96$	$94.06$	$88.52$	$94.68$
	110	$-0.0193$	$0.0105$	$-0.1033$	$0.0010$	$0.2969$	$0.3773$	$0.8324$	$0.1422$	$94.08$	$93.78$	$89.44$	$94.92$
	130	$-0.0167$	$0.0086$	$-0.0670$	$0.0021$	$0.2658$	$0.3404$	$0.7947$	$0.1281$	$94.34$	$94.20$	$90.35$	$95.34$
	150	$-0.0154$	$0.0074$	$-0.0553$	$0.0011$	$0.2481$	$0.3166$	$0.7600$	$0.1181$	$94.38$	$94.68$	$90.49$	$95.56$
0.75	50	$-0.0865$	$0.0177$	$-0.1758$	$0.0249$	$0.4541$	$0.5560$	$1.0098$	$0.2748$	$93.62$	$93.78$	$88.66$	$90.54$
	70	$-0.0668$	$0.0152$	$-0.1061$	$0.0117$	$0.3916$	$0.4751$	$0.9578$	$0.2395$	$93.58$	$94.12$	$90.27$	$91.44$
	90	$-0.0544$	$0.0158$	$-0.0698$	$0.0090$	$0.3540$	$0.4226$	$0.9015$	$0.2143$	$93.96$	$94.16$	$91.88$	$92.20$
	110	$-0.0447$	$0.0145$	$-0.0337$	$0.0065$	$0.3127$	$0.3823$	$0.8656$	$0.1953$	$93.98$	$93.84$	$92.52$	$92.62$
	130	$-0.0389$	$0.0123$	$-0.0154$	$0.0051$	$0.2806$	$0.3451$	$0.8285$	$0.1827$	$94.46$	$94.26$	$93.77$	$92.78$
	150	$-0.0339$	$0.0107$	$-0.0030$	$0.0030$	$0.2619$	$0.3203$	$0.7931$	$0.1692$	$94.10$	$94.64$	$94.29$	$93.78$

and

Table 10. Empirical RMSE, bias, and $95\%$ CP (true values: ${\beta }_0=0.50$, ${\beta }_1=0.50$, $\varsigma =2.00$, and $\nu =2.00$) of the ML estimator for the indicated parameter with simulated data.

		Bias				RMSE				CP${}_{95\%}$
$\mathbf{\tau }$	$\mathit{n}$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$	${\mathbf{\beta }}_{\mathbf{0}}$	${\mathbf{\beta }}_{\mathbf{1}}$	$\mathbf{\varsigma }$	$\mathbf{\nu }$
0.25	50	$0.0034$	$-0.0084$	$-0.6018$	$-0.0170$	$0.2387$	$0.2891$	$1.0116$	$0.2983$	$92.78$	$93.74$	$61.57$	$95.40$
	70	$0.0004$	$-0.0058$	$-0.5062$	$-0.0169$	$0.2070$	$0.2482$	$0.9468$	$0.2443$	$93.48$	$93.76$	$64.29$	$95.84$
	90	$-0.0001$	$-0.0031$	$-0.4392$	$-0.0124$	$0.1882$	$0.2223$	$0.8935$	$0.2109$	$93.80$	$93.80$	$67.41$	$95.52$
	110	$-0.0000$	$-0.0020$	$-0.3802$	$-0.0104$	$0.1670$	$0.2013$	$0.8523$	$0.1858$	$93.60$	$94.02$	$69.84$	$95.60$
	130	$-0.0005$	$-0.0018$	$-0.3281$	$-0.0092$	$0.1499$	$0.1817$	$0.8198$	$0.1701$	$94.30$	$94.30$	$71.35$	$95.24$
	150	$-0.0013$	$-0.0016$	$-0.2976$	$-0.0094$	$0.1400$	$0.1693$	$0.7922$	$0.1575$	$94.00$	$94.82$	$72.83$	$95.32$
0.50	50	$-0.0257$	$0.0124$	$-0.3116$	$-0.0083$	$0.2331$	$0.2995$	$0.8357$	$0.3178$	$93.72$	$93.76$	$75.51$	$95.88$
	70	$-0.0204$	$0.0102$	$-0.2419$	$-0.0096$	$0.2020$	$0.2552$	$0.7673$	$0.2630$	$93.84$	$93.92$	$78.46$	$96.12$
	90	$-0.0166$	$0.0101$	$-0.1884$	$-0.0056$	$0.1836$	$0.2275$	$0.7207$	$0.2259$	$94.14$	$94.06$	$79.48$	$95.70$
	110	$-0.0132$	$0.0089$	$-0.1523$	$-0.0047$	$0.1621$	$0.2054$	$0.6822$	$0.2023$	$94.30$	$93.88$	$81.48$	$95.66$
	130	$-0.0114$	$0.0074$	$-0.1222$	$-0.0037$	$0.1449$	$0.1851$	$0.6500$	$0.1840$	$94.34$	$94.30$	$82.84$	$95.42$
	150	$-0.0104$	$0.0065$	$-0.1034$	$-0.0048$	$0.1353$	$0.1721$	$0.6291$	$0.1706$	$94.28$	$94.76$	$83.36$	$95.36$
0.75	50	$-0.0412$	$0.0239$	$-0.1298$	$0.0013$	$0.2505$	$0.3089$	$0.7296$	$0.3772$	$93.70$	$93.86$	$87.93$	$95.06$
	70	$-0.0307$	$0.0188$	$-0.0890$	$-0.0012$	$0.2154$	$0.2618$	$0.6529$	$0.3100$	$93.68$	$94.00$	$88.90$	$95.78$
	90	$-0.0246$	$0.0168$	$-0.0590$	$0.0026$	$0.1945$	$0.2324$	$0.5991$	$0.2637$	$94.00$	$94.06$	$90.12$	$95.48$
	110	$-0.0196$	$0.0145$	$-0.0412$	$0.0022$	$0.1714$	$0.2093$	$0.5622$	$0.2360$	$93.84$	$93.88$	$90.62$	$95.46$
	130	$-0.0168$	$0.0121$	$-0.0303$	$0.0013$	$0.1537$	$0.1886$	$0.5300$	$0.2227$	$94.48$	$94.16$	$91.40$	$95.56$
	150	$-0.0146$	$0.0106$	$-0.0264$	$-0.0000$	$0.1436$	$0.1749$	$0.5003$	$0.2036$	$94.12$	$94.74$	$91.16$	$95.38$

indicate the results of the Monte-Carlo experiments for our QR model. According to these results, when estimating $\mathbf{\beta }$, $\varsigma$, and $\nu$, we report a low bias. The value of RMSE is also low and rapidly decreases as the sample size increases. Note that, for each parameter, the CPs approach the nominal level of confidence as n increases. Hence, the results obtained from the simulation study confirm the desirable properties of the ML estimators.

In addition, we report that the results of simulation studies involving two or more covariates are similar to studies with a single covariate. Thus, we have omitted them due to restrictions of space.

3.3. Application to Sports Medicine

Continuing with the motivating example presented in Section 1.2, we utilize several QR models and detect the relationships among the explanatory variables and the fat percentage at the legs. Let us assume that, for a constant quantile value $\tau =0.5$, the relationship of $\mu$, $\varsigma$, $\nu$, and the covariates is provided by

$$\begin{array}{ccc}\hfill \mathrm{logit}\left({\mu }_i\right)& =& {\beta }_0+{\beta }_1{\mathrm{bmi}}_i+{\beta }_2{\mathrm{age}}_i+{\beta }_3{\mathrm{sexmale}}_i+{\beta }_4{\mathrm{ipaqinsufficientlyactive}}_i+{\beta }_5{\mathrm{ipaqactive}}_i;\hfill \\ \hfill \log \left({\varsigma }_i\right)& =& {\delta }_0;\log \left({\nu }_i\right)={\gamma }_0;\hfill \end{array}$$

where “age${}_i$ = (age${}_i$− 46.00)” with 46.00 being the average age; “bmi${}_i$ = (bmi${}_i$− 24.72)” with 24.72 being the average BMI; “sexmale = 1” is employed for “sex = male” and “sexmale = 0” for “sex = female”; “ipaqinsufficientlyactive = 1” is used for “ipaq = insufficiently active’’ and “ipaqinsufficientlyactive = 0’’ for “ipaq = sedentary”, while “ipaqactive = 1” is utilized for “ipaq = active”, and “ipaqactive = 0” for “ipaq = sedentary”.

As mentioned, we estimate the model parameters with a SAS macro, which is restricted to analyzing data with logit link for $\mu$ and log link for $\varsigma$ and $\nu$.

Table 11. ML estimates, SEs, and 95% CI for the indicated $\tau$ and parameter with data related to fat percentage at the legs.

				95% CI
$\mathit{\tau }$	Parameter	Estimate	SE	Lower	Upper
0.25	${\beta }_0$	−0.4084	0.0400	−0.4871	−0.3296
	${\beta }_1$	0.0006	0.0011	−0.0015	0.0027
	${\beta }_2$	0.0664	0.0062	0.0542	0.0787
	${\beta }_3$	−0.9202	0.0352	−0.9895	−0.8509
	${\beta }_4$	−0.0459	0.0484	−0.1411	0.0494
	${\beta }_5$	−0.1249	0.0454	−0.2144	−0.0355
	${\delta }_0$	1.6274	0.3308	0.9763	2.2784
	${\gamma }_0$	2.3707	0.0481	2.2761	2.4654
0.50	${\beta }_0$	−0.2537	0.0370	−0.3265	−0.1809
	${\beta }_1$	0.0006	0.0010	−0.0014	0.0026
	${\beta }_2$	0.0637	0.0059	0.0520	0.0754
	${\beta }_3$	−0.8776	0.0335	−0.9435	−0.8117
	${\beta }_4$	−0.0444	0.0451	−0.1331	0.0443
	${\beta }_5$	−0.1201	0.0427	−0.2042	−0.0360
	${\delta }_0$	1.0004	0.2096	0.5880	1.4128
	${\gamma }_0$	2.3688	0.0487	2.2730	2.4646
0.75	${\beta }_0$	−0.1047	0.0361	−0.1759	−0.0336
	${\beta }_1$	0.0006	0.0010	−0.0014	0.0025
	${\beta }_2$	0.0612	0.0057	0.0501	0.0724
	${\beta }_3$	−0.8396	0.0320	−0.9026	−0.7766
	${\beta }_4$	−0.0431	0.0432	−0.1281	0.0420
	${\beta }_5$	−0.1155	0.0409	−0.1960	−0.0350
	${\delta }_0$	1.0087	0.2280	0.5600	1.4573
	${\gamma }_0$	2.3695	0.0486	2.2739	2.4651

reports the SEs, 95% CIs, and ML estimates of the corresponding parameters.

Table 12. Values of the indicated likelihood-based statistic and distribution for $\tau =0.25$ with data related to fat percentage at the legs.

Distribution	$-2$ LL	AIC	AICC	BIC
ASHW	−748.1069	−734.1069	−733.7207	−708.2273
JOSB	−841.6750	−827.6750	−827.2888	−801.7953
KUMA	−823.3698	−809.3698	−808.9836	−783.4902
LEEG	−816.0419	−802.0419	−801.6557	−776.1623
UBSA	−854.4446	−840.4446	−840.0584	−814.5650
UBUR	−667.9945	−653.9945	−653.6083	−628.1148
UCHE	−632.5113	−618.5113	−618.1250	−592.6316
UGHE	−702.0870	−688.0870	−687.7008	−662.2074
UGHX	−819.2782	−805.2782	−804.8920	−779.3986
UGOM	−680.6044	−666.6044	−666.2182	−640.7248
UGUM	−727.1883	−713.1883	−712.8021	−687.3086
ULOG	−849.4922	−835.4922	−835.1060	−809.6125
UWEE	−757.5552	−743.5552	−743.1689	−717.6755
UWEM	−844.5334	−830.5334	−830.1472	−804.6538
UWMO	−859.8022	−843.8022	−843.3039	−814.2254
VASI	−847.9083	−833.9083	−833.5220	−808.0286

Where AIC: Akaike information criterion; AICC: Akaike information corrected criterion; BIC: Bayesian information criterion; LL: log-likelihood. See acronyms of the distributions in Table A1.

,

Table 13. Values of the indicated likelihood-based statistic and distribution for $\tau =0.50$ with data related to fat percentage at the legs.

Distribution	$-2$ LL	AIC	AICC	BIC
ASHW	−749.0414	−735.0414	−734.6551	−709.1617
JOSB	−841.6750	−827.6750	−827.2888	−801.7953
KUMA	−825.4268	−811.4268	−811.0406	−785.5471
LEEG	−818.5204	−804.5204	−804.1342	−778.6408
UBSA	−855.4990	−841.4990	−841.1128	−815.6194
UBUR	−672.1146	−658.1146	−657.7284	−632.2350
UCHE	−630.8162	−616.8162	−616.4300	−590.9365
UGHE	−703.1626	−689.1626	−688.7764	−663.2829
UGHX	−819.2782	−805.2782	−804.8920	−779.3986
UGOM	−684.6376	−670.6376	−670.2514	−644.7580
UGUM	−727.1883	−713.1883	−712.8021	−687.3086
ULOG	−849.4922	−835.4922	−835.1060	−809.6125
UWEE	−758.6517	−744.6517	−744.2655	−718.7720
UWEX	−844.5334	−830.5334	−830.1472	−804.6538
UWMO	−860.9117	−844.9117	−844.4134	−815.3350
VASI	−848.7302	−834.7302	−834.3440	−808.8505

and

Table 14. Values of the indicated likelihood-based statistic and distribution for $\tau =0.75$ with data related to fat percentage at the legs.

Distribution	$-2$ LL	AIC	AICC	BIC
ASHW	−750.1398	−736.1398	−735.7535	−710.2601
JOSB	−841.6750	−827.6750	−827.2888	−801.7953
KUMA	−827.3404	−813.3404	−812.9542	−787.4607
LEEG	−821.4608	−807.4608	−807.0746	−781.5811
UBSA	−856.4866	−842.4866	−842.1004	−816.6069
UBUR	−674.3747	−660.3747	−659.9885	−634.4951
UCHE	−623.7983	−609.7983	−609.4121	−583.9187
UGHE	−704.5493	−690.5493	−690.1631	−664.6696
UGHX	−819.2782	−805.2782	−804.8920	−779.3986
UGOM	−690.5561	−676.5561	−676.1699	−650.6765
UGUM	−727.1883	−713.1883	−712.8021	−687.3086
ULOG	−849.4922	−835.4922	−835.1060	−809.6125
UWEE	−759.9440	−745.9440	−745.5578	−720.0644
UWEX	−844.5334	−830.5334	−830.1472	−804.6538
UWMO	−861.7335	−845.7335	−845.2352	−816.1567
VASI	−849.5840	−835.5840	−835.1978	−809.7044

presents values of likelihood-based statistics for model selection considering different QR models. Such statistics correspond to values of the Akaike (AIC) and (BIC) Bayesian information criteria and log-likelihood (LL). Note that smaller values of AIC, BIC, and LL indicate a better concordance between a model and the data set. These statistics report that the UWMOQ model has the best performance to describe the fat percentage at the legs with covariates, as expected, due to the mentioned flexibility of the Weibull–Marshall–Olkin distribution. Such conclusions are supported by the probability plots of Figure 5.

One can use a coefficient of determination, $R^2$, namely, such as in ordinary regression models, to state goodness-of-fit in QR structures [4]. The coefficient of determination is a global indicator of the explained variation for the dependent (response) variable, expressed as

$$R^2=1-{\left(\frac{L_0}{L_{p+q}}\right)}^{2/n},$$

where $L_0$ and $L_{p+q}$ are the likelihood functions for statistical formulations considering only the intercept or this plus $p+q$ explanatory variables.

Table 15. Coefficient of determination ($R^2$) for the indicated distribution with data related to fat percentage at the legs.

Distribution	${\mathit{R}}^2$
ASHW	$0.7641$
JOSB	$0.7689$
KUMA	$0.7219$
LEEG	$0.7678$
UBSA	$0.7751$
UBUR	$0.5202$
UCHE	$0.6967$
UGHE	$0.7465$
UGHX	$0.7323$
UGOM	$0.7558$
UGUM	$0.7715$
ULOG	$0.7773$
UWEX	$0.7578$
UWMOQ	$0.7808$

presents the values of the coefficient of determination $R^2$ for all distributions with $\tau =0.5$ (median). Such as in the case of AIC and BIC, the coefficient of determination enables us to compare statistical formulations not necessarily nested, but in this case a larger value states a better model to describe a data set. Based on the results from this table, once again the UWMOQ model presents the best performance.

As mentioned, to detect the adequacy of the model to the data, we also employ probability plots with envelopes based on the Cox-Snell and randomized quantile residuals. Figure 5 shows the good performance of the UWMOQ model with the data set associated with body fat. Hence, we continue with our QR analysis utilizing this model, where the UBSA [47,48] and KUMA [49] models are demonstrated to be competitive.

Note that ${\widehat{\beta }}_0$ is the estimate of a sedentary female of 46 years old, with a BMI which equals $24.72$ kg/m${}^2$. The values of estimated ${\beta }_1$ and ${\beta }_2$ establish that the age and BMI positively affect the fat percentage at the legs. In contrast, the estimates of ${\beta }_3$, ${\beta }_4$, and ${\beta }_5$ are negative, indicating that insufficiently active and active men have less fat percentage at the legs.

For the other response variables available in the data set, that is, the fat percentage at the android, arms, body, and gynecoid, the likelihood-based statistics for model selection and $\tau =0.5$ are in

Table 16. Values of the indicated likelihood-based statistic and distribution for $\tau =0.50$ with data related to fat percentage at the android.

Distribution	$-2$ LL	AIC	AICC	BIC
ASHW	−656.6639	−642.6639	−642.2777	−616.7843
JOSB	−733.4400	−719.4400	−719.0538	−693.5603
KUMA	−731.5464	−717.5464	−717.1602	−691.6667
LEEG	−703.3988	−689.3988	−689.0126	−663.5192
UBUR	−804.1466	−790.1466	−789.7604	−764.2669
UBSA	−786.2243	−772.2243	−771.8381	−746.3446
UCHE	−472.4102	−458.4102	−458.0240	−432.5306
UGHE	−635.5863	−621.5863	−621.2001	−595.7066
UGHX	−757.9927	−743.9927	−743.6065	−718.1131
UGOM	−509.3101	−495.3101	−494.9239	−469.4305
UGUM	−601.3444	−587.3444	−586.9581	−561.4647
ULOG	−764.8452	−750.8452	−750.4590	−724.9656
UWEE	−689.5933	−675.5933	−675.2071	−649.7137
UWEX	−772.4956	−758.4956	−758.1094	−732.6159
UWMO	−800.0696	−784.0696	−783.5713	−754.4928
VASI	−751.1870	−737.1870	−736.8008	−711.3073

,

Table 17. Values of the indicated likelihood-based statistic and distribution for $\tau =0.50$ with data related to fat percentage at the arms.

Distribution	$-2$ LL	AIC	AICC	BIC
ASHW	−844.2975	−830.2975	−829.9113	−804.4179
JOSB	−897.9673	−883.9673	−883.5811	−858.0876
KUMA	−859.0303	−845.0303	−844.6441	−819.1507
LEEG	−867.8135	−853.8135	−853.4273	−827.9339
UBSA	−923.3318	−909.3318	−908.9456	−883.4522
UBUR	−661.0975	−647.0975	−646.7113	−621.2179
UCHE	−705.7224	−691.7224	−691.3362	−665.8427
UGHE	−808.1356	−794.1356	−793.7494	−768.2560
UGHX	−858.1305	−844.1305	−843.7443	−818.2509
UGOM	−771.3000	−757.3000	−756.9138	−731.4203
UGUM	−814.9471	−800.9471	−800.5609	−775.0674
ULOG	−902.4015	−888.4015	−888.0153	−862.5218
UWEE	−854.9321	−840.9321	−840.5459	−815.0525
UWEX	−885.2879	−871.2879	−870.9016	−845.4082
UWMO	−923.3566	−907.3566	−906.8583	−877.7798
VASI	−910.9891	−896.9891	−896.6029	−871.1094

,

Table 18. Values of the indicated likelihood-based statistic and distribution for $\tau =0.25$ with data related to fat percentage at the body.

Distribution	$-2$ LL	AIC	AICC	BIC
ASHW	−802.6541	−788.6541	−788.2679	−762.7745
JOSB	−859.1252	−845.1252	−844.7389	−819.2455
KUMA	−837.0105	−823.0105	−822.6243	−797.1309
LEEG	−834.1941	−820.1941	−819.8079	−794.3144
UBSA	−902.1138	−888.1138	−887.7276	−862.2342
UBUR	−864.2166	−850.2166	−849.8304	−824.3369
UCHE	−601.4599	−587.4599	−587.0736	−561.5802
UGHE	−779.5107	−765.5107	−765.1245	−739.6310
UGHX	−851.4477	−837.4477	−837.0615	−811.5680
UGOM	−685.0341	−671.0341	−670.6479	−645.1545
UGUM	−756.7462	−742.7462	−742.3599	−716.8665
ULOG	−879.6604	−865.6604	−865.2742	−839.7808
UWEX	−870.3416	−856.3416	−855.9554	−830.4619
UWEE	−827.4140	−813.4140	−813.0278	−787.5344
UWMO	−909.8991	−893.8991	−893.4008	−864.3223
VASI	−875.2501	−861.2501	−860.8639	−835.3704

and

Table 19. Values of the indicated likelihood-based statistic and distribution for $\tau =0.25$ with data related to fat percentage at the gynecoid.

Distribution	$-2$ LL	AIC	AICC	BIC
ASHW	−782.8493	−768.8493	−768.4631	−742.9697
JOSB	−874.0575	−860.0575	−859.6713	−834.1779
KUMA	−859.6809	−845.6809	−845.2947	−819.8013
LEEG	−847.9210	−833.9210	−833.5348	−808.0414
UBSA	−893.3770	−879.3770	−878.9908	−853.4973
UBUR	−778.2260	−764.2260	−763.8398	−738.3463
UCHE	−626.8613	−612.8613	−612.4750	−586.9816
UGHE	−739.2528	−725.2528	−724.8666	−699.3731
UGHX	−861.7670	−847.7670	−847.3808	−821.8873
UGOM	−705.0838	−691.0838	−690.6976	−665.2041
UGUM	−760.1826	−746.1826	−745.7964	−720.3030
ULOG	−884.1771	−870.1771	−869.7909	−844.2975
UWEE	−795.4313	−781.4313	−781.0451	−755.5516
UWEX	−883.5176	−869.5176	−869.1314	−843.6379
UWMO	−898.7473	−882.7473	−882.2490	−853.1706
VASI	−880.3849	−866.3849	−865.9986	−840.5052

. Note that, for the variable body and gynecoid, our proposal is the best.

3.4. COVID-19 Application with Data from the US

Our second illustration uses a data set related to COVID-19. These data can be secured from https://github.com/tatianefribeiro/RUBXII_Regression_COVID-19/tree/master (accessed on 30 January 2023). For purposes of comparison, we consider the distributions of the unitquantreg package. For details about the characteristics of these distributions, see

Table A1. PDF and reparameterization of the indicated distribution and its acronym.

Distribution	Acronym	PDF and Reparameterization
Arc-secant hyperbolic Weibull	ASHW	$f(z;\alpha ,\beta )=\frac{\alpha \beta }{z\sqrt{1-z^2}}\mathrm{arcsech}{\left(z\right)}^{\beta -1}exp\left[-\alpha \mathrm{arcsech}{\left(z\right)}^{\beta }\right]$,
		where $\alpha =-log\left(\tau \right)/\mathrm{arcsech}{\left(\mu \right)}^{\beta }$
Johnson–SB	JOSB	$f(z;\alpha ,\beta )={\displaystyle \frac{\beta }{\sqrt{2\pi }}}\frac{1}{z(1-z)}exp\left\{-{\displaystyle \frac{1}{2}}{\left[\alpha +\beta log\left(\frac{z}{1-z}\right)\right]}^2\right\}$,
		where $\alpha ={\mathrm{\Phi }}^{-1}\left(\tau \right)-\beta log\left(\mu /\left(1-\mu \right)\right)$
Kumaraswamy	KUMA	$f(z;\alpha ,\beta )=\alpha \beta z^{\beta -1}{(1-z^{\beta })}^{\alpha -1}$,
		where $\alpha =log(1-\tau )/log(1-{\mu }^{\beta })$
Log-extended exponential–geometric	LEEG	$f(z;\alpha ,\beta )=\frac{\beta \left(1+\alpha \right)z^{\beta -1}}{{\left(1+\alpha z^{\beta }\right)}^2}$
		where $\alpha =-\left(1-\tau {\mu }^{-\beta }\right)/\left(1-\tau \right)$
Unit–Birnbaum–Saunders	UBSA	$f(z;\alpha ,\beta )=\frac{1}{2z\alpha \beta \sqrt{2\pi }}\left[{\left(-\frac{\alpha }{log\left(z\right)}\right)}^{\frac{1}{2}}+{\left(-\frac{\alpha }{log\left(z\right)}\right)}^{\frac{3}{2}}\right]exp\left[\frac{1}{2{\beta }^2}\left(2+\frac{log\left(z\right)}{\alpha }+\frac{\alpha }{log\left(z\right)}\right)\right]$,
		where $\alpha =log\left(\mu \right)\{\beta {\mathrm{\Phi }}^{-1}(1-\tau ){\left(4+\beta {\mathrm{\Phi }}^{-1}(1-\tau )\right)}^{\frac{1}{2}}-2-{\left[\beta {\mathrm{\Phi }}^{-1}(1-\tau )\right]}^2\}/2)$
Unit–Burr-XII	UBUR	$f(z;\alpha ,\beta )=\frac{\alpha \beta }{z}{\left[-log\left(z\right)\right]}^{\alpha -1}{\left\{1+{\left[-log\left(z\right)\right]}^{\alpha }\right\}}^{(-\beta -1)}$,
		where $\alpha =log({\tau }^{-1/\beta }-1)/log[-log(\mu \left)\right]$
Unit–Chen	UCHE	$f(z;\alpha ,\beta )=\frac{\alpha }{z}\beta {\left[-log\left(z\right)\right]}^{\beta -1}exp\left\{{\left[-log\left(z\right)\right]}^{\beta }\right\}exp\left[\alpha \left(1-exp\left\{{\left[-log\left(z\right)\right]}^{\beta }\right\}\right)\right]$,
		where $\alpha =log\left(\tau \right)/\left(1-exp\left\{{[-log\left(\mu \right)]}^{\beta }\right\}\right)$
Unit-generalized half-normal-E	UGHE	$f(z;\alpha ,\beta )=\sqrt{\frac{2}{\pi }}\frac{\beta }{z\left[-log\left(z\right)\right]}{\left[-\frac{log\left(z\right)}{\alpha }\right]}^{\beta }exp\left\{-\frac{1}{2}{\left[-\frac{log\left(z\right)}{\alpha }\right]}^{2\beta }\right\}$,
		where $\alpha =-log\left(\mu \right){\left[{\mathrm{\Phi }}^{-1}(\tau /2)\right]}^{-1/\beta }$
Unit-generalized half-normal-X	UGHX	$f(z;\alpha ,\beta )=\sqrt{\frac{2}{\pi }}\frac{\beta }{z\left(1-z\right)}{\left[\frac{z}{\alpha \left(1-z\right)}\right]}^{\beta }exp\left\{-\frac{1}{2}{\left[\frac{z}{\alpha \left(1-z\right)}\right]}^{2\beta }\right\}$,
		where $\alpha =[\mu /(1-\mu )]{\left\{{\mathrm{\Phi }}^{-1}[\left(\tau +1\right)/2]\right\}}^{1/\beta }$
Unit–Gompertz	UGOM	$f(z;\alpha ,\beta )=\alpha \beta z^{-\left(1+\beta \right)}exp\left[\alpha \left(1-z^{-\beta }\right)\right]$,
		where $\alpha =log\left(\tau \right)/\left(1-{\mu }^{-\beta }\right)$
Unit–Gumbel	UGUM	$f(z;\alpha ,\beta )=\frac{\beta }{z(1-z)}exp\left\{-\alpha -\beta log\left(\frac{z}{1-z}\right)-exp\left[-\alpha -\beta log\left(\frac{z}{1-z}\right)\right]\right\}$,
		where $\alpha =\beta log\left[\right(1-\mu )/\mu ]+log[-1/log\left(\tau \right)]$
Unit-logistic	ULOG	$f(z;\alpha ,\beta )=\frac{\beta exp\left(\alpha \right){\left(\frac{z}{1-z}\right)}^{\beta -1}}{{\left[1+exp\left(\alpha \right){\left(\frac{z}{1-z}\right)}^{\beta }\right]}^2}$,
		where $\alpha =log[\tau /(1-\tau \left)\right]-\beta log[\mu /(1-\mu \left)\right]$
Unit–Weibull–E	UWEE	$f(z;\alpha ,\beta )=\frac{1}{z}\alpha \beta {\left[-log\left(z\right)\right]}^{\beta -1}exp\left\{-\alpha {\left[-log\left(z\right)\right]}^{\beta }\right\}$,
		where $\alpha =-log\left(\tau \right)/{[-log\left(\mu \right)]}^{\beta }$
Unit–Weibull–X	UWEX	$f(z;\alpha ,\beta )=\frac{1}{{(1-z)}^2}\alpha \beta {\left(\frac{z}{1-z}\right)}^{\beta -1}exp\left[-\alpha {\left(\frac{z}{1-z}\right)}^{\beta }\right]$,
		where $\alpha =-log(1-\tau ){[\mu /\left(1-\mu \right)]}^{-\beta }$
Vasicek	VASI	$f(z;\alpha ,\beta )=\sqrt{\frac{1-\beta }{\beta }}exp\left\{\frac{1}{2}\left[{\mathrm{\Phi }}^{-1}{\left(z\right)}^2-{\left(\frac{{\mathrm{\Phi }}^{-1}\left(z\right)\sqrt{1-\beta }-{\mathrm{\Phi }}^{-1}\left(\alpha \right)}{\sqrt{\beta }}\right)}^2\right]\right\}$,
		where $\alpha =\mathrm{\Phi }\left[{\mathrm{\Phi }}^{-1}\left(\mu \right)\sqrt{1-\beta }-{\mathrm{\Phi }}^{-1}\left(\tau \right)\sqrt{\beta }\right]$

in Appendix A.

We define the response variable as: $\mathrm{recovery}\mathrm{rate}=1-\mathrm{mortality}\mathrm{rate}$, throughout all 50 USA states, and the subsequent model is formulated as

$$\begin{array}{ccc}\hfill \mathrm{logit}\left({\mu }_i\right)& =& {\beta }_0+{\beta }_1{\mathrm{PD}}_i+{\beta }_2{\mathrm{GINI}}_i+{\beta }_3{\mathrm{BEDS}}_i+{\beta }_4{\mathrm{SR}}_i+{\beta }_5{\mathrm{PR}}_i+{\beta }_6{\mathrm{LE}}_i+{\beta }_7\mathrm{T}90+{\beta }_8{\mathrm{T}180}_i,\hfill \end{array}$$

(11)

with GINI being the GINI indicator in 2017; PD relating to the population density in 2020 (people/m${}^2$); BEDS being the number of hospital beds per 100,000 people in 2018; PR corresponding to the level of poverty in 2020; SR being the state-by-state smoking rate in 2020; LE indicating the life expectancy in 2018; T90 being a dummy variable that is set as one if the response reflects the recovery rate after 90 days for the 10th confirmed case, and it is set as zero, otherwise. In contrast, T180 also is a dummy variable that is set as one if the response reflects the recovery rate after 180 days for the 10th confirmed case; otherwise, it is zero. Additional information about the covariates related to the indicators GINI, BEDS, and LE in 2020 is available in [50], where the mortality rate, defined as $\mathrm{mortality}\mathrm{rate}=(1-\mathrm{recovery}\mathrm{rate})$, was considered as the response variable.

Note that, in the three time periods, the response presents a positive coefficient of skewness (CS), and the average is far from the median. At 30 and 90 days, the coefficient of kurtosis (CK) is $\mathrm{CK}>3$, informing a leptokurtic distribution. Histograms and scatter plots are shown in Figure 6 for the variables BEDS, GINI, PD, PR, RR, SR, and LE. The empirical distribution of the response is leptokurtic, asymmetric, and ranges from 0.820 to 0.999. Several models implemented in our R package work well to model the data. In addition, note that the response variable only has a statistically significant correlation at 1% with the covariates PD and GINI, allowing the model to leave out the other covariates. However, when the regression models are specified, substantial correlations between some covariates may point to multicollinearity problems.

As opposed to what occurred with the body fat data, the results acquired for the COVID-19 data set in terms of significance at 5% are somewhat different among the models. We use the measure $R^2$ once again as a criterion for comparing all models, with the UGOM QR having the best fit closely followed by the UWMOQ model.

Correlation coefficients reveal pairwise correlations. The VIF is employed to identify which explanatory variables are collinear and then they must be removed before the analysis of the model established in (11), which we call M0. The VIF values for model M0 are:

• > car::vif(fitcovidULOG)
• PD     GINI   BEDS    SR    PR    LE    T90    T180
• 2.136667 2.551950 1.740564 6.988683 4.176044 6.791217 1.423759 1.419650

As previously stated, VIF $>10$ suggests potential collinearity. Here, the covariates SR and LE have VIF scores of 6.989 and 6.791, respectively. Thus, these covariates could be considered, but the models considering such covariates must be carefully analyzed. If one covariate is removed, the VIF values must be recalculated, and the procedure is repeated until all VIF values are sufficiently small (less than five, for example) to ensure no collinearity issues. Then, we state three QR models to describe the recovery rate:

• (M1 –reduced 1–) with covariates PD, GINI, BEDS, PR, LE, T90, and T180;
• (M2 –reduced 2–) with covariates PD, GINI, BEDS, SR, PR, T90, and T180;
• (M1 –reduced 3–) with covariates PD, GINI, BEDS, PR, T90, and T180.

The corresponding VIF values for models M1, M2, and M3 are:

• > car::vif(reduced.1)
• PD      GINI    BEDS    PR     LE     T90    T180
• 2.068567  2.516064  1.183617  4.169128  2.701867  1.393607  1.374571
• > car::vif(reduced.2)
• PD      GINI    BEDS    SR    PR    T90    T180
• 2.016583 2.253713 1.637897 2.888653 3.165383 1.433448 1.422845
• > car::vif(reduced.3)
• PD      GINI   BEDS     PR     T90    T180
• 2.005546 2.224827 1.120871 2.155601 1.401250 1.379976

Now, observe that the VIF is small for all reduced models (M1, M2, and M3). We can conclude from these results that the difference between the four models (M0, M1, M2, and M3) is not significant at 5%. M3 is chosen because of the parsimony principle. Essential details about the final model are the following: the model is congruent with the hypotheses established from the descriptive statistical analysis, which recommended to only include the explanatory variables T180, T90, GINI, and PD at a significant level of 1%. Observe that ${\widehat{\beta }}_0$ represents the percentage of recovered COVID-19 cases when all covariates take a value equal to zero. According to the estimate of the parameter ${\beta }_3$, hospital beds have a favorable impact on the recovery rates, meaning that recovery rates are higher in states with several hospital beds in the USA. The estimates of the parameters ${\beta }_1$, ${\beta }_2$, and ${\beta }_4$ have an unfavorable consequence on the rate of recovery, showing that the recovered COVID-19 cases are less in USA states with higher levels of population, GINI indicator, and level of poverty. Moreover, ${\widehat{\beta }}_5$ and ${\widehat{\beta }}_6$ are also negative, with ${\widehat{\beta }}_6$ being less than ${\widehat{\beta }}_5$. This suggests that the period after the 10th confirmed case has a negative impact on the recovery rate.

4. Concluding Remarks

The fractile or quantile regression models use conditional quantile functions and provide a method for stating the relationship between a response variable or outcome and covariates or explanatory variables. This method permits a more in-depth investigation of the response conditional distribution for various quantiles and provides a robust alternative for estimating the response central tendency. Although quantile regression structures were applied in various fields, parametric approaches have only been investigated in the past ten years. As mentioned, several parametric quantile regressions were postulated based on the arc-secant hyperbolic Weibull, Johnson SB, Kumaraswamy, log-extended exponential-geometric, unit-Birnbaum-Saunders, unit-Burr-XII, unit-Chen, unit-Gompertz, unit-Gumbel, unit-logistic, unit-generalized half-normal-E, unit-generalized half-normal-X, unit-Weibull-E, unit-Weibull-X, and Vasicek distributions [11,12,13,14,15,16,17,18,19,20,21,23,24,25]. In [10], the authors provided a number of parametric quantile regressions, as well as examples and software packages, which can be employed to model rates and indices. These indicators are particularly of interest when analyzing COVID-19 data, as we did in one of our applications. Therefore, new and more precise information based on our methodology was obtained, due to our model fitting the COVID-19 data in a better form than in other existing models. Our quantile regression formulation under the Weibull–Marshall–Olkin distribution had not been established until now.

We use two medical data sets and a computational library written in the R statistical software. By reparameterizing a distribution as a quantile function, we made an updated examination of the parametric quantile regression developed. To describe continuous variables with unit interval bounds, several distributions were used. We summarized their essential characteristics in this work. The mentioned R library was introduced for parameter estimation and model validation with unit interval distributions. Through the mentioned medical applications, we have demonstrated how to use the techniques and capabilities of the package.

A future work is related to the inclusion of quantile regression for inflation levels of zero, one, or zero-one. Considering that in such a case the likelihood function factors in two parts, one exclusively dependent on the discrete element and another solely dependent on the continuous element, it must be noted that the computation execution of zero or one is evidently enhanced. Therefore, the discrete elements can be estimated using the function of the R statistical software: stats::glm(), while the continuous elements may be determined employing the uniquantreg::uniquantreg() function. In future releases of the uniquantreg package, we will intend to add a wrapper function for estimation and hypothesis testing in other quantile regression models, as well as the model presented in this investigation.

Explanatory variables may affect the quantiles and other parameters. Then, one can study the joint effect that the explanatory variables can have on both quantiles and another parameters in regression formulations. Moreover, we want to study statistical structures generated from the current research under settings related to partial least squares, temporal, spatial, and multivariate frameworks [51,52,53,54,55,56]. Similarly, considering censored data may also be of interest to study [57], as well as control charts for quantiles based on covariates [58]. We are planning to conduct studies on these issues in the future.