Slash-Weighted Lindley Distribution: Properties, Inference, and Applications

Abstract

The slash-weighted Lindley model is introduced due to the need to obtain a model with more kurtosis than the weighted Lindley distribution. Several expressions for the pdf of this model are given. Its cumulative distribution function is expressed in terms of a generalized hypergeometric function and the incomplete gamma function. Moments and maximum likelihood estimation were studied. A simulation study was carried out to illustrate the good performance of the estimates. Finally, two real applications are included.

1. Introduction

The Lindley distribution was introduced by Lindley [1], and it has been widely used for modeling survival or reliability data. A treatment of the mathematical properties of this model can be seen in Ghitany et al. [2]. Recall that a nonnegative random variable (rv) T follows a Lindley distribution with shape parameter $\alpha >0$, $T\sim L\left(\alpha \right)$, if its probability density function (pdf) is given by

$$f_T(t;\alpha )=\frac{{\alpha }^2}{1+\alpha }(1+t)\exp (-\alpha t),t>0.$$

On the other hand, Patil and Rao [3] proposed weighted distributions. This is a well-known technique used to generate new models. The weighted distributions are used to describe skewed data and data coming from hidden truncated models. In this paper, we focused on the weighted version of the Lindley distribution proposed by Ghitany et al. [4], which is of interest in survival analysis. A nonnegative rv follows a Weighted Lindley (WL) distribution with parameters $\alpha >0$ and $\beta >0$, $X\sim WL(\alpha ,\beta )$, if its pdf is given by

$$f_X(x;\alpha ,\beta )=\frac{{\alpha }^{\beta +1}}{(\alpha +\beta )\mathsf{\Gamma }\left(\beta \right)}{x}^{\beta -1}(1+x)\exp (-\alpha x),x>0,$$

where $\mathsf{\Gamma }(·)$ denotes the gamma function.

The aim of our paper was to introduce a new extension of the WL distribution by using the slash methodology, in such a way that the new model has a heavier right tail than the original WL one. To reach this end, the Slash-Weighted Lindley (SWL) distribution is defined as follows:

$$Z=X{Y}^{-1},$$

(1)

where $X\sim WL(\alpha ,\beta ),$$\alpha >0$, $\beta >0$, $Y\sim Beta(q,1)$, and $q>0$ are independent. (1) is denoted as $Z\sim SWL(\alpha ,\beta ,q)$.

We highlight that the slash methodology is appropriate to obtain more-flexible models in terms of kurtosis and skewness than the original distribution. Since the pioneering works by Rogers and Tukey [5] and Mosteller and Tukey [6], it has been applied to a number of symmetrical and skewed distributions. We can cite Gómez et al. [7] for distributions with elliptical contours, Olmos et al. [8] for the generalized half-normal distribution, Del Castillo [9] for the sum of independent logistic distributions, Reyes et al. for the Birnbaum–Saunders distribution in [10,11] and for the normal distribution in [12], Barranco-Chamorro et al. [13] for the generalized Rayleigh distribution in Wang and Genton [14] and in Arslan and Genc [15], who proposed multivariate versions based on the normal multivariate distribution, to cite only a few.

On the other hand, generalizations of the Lindley and weighted Lindley distribution, along with practical applications, can be seen in Nadarajah et al. [16], Gui [17], Shanker et al. [18,19], Asgharzadeh et al. [20], and Arslan et al. [21], among others. Due to the interest in these models and since this issue has not yet been studied, we focused in this paper on the application of the slash methodology to the weighted Lindley model. It is also worth noting that our proposal is appropriate to deal with skewed positive data, quite common in medical research; see for instance, Liu et al. [22] or Ren et al. [23]. In this context, the slash-weighted Lindley distribution is an alternative to lifetime distributions, such as the Weibull distribution [24] or the generalized Gamma distribution introduced by Stacy [25]. This issue will be addressed in our applications.

The outline of this paper is as follows. In Section 2, several expressions for the pdf of the SWL model are given. Next, its cumulative distribution function is obtained in terms of the generalized hypergeometric function and the incomplete gamma function. Moments are also obtained. Section 3 is devoted to maximum likelihood estimation. A simulation study was carried out based on the definition of the SWL model and the representation of the WL model as a mixture proposed by Ghitany [4]. In Section 4, two real applications are given. The final conclusions are included in Section 5.

2. Materials and Methods in the Slash-Weighted Lindley Distribution

In this section, the main results for the SWL model introduced in (1) are given. Special functions are used in our results. Other applications of interest can be seen in Wani [26] or Zayed [27].

2.1. Probability Density Function and Properties

In Proposition 1, the pdf of the SWL distribution will be obtained by using the stochastic representation given in (1). In this result, the confluent hypergeometric function, ${}_1{F}_1$, appears. It can be seen in Abramowitz and Stegun [28] that this function has the following integral representation:

$${}_1{F}_1(a;b;x)=\frac{\mathsf{\Gamma }\left(b\right)}{\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }(b-a)}{\int }_0^1{v}^{a-1}{(1-v)}^{b-a-1}{e}^{xv}dv,b>a>0.$$

(2)

Moreover, under suitable conditions, we have

$${\int }_0^1{u}^{{\alpha }_1-1}{(1-u)}^{{\alpha }_2-1}{}_1{F}_1(a;b;uy)du=\frac{\mathsf{\Gamma }\left({\alpha }_1\right)\mathsf{\Gamma }\left({\alpha }_2\right)}{\mathsf{\Gamma }({\alpha }_1+{\alpha }_2)}{}_2{F}_2(a,{\alpha }_1;b,{\alpha }_1+{\alpha }_2;y),$$

(3)

where ${}_2{F}_2$ denotes the generalized hypergeometric function ${}_p{F}_q$ with $p=q=2$, and the details can be seen in Luke [29] or Zörnig [30]. This result will be used to obtain the cdf of a SWL model.

Next, it is proven that the pdf of SWL$(\alpha ,\beta ,q)$ can be expressed in terms of the confluent hypergeometric function, ${}_1{F}_1$.

Proposition 1.

Let $Z\sim SWL(\alpha ,\beta ,q)$ with α, β, and $q>0$. Then, the pdf of Z is given by  

$$f_Z(z;\alpha ,\beta ,q)=\frac{q{\alpha }^{\beta }{z}^{\beta -1}}{(\alpha +\beta )\mathsf{\Gamma }\left(\beta \right)}\left\{\frac{\alpha +\beta +q}{\beta +q}{}_1{F}_1(\beta +q;\beta +q+1;-\alpha z)-e^{-\alpha z}\right\},$$

(4)

where $z>0$.

Proof. 

From (1), we can write $Z=X{Y}^{-1}$ with $X\sim WL\left(\alpha \right)$ and $Y\sim Beta(q,1)$ independent. By applying the Jacobian technique, the joint pdf of $(Z,W)$ with $W=Y$ is

$$\begin{array}{ccc}\hfill f_{Z,W}(z,w)& =& \hfill w{f}_X\left(zw\right)f_Y\left(w\right),z>0,0<w<1.\hfill \end{array}$$

Marginalizing with respect to W, the pdf of Z is

$$\begin{array}{ccc}{f}_Z(z;\alpha ,\beta ,q)& =& \frac{q{\alpha }^{1+\beta }}{(\alpha +\beta )\mathsf{\Gamma }\left(\beta \right)}{z}^{\beta -1}{\int }_0^1{w}^{\beta +q-1}(1+zw)\exp (-\alpha zw)dw,z>0.\end{array}$$

Finally, by integration by parts and applying (2), (4) is obtained.    □

Next, several alternative expressions for the pdf of the SWL distribution are given. They will be useful for estimation purposes.

Proposition 2.

Let $Z\sim SWL(\alpha ,\beta ,q)$. Then, the pdf of Z can be obtained:

$$f_Z(z;\alpha ,\beta ,q)=\frac{q}{(\alpha +\beta )\mathsf{\Gamma }\left(\beta \right){\alpha }^q{z}^{q+1}}\left\{\alpha \gamma (\beta +q,\alpha z)+\gamma (\beta +q+1,\alpha z)\right\},z>0,$$

(5)

where $z>0$, $\alpha >0$, $\beta >0$, $q>0$, and $\gamma (·,·)$ denotes the lower incomplete gamma function defined as $\gamma (s,x)={\int }_0^x{t}^{s-1}{e}^{-t}dt,$$x\ge 0$, and $s>0$.

Proof. 

Similar to Proposition 1, by using the definition of the gamma incomplete function and integration by parts, the proposed result is obtained.    □

Remark 1.

The following notation will be used throughout this paper:

1. 

An rv T follows a gamma distribution, $T\sim Ga(\beta ,\alpha )$, if its pdf is $f\left(t\right)=\frac{{\alpha }^{\beta }}{\mathsf{\Gamma }\left(\beta \right)}{t}^{\beta -1}{e}^{-\alpha t}$, α, $\beta >0$, and $t>0$.

2. 

If the pdf of T is $f(·)$, then we may write $T\sim f$.

Next, a hierarchical representation for the SWL model is given. Specifically, it is proven that, if the conditional distribution of Z given $Y=y$, $Z|Y=y$, is the mixture of gamma distributions proposed in Proposition 3 and $Y\sim Beta(q,1)$, then the marginal distribution of Z is an SWL.

Proposition 3.

If $Z|Y=y\sim \left(\frac{\alpha }{\alpha +\beta }\right)f_1\left(z\right)+\left(\frac{\beta }{\alpha +\beta }\right)f_2\left(z\right)$ with $f_i$ the pdf of a gamma distribution, $Ga(\beta +i-1,\alpha y)$, for $i=1,2$, and $Y\sim Beta(q,1)$, then $Z\sim SWL(\alpha ,\beta ,q)$.

Proof. 

Note that the marginal pdf of Z is given by

$$f_Z\left(z\right)=\int f_{Z|Y=y}\left(z\right)f_y\left(y\right)dy={\int }_0^1\frac{q{\alpha }^{\beta +1}}{(\alpha +\beta )\mathsf{\Gamma }\left(\beta \right)}{y}^{\beta +q-1}{z}^{\beta -1}(1+yz)e^{-\alpha yz}dy.$$

By proceeding similarly to Proposition 1, the proposed result is obtained.    □

Particular Case of Interest: The Slash Lindley Distribution

Note that, for $\beta =1$, the SWL model reduces to the Slash Lindley (SL) distribution introduced in [17], that is SWL$(\alpha ,1,q)=$ SL$(\alpha ,q)$.

On the other hand, if $q\to \infty$, it will be proven in Proposition 6 that SWL$(\alpha ,\beta ,q)$ converges to the WL$(\alpha ,\beta )$ model.

2.2. Cumulative Distribution Function

The following proposition gives the cumulative distribution function (cdf) of the SWL distribution.

Proposition 4.

Let $Z\sim SWL(\alpha ,\beta ,q)$. Then, the cdf of Z is given by

$$F_Z(z;\theta )=\frac{q}{(\alpha +\beta )\mathsf{\Gamma }\left(\beta \right)}\left\{\frac{\alpha +\beta +q}{\beta (\beta +q)}{\left(\alpha z\right)}^{\beta }{}_2{F}_2(\beta +q,\beta ;\beta +q+1,\beta +1;-\alpha z)-\gamma (\beta ,\alpha z)\right\},$$

where $\theta =(\alpha ,\beta ,q)$ with α, β, and $q>0$.

Proof. 

The cdf of Z is given by

$${\displaystyle F_Z(z;\alpha ,\beta ,q)={\int }_0^z{f}_Z(t;\alpha ,\beta ,q)dt.}$$

It can be seen in Luke [29] that, using the expression of the pdf of Z given in (4) and applying (3), the result proposed is obtained.    □

Next, an expression for the cdf of the SWL distribution is given in terms of the incomplete gamma function.

Proposition 5.

Let $Z\sim SWL(\alpha ,\beta ,q)$. Then, the cdf of Z is given by

$$F_Z(z;\mathit{\theta })=\frac{1}{(\alpha +\beta )\mathsf{\Gamma }\left(\beta \right)}\left\{\alpha \gamma (\beta ,\alpha z)+\gamma (\beta +1,\alpha z)-\frac{1}{{\alpha }^q{z}^q}\left[\alpha \gamma (\beta +q,\alpha z)+\gamma (\beta +q+1,\alpha z)\right]\right\},$$

where $\mathit{\theta }=(\alpha ,\beta ,q)$ with α, β, and $q>0$.

Proof. 

For $a=\beta +q$ and $\beta +q+1$, the proposed result is obtained by applying integration by parts to

$${\int }_0^z{t}^{-(q+1)}\gamma (a,\alpha t)dt.$$

   □

Next, we briefly illustrate that the right tail in the SWL is heavier than the one in the WL model. Figure 1 compares the pdfs of the WL and SWL models with $q=1,5$.

Table 1. Right tail comparison for the SWL and WL distributions.

Distribution	$\mathit{P}(\mathit{Z}>3)$	$\mathit{P}(\mathit{Z}>4)$	$\mathit{P}(\mathit{Z}>5)$
SWL(2,2,1)	$0.4082$	$0.3112$	$0.2498$
SWL(2,2,5)	$0.0994$	$0.0370$	$0.0146$
WL(2,2)	$0.0397$	$0.0084$	$0.0016$

provides $P(Z>z)$ for several values of z. Figure 1 and Table 1 show that the right tail of the SWL is much heavier than the WL model for decreasing values of q. We will go deeper into these appreciations.

2.3. Properties

Next, we study the convergence in law of an SWL model to a WL distribution. To emphasize the relevance of parameter q, in this section, we include it in the notation, i.e., $Z_q\sim SWL(\alpha ,\beta ,q)$.

Proposition 6.

Let $Z_q\sim SWL(\alpha ,\beta ,q)$. If $q\to \infty$, then $Z_q$ converges in law to an rv $X\sim WL(\alpha ,\beta )$.

Proof. 

Let $Z_q\sim SWL(\alpha ,\beta ,q)$. Then, we can write $Z_q=\frac{X}{Y_q}$ with $X\sim WL(\alpha ,\beta )$ and $Y_q\sim Beta(q,1)$.

Since $Y_q\sim Beta(q,1)$, $\mathbb{E}\left[Y_q\right]=\frac{q}{1+q}$ and $Var\left[Y_q\right]=\frac{q}{(q+2){(q+1)}^2}$. By applying Chebyshev’s inequality to $Y_q$, we have that, for all $ϵ>0$,

$$P\left[|Y_q-\mathbb{E}\left[Y_q\right]|>ϵ\right]\le \frac{Var\left(Y_q\right)}{{ϵ}^2}=\frac{q}{(q+2){(q+1)}^2{ϵ}^2}.$$

(6)

If $q\to \infty$, then the right-hand side of (6) tends to zero, i.e., $Y_q-\mathbb{E}\left[Y_q\right]$ converges in probability to 0. Moreover, since $\mathbb{E}\left[Y_q\right]=\frac{q}{1+q}⟶1,q\to \infty$, then we have

$$Y_q=Y_q-\mathbb{E}\left[Y_q\right]+\mathbb{E}\left[Y_q\right]\stackrel{\mathcal{P}}{⟶}1,q\to \infty .$$

By applying Slutsky’s lemma to $Z_q=\frac{X}{Y}$ and since $X\sim WL(\alpha ,\beta )$, it follows that

$$Z_q\stackrel{\mathcal{L}}{⟶}X\sim WL(\alpha ,\beta ),q\to \infty .$$

That is, for increasing values of q, $Z_q$ converges in law to a WL$(\alpha ,\beta )$ distribution.    □

Proposition 6 implies that, for large q, the pdf of an SWL$(\alpha ,\beta ,q)$ distribution can be approached by the pdf of a WL$(\alpha ,\beta )$ distribution.

2.4. Moments

Proposition 7.

Let $Z\sim SWL(\alpha ,\beta ,q)$. Then, for r a positive integer, $E\left[Z^r\right]$ exists, if and only if $r<q$, and in this case,

$$E\left[Z^r\right]=\frac{(\alpha +\beta +r)\mathsf{\Gamma }(\beta +r)q}{{\alpha }^r(\alpha +\beta )\mathsf{\Gamma }\left(\beta \right)(q-r)}.$$

Proof. 

By using the stochastic representation given in (1) and since X and Y are independent, we have

$$E\left[Z^r\right]=E\left[X^r\right]E\left[Y^{-r}\right].$$

(7)

It can be seen in Ghitany et al. [4] that $E\left[X^r\right]$ exists for $r=1,2,\dots$ and is given by

$$E\left[X^r\right]=\frac{(\alpha +\beta +r)\mathsf{\Gamma }(\beta +r)}{{\alpha }^r(\alpha +\beta )\mathsf{\Gamma }\left(\beta \right)}.$$

(8)

As for $Y\sim Beta(q,1)$, we have that $E\left[Y^{-r}\right]$ exists if and only if $r<q$ and

$$E\left[Y^{-r}\right]=\frac{q}{q-r},q>r.$$

(9)

From (8) and (9), the proposed result follows.    □

From Proposition 7, the next corollary follows.

Corollary 1.

Let $Z\sim SWL(\alpha ,\beta ,q)$. Then:

1. 

$E\left[Z\right]=\frac{\beta (\alpha +\beta +1)q}{\alpha (\alpha +\beta )(q-1)},$ provided that $q>1$.

2. 

$Var\left[Z\right]=\frac{\beta q}{{\alpha }^2(\alpha +\beta )}\left[\frac{(\alpha +\beta +2)(\beta +1)}{(q-2)}-\frac{\beta q{(\alpha +\beta +1)}^2}{(\alpha +\beta ){(q-1)}^2}\right],$ provided that $q>2$.

3. 

Let ${\mu }_j=E\left[Z^j\right]$. Then, the skewness, $\sqrt{{\beta }_1}$, and kurtosis, ${\beta }_2$, coefficients can be obtained by using

$$\begin{array}{ccc}\sqrt{{\beta }_1}& =& \frac{{\mu }_3-3{\mu }_1{\mu }_2+2{\mu }_1^3}{{({\mu }_2-{\mu }_1^2)}^{\frac{3}{2}}},q>3,\hfill \\ {\beta }_2& =& \frac{{\mu }_4-4{\mu }_1{\mu }_3+6{\mu }_2{\mu }_1^2-3{\mu }_1^4}{{({\mu }_2-{\mu }_1^2)}^2},q>4.\end{array}$$

Figure 2 shows the plot of the skewness and kurtosis coefficients for the SWL model when $\beta =2$.

Corollary 2.

From (7) and (9), the following relationships between the moments of $Z\sim SWL(\alpha ,\beta ,q)$ and $X\sim WL(\alpha ,\beta )$ are obtained:

1. 

$E\left[Z\right]={\mu }_X\frac{q}{(q-1)},$ where ${\mu }_X=E\left[X\right]$ and $q>1$.

2. 

$$Var\left[Z\right]=\left(1+\frac{2}{q-2}\right)Var\left[X\right]+\left(\frac{q}{{(q-1)}_2(q-1)}\right){\mu }_X^2,$$

where $q>2$ and ${(q-1)}_2$ denotes the falling factorial, ${(q-1)}_2=(q-1)(q-2)$.

3. 

$$E\left[{(Z-{\mu }_Z)}^3\right]=\left(1+\frac{3}{q-3}\right)E\left[{(X-{\mu }_X)}^3\right]+\frac{6q}{{(q-1)}_3}{\mu }_{X}Var\left[X\right]+\frac{2q(q+1)}{{(q-1)}_3{(q-1)}^2}{\mu }_X^3,$$

where $q>3$ and ${(q-1)}_3=(q-1)(q-2)(q-3)$.

The expressions given in Corollary 2 can be used to obtain the central moments of an SWL distribution from the ones in a WL model. They also show that, in general, the central moments of the SWL distribution, $E\left[{(Z-{\mu }_Z)}^j\right]$, are greater than the ones in the WL model, $E\left[{(X-{\mu }_X)}^j\right]$. Moreover, although, in general, the convergence in law does not imply the convergence of moments [31], we have that, for SWL$(\alpha ,\beta ,q)$, the moments converge to the moments of WL$(\alpha ,\beta )$ when $q\to \infty$.

3. Inference

3.1. Maximum Likelihood Estimation

Given $Z_1,Z_2,\dots ,Z_n$ a random sample of size n from SWL$(\alpha ,\beta ,q)$, then from (5), the log-likelihood function is given by

$$\begin{array}{ccc}\hfill \ell \left(\mathit{\theta }\right)& \propto & \hfill {\displaystyle n\log q-n\log (\alpha +\beta )-n\log \mathsf{\Gamma }\left(\beta \right)-nq\log \left(\alpha \right)-(q+1)\sum _{i=1}^n\log \left(z_i\right)}\hfill \\ & +& {\displaystyle \sum _{i=1}^n\log \left[\alpha \gamma (\beta +q,\alpha z_i)+\gamma (\beta +q+1,\alpha z_i)\right],}\hfill \end{array}$$

(10)

where $\mathit{\theta }=(\alpha ,\beta ,q)$ and ∝ means proportional to.

Taking partial derivatives with respect to $\alpha$, $\beta$, and q, the elements of the score vector are obtained, $S\left(\mathit{\theta }\right)=\left(\frac{\partial \ell }{\partial \alpha },\frac{\partial \ell }{\partial \beta },\frac{\partial \ell }{\partial q}\right)$, that is

$$\begin{array}{ccc}\hfill \frac{\partial \ell }{\partial \alpha }& =& -\frac{n}{\alpha +\beta }-\frac{nq}{\alpha }+\sum _{i=1}^n\frac{\gamma (\beta +q,\alpha z_i)+{\alpha }^{\beta +q}{z}_i^{\beta +q}{e}^{-\alpha z_i}(1+z_i)}{\alpha \gamma (\beta +q,\alpha z_i)+\gamma (\beta +q+1,\alpha z_i)},\hfill \\ \hfill \frac{\partial \ell }{\partial \beta }& =& -\frac{n}{\alpha +\beta }-n\psi \left(\beta \right)+\sum _{i=1}^n\frac{\alpha I(\beta +q,\alpha z_i)+I(\beta +q+1,\alpha z_i)}{\alpha \gamma (\beta +q,\alpha z_i)+\gamma (\beta +q+1,\alpha z_i)},\hfill \\ \hfill \frac{\partial \ell }{\partial q}& =& \frac{n}{q}-n\log \left(\alpha \right)-\sum _{i=1}^n\log \left(z_i\right)+\sum _{i=1}^n\frac{\alpha I(\beta +q,\alpha z_i)+I(\beta +q+1,\alpha z_i)}{\alpha \gamma (\beta +q,\alpha z_i)+\gamma (\beta +q+1,\alpha z_i)},\hfill \end{array}$$

where $\psi (·)$ denotes the digamma function and ${\displaystyle I(a,v)={\int }_0^v{t}^{a-1}\log \left(t\right)e^{-t}dt}$, $a>0$, and $v>0$, which is related to the generalized integro-exponential function when $v=\infty$ (see Milgram [32]).

The MLEs of $\widehat{\mathit{\theta }}$ are obtained as the solution of $S\left(\mathit{\theta }\right)=\mathbf{0}$. Numerical methods must be used to solve this system. As the initial values to start the recursion, we propose $({\widehat{\alpha }}_0,{\widehat{\beta }}_0)$, the MLEs of $(\alpha ,\beta )$ in a WL$(\alpha ,\beta )$ model. As for q, note that, using $({\widehat{\alpha }}_0,{\widehat{\beta }}_0)$ and $\frac{\partial \ell }{\partial \beta }=\frac{\partial \ell }{\partial q}=0$, an initial estimate of q, ${\widehat{q}}_0$, can be obtained. The existence of the ML estimators for the vector of parameters $\mathit{\theta }=(\alpha ,\beta ,q)$ does not present any difficulties and follows from the properties of the pdf f. Since f is a smooth and continuous function, with first and second derivatives that exist and are finite, then the existence of the roots of the equation $S\left(\mathit{\theta }\right)=\mathbf{0}$ is guaranteed, and they correspond to the ML estimator of the vector $\mathit{\theta }$. It can be checked that the solutions are a maximum using appropriate calculus techniques.

3.2. Observed Fisher Information Matrix

The asymptotic variance of MLEs, $\widehat{\mathit{\theta }}=(\widehat{\alpha },\widehat{\beta },\widehat{q})$, can be estimated from the Fisher information matrix, given by $\mathcal{I}\left(\mathit{\theta }\right)=-E\left[{\partial }^2\ell \left(\mathit{\theta }\right)/\partial \mathit{\theta }\partial {\mathit{\theta }}^{\top }\right]$, where $\ell \left(\mathit{\theta }\right)$ was given in (10). Under regularity conditions, the MLEs are asymptotically normal, that is

$$\mathcal{I}{\left(\mathit{\theta }\right)}^{-1/2}\left(\widehat{\mathit{\theta }}-\mathit{\theta }\right)\stackrel{\mathcal{D}}{⟶}{N}_3({\mathbf{0}}_3,{\mathit{I}}_3),\mathrm{as}n\to +\infty ,$$

where $\mathcal{D}$ stands for convergence in distribution and $N_3({\mathbf{0}}_3,{\mathit{I}}_3)$ denotes the standard trivariate normal distribution. $\mathcal{I}\left(\mathit{\theta }\right)$ is obtained from the matrix $-{\partial }^2\ell \left(\mathit{\theta }\right)/\partial \mathit{\theta }\partial {\mathit{\theta }}^{\top }$, whose elements are given by $I_{\alpha \alpha }=-{\partial }^2\ell \left(\mathit{\theta }\right)/\partial {\alpha }^2$, $I_{\alpha \beta }=-{\partial }^2\ell \left(\mathit{\theta }\right)/\partial \alpha \partial \beta$, and so on.

Usually, it is not possible to obtain a closer form to the expected value of previous expressions. Therefore, the covariance matrix of the MLEs, $\mathcal{I}{\left(\mathit{\theta }\right)}^{-1}$, can be estimated by $I{\left(\widehat{\mathit{\theta }}\right)}^{-1}$, where $I\left(\widehat{\mathit{\theta }}\right)$ denotes the observed information matrix, which is obtained by evaluating the previous derivatives at the MLE $\widehat{\mathit{\theta }}$:

$$I\left(\widehat{\mathit{\theta }}\right)=-{\partial }^2\ell \left(\mathit{\theta }\right)/\partial \mathit{\theta }\partial {\mathit{\theta }}^{\top }{|}_{\mathit{\theta }=\widehat{\mathit{\theta }}}.$$

The asymptotic variances of $\widehat{\alpha }$, $\widehat{\beta }$, and $\widehat{q}$ are estimated by the diagonal elements of $I{\left(\widehat{\mathit{\theta }}\right)}^{-1}$ and their standard errors by the square root of asymptotic variances. Details about the theoretical results used in this subsection can be seen in [31,33].

3.3. Simulation Study

Next, the acceptance–rejection algorithm proposed by Ghitany et al. [4] to generate the values of the WL model is tailored to this situation.

Using Algorithm 1, 1000 samples from an SWL distribution were generated, with sample sizes of 100, 150, 200, 300, and 500 and different values of the parameters. For every sample, the MLEs were obtained by using the Newton–Raphson numerical method. The mean of the estimates, standard errors, and 95% empirical coverage probabilities, $C(·)$, (in percentages) are given in

Table 2. Simulations of the SWL$(\alpha ,\beta ,q)$ model.

n	$\mathit{\alpha }$	$\mathit{\beta }$	q	$\widehat{\mathit{\alpha }}$	SD$(\widehat{\mathit{\alpha }})$	C$(\widehat{\mathit{\alpha }})$	$\widehat{\mathit{\beta }}$	SD$(\widehat{\mathit{\beta }})$	C$(\widehat{\mathit{\beta }})$	$\widehat{\mathit{q}}$	SD$(\widehat{\mathit{q}})$	C$(\widehat{\mathit{q}})$
100				0.3287	0.0997	93.4	1.6667	0.4718	93.7	1.3837	0.2909	95.1
150				0.3135	0.0748	92.7	1.5758	0.3533	94.7	1.3454	0.2106	96.2
200	0.3	1.5	1.3	0.3098	0.0631	92.3	1.5550	0.2978	93.8	1.3275	0.1755	95.6
300				0.3066	0.0517	94.4	1.5300	0.2430	94.8	1.3197	0.1417	95.5
500				0.3030	0.0392	96.2	1.5209	0.1859	95.6	1.3132	0.1081	94.8
100				1.0621	0.3075	92.9	2.1282	0.5232	94.6	1.7362	2.6924	95,7
150				1.0504	0.2457	90.9	2.1005	0.4175	92.2	1.5607	0.2590	96.1
200	1	2	1.5	1.0482	0.2088	93.4	2.0857	0.3543	95.0	1.5262	0.2114	92.9
300				1.0122	0.1659	93.7	2.0276	0.2829	94.1	1.5421	0.1766	95.4
500				1.0058	0.1270	94.6	2.0148	0.2168	94.8	1.5198	0.1327	95.7
100				5.4999	1.6739	92.5	3.2666	0.7710	94.6	2.4591	4.4267	94.4
150				5.2170	1.2349	92.2	3.1119	0.5745	93.4	2.1412	0.4912	95.5
200	5	3	2	5.1907	1.0507	93.3	3.0994	0.4906	93.5	2.0893	0.3552	95.9
300				5.0983	0.8443	93.1	3.0518	0.3943	94.4	2.0551	0.2711	95.9
500				5.0874	0.6450	94.9	3.0443	0.3018	95.6	2.0322	0.2024	96.4
100				2.1336	0.6727	93.6	6.4042	1.7919	95.1	3.2994	24.8036	95.8
150				2.0935	0.5267	93.6	6.2940	1.4101	94.9	2.6522	0.5101	95.6
200	2	6	2.5	2.0754	0.4459	93.5	6.2343	1.1953	94.2	2.5915	0.4001	94.6
300				2.0683	0.3612	94.9	6.2025	0.9677	96.0	2.5567	0.3135	94.8
500				2.0405	0.2741	95.8	6.1227	0.7353	95.2	2.5248	0.2346	95.3
100				1.8016	0.5129	93.6	2.1186	0.4809	95.4	2.0067	2.6226	95.5
150				1.7817	0.4048	91.8	2.0977	0.3819	93.9	1.8022	0.3460	95.8
200	1.7	2	1.7	1.7800	0.3473	94.3	2.0792	0.3255	94.6	1.7408	0.2706	93.5
300				1.7368	0.2767	93.5	2.0418	0.2604	94.4	1.7413	0.2209	95.2
500				1.7197	0.2097	96.5	2.0291	0.1986	96.3	1.7304	0.1669	96.5
100				1.0959	0.3507	92.6	4.3779	1.2543	94.9	2.1574	0.5247	95.7
150				1.0535	0.2656	92.5	4.2018	0.9508	93.8	2.0752	0.3600	93.8
200	1	4	2	1.0361	0.2255	93.4	4.1356	0.8087	94.2	2.0643	0.3053	95.2
300				1.0186	0.1793	96.0	4.0718	0.6451	95.7	2.0373	0.2369	97.1
500				1.0168	0.1371	95.4	4.0656	0.4945	95.4	2.0171	0.1782	95.4

. We observed that, in general, the bias and standard deviation of estimates decreased if the sample size increased n, which suggests that the MLEs are consistent. The empirical coverage probabilities approached the 95% nominal level. As for this point, note that $\widehat{\mathit{\beta }}$ and $\widehat{\mathit{q}}$ behaved better than $\widehat{\mathit{\alpha }}$. The results in Table 2 suggest that it is necessary to have a larger sample size ($n=300$ or 500) so that C$\left(\widehat{\mathit{\alpha }}\right)$ approaches 0.95.

Algorithm 1 Algorithm to simulate values from the $Z\sim SWL(\alpha ,\beta ,q)$ distribution.

1: Simulate $R\sim U(0,1)$. 2: If $R\le \frac{\alpha }{\alpha +\beta }$, then generate $X\sim Gamma(\beta ,\alpha )$. 3: If $R>\frac{\alpha }{\alpha +\beta }$, then generate $X\sim Gamma(\beta +1,\alpha )$. 4: Simulate $Y\sim Beta(q,1)$. 5: Compute $Z=\frac{X}{Y}$.

4. Applications

In this section, applications to two real datasets are given. Explicitly, the SWL model was compared to the SL, WL, Generalized Gamma (GG), and Weibull (W) distributions. It was proven that the SWL model provides a better fit to these datasets since it is able to model more kurtosis. The pdf’s of GG and W models can be seen in Appendix A.

4.1. Application 1 (Patients with Liver Transplants)

The first dataset considers the survival time for 54 patients before undergoing a liver transplant. This dataset was studied by Freund [34]. Maximum likelihood estimates were obtained with the optim function of the R software [35].

Table 3. Descriptive summaries for patients with liver transplants.

n	$\overline{\mathit{X}}$	S	${\mathit{b}}_1$	${\mathit{b}}_2$
54	197.1667	145.2994	2.1106	5.43881

provides the descriptive summaries for this dataset: sample mean, sample standard deviation, sample skewness ($b_1$), and sample kurtosis coefficient ($b_2$). On the other hand, Figure 3 shows the boxplot of this dataset, where we can see the existence of several outliers.

Table 4. Estimates, SE in parentheses, log-likelihood, AIC, and BIC for patients with liver transplants.

Parameters	W (SE)	WL (SE)	SL (SE)	GG (SE)	SWL (SE)
$\alpha$	1.5375 (0.1477)	0.0134 (0.0026)	-	0.0382 (0.0706)	0.0441 (0.0201)
$\beta$	220.7527 (20.7315)	1.6483 (0.4664)	-	0.3553 (0.0487)	4.0877 (1.7445)
$\theta$	-	-	0.0123 (0.0102)	19.8861 (5.6518)	-
$\sigma$	-	-	0.9934 (0.8532)	-	-
q	-	-	5.2851 (4.3762)	-	2.2975 (0.7001)
log-likelihood	−331.5893	−328.5691	−329.6107	−326.2089	−325.5083
AIC	667.1785	661.1383	665.2214	658.4178	657.0167
BIC	671.1565	665.1162	671.1884	664.3847	662.9836

provides the estimates of the parameters and their Standard Errors (SE) in parentheses for the W, WL, SL, GG, and SWL models, along with the comparison of these models by using the Akaike Information Criterion (AIC) (see Akaike [36]) and the Bayesian Information Criterion (BIC) (see Schwarz [37]). Both criteria support the fact that the SWL model provides a better fit to this dataset than the other ones.

Figure 4 shows the histogram for this dataset along with the fit provided by the pdfs of these models. Moreover, Figure 5 gives the associated QQ-plots. Both plots support the fact that the SWL distribution provides a better fit than the SL, WL, GG, and W models.

4.2. Application 2 (Survival Times of Guinea Pigs)

The second dataset considers the survival time of 72 guinea pigs injected with different doses of tubercle bacilli. This dataset was analyzed in Kundu [38] and Gómez and Bolfarine [39], among others.

Table 5. Descriptive summaries for survival times of guinea pigs.

n	$\overline{\mathit{X}}$	S	${\mathit{b}}_1$	${\mathit{b}}_2$
72	99.82	81.12	1.80	5.61

gives the statistical summaries of this dataset. Figure 6 shows the boxplot, where, again, outliers are found.

Table 6. Estimates, SE in parentheses, log-likelihood, AIC, and BIC for survival times of guinea pigs.

Parameters	W (SE)	WL (SE)	SL (SE)	GG (SE)	SWL (SE)
$\alpha$	1.3924 (0.1184)	0.0213 (0.0036)	-	0.0314 (0.0463)	0.0876 (0.0319)
$\beta$	110.3530 (9.9121)	1.1434 (0.3156)	-	0.3507 (0.0401)	3.5063 (1.2389)
$\theta$	-	-	0.0126 (0.0094)	15.9390 (3.7380)	-
$\sigma$	-	-	0.4532 (0.3542)	-	-
q	-	-	3.4264 (1.7450)	-	1.9025 (0.4206)
log-likelihood	−397.1479	−394.4175	−393.6876	−391.0979	−389.8986
AIC	798.2958	792.8350	793.3751	788.1959	785.7972
BIC	802.8491	797.3883	800.2051	795.0258	792.6272

shows the MLEs of the parameters in the W, WL, SL, GG, and SWL models along with their standard errors in parentheses. The AIC and BIC were obtained, which supported the fact that the SWL model provided a better fit than the other distributions in comparison. Moreover, the histogram and the fit provided by the densities of these models are given in Figure 7. The QQ-plots can be seen in Figure 8. Both plots support the good performance of SWL model in this setting.

5. Conclusions

This paper presented a study of the SWL distribution, which was generated based on the slash methodology using the WL and beta distributions. The SWL distribution had heavier tails than the WL distribution, and this characteristic made it an interesting distribution for modeling datasets that have atypical observations. Some further characteristics of the model are as follows:

• The SWL distribution can be represented based on the confluent hypergeometric and lower incomplete gamma functions.
• The cdf is represented based on the hypergeometric function.
• A limiting case of the SWL distribution happens to be the WL distribution.
• The moments of the SWL distribution are explicit and depend on the gamma function.
• The applications showed that the SWL distribution performed well in fitting two real datasets.
• In these applications, it was shown that the SWL distribution outperformed the Lindley and weighted Lindley and other important lifetime parametric models, such as the Weibull distribution and generalized gamma distribution.