**4. Applications**

In this section, we carry out two applications to real data, the first using the GSC model without covariates and the second applying quantile regression to uni- and bimodal data.

### *4.1. Application 1: Without Covariates*

The first application reported is for the data set consisting of 1150 heights measured at 1 micron intervals along the drum of a roller (i.e., parallel to the axis of the roller). This was part of an extensive study of roller surface roughness. It is available for downloading at http://lib.stat,emu.edu/jasadata/ laslett. Table 3 presents summary statistics for the data set where *b*1 and *b*2 correspond to the sample asymmetry and kurtosis coefficients, respectively.

**Table 3.** Descriptive statistics for the data set


We fitted this using the SN model, the exponentiated sinh Cauchy (ESC) model (see Cooray [23]), and the GSC model. A summary of these fits is presented in Table 4. Based on the AIC criteria, the GSC provided the best fit for the height data set. Figure 4 shows plots of the density functions for the fitted models using the MLEs for SN, ECG and GSC distributions.


**Table 4.** Maximum likelihood (ML) estimates for the GSC, exponentiated sinh Cauchy (ECG), and skew-normal (SN) models for the roller data set.

**Figure 4.** Fitted models for roller data set.

### *4.2. Data Set 2: Quantile Regression to Bimodal Data*

The second application we consider is the Australian data set available in the sn package in R. This data set is related to 102 male and 100 female athletes collected at the Australian Institute of Sport. The linear model considered is

$$\mathbf{Bf at\_i} = \beta\_0(q) + \beta\_1(q)\mathbf{bm1\_i} + \beta\_2(q)\mathbf{1bm\_i} + \epsilon\_i(q), \quad i = 1, \ldots, 202,\tag{9}$$

where Bfat*i* is the body fat percentage for the *i*-th athlete and bmi*i* and lbm*i* are the covariates body mass index and lean body mass for the *i*-th athlete, respectively. In addition, *i*(*q*) ∼ *GSC*(0, *σ*, *λ*, *φ*(*q*)), *φ*(*q*) satisfies Equation (7), and *q* ∈ (0, 1) is the fixed quantile that is being modeled. This data set was also analyzed in Martínez-Flórez et al. [24] using a bimodal regression model. However, the authors modeled the mean of the distribution. In our approach, we model the 0.1, 0.25, 0.5, 0.75, and 0.9 percentiles of the distribution, which provides a more informative scenario to explain body fat in terms of the body mass index and lean body mass. Our approach is compared with the skewed Laplace (SKL) and skewed Student-t (SKT) models discussed in Galarza et al. [25], where the authors proposed a flexible model in a quantile regression model context. Table 5 shows the AIC for those models considering different quantiles. We also present the *p*-value for the Kolmogorov–Smirnov (K–S) test of the hypothesis that the respective quantile residuals came from the standard normal distribution. *P*-values greater than 5% sugges<sup>t</sup> that with this significance level, the standard normal assumption is reasonable for those residuals, in which case the model would be appropriate for this

data set. Note that based on the AIC criteria, the GSC presents a better fit for this data set, except for the median regression (*q* = 0.5). On the other hand, based on the *p*-value for the K–S test applied to the quantile residuals, we conclude that GSC, SKL, and SKT are appropriate models for *q* = 0.25 and *q* = 0.5 (the GSC and SKT provide a better fit based on the AIC criteria). However, for *q* = 0.1 and 0.75, GSC provides the better fit because the *p*-values are (significantly) greater than 0.05. Finally, for *q* = 0.9 no model seems appropriate, but based on the *p*-values, the GSC provides a better fit than SKL and SKT distributions.

**Table 5.** AIC and *p*-value for K–S test in the ais data set for the GSC, skewed Laplace (SKL), andskewed Student-t (SKT) models and different quantiles.


Figure 5 shows the regression coefficients for the quantile regression presented in Equation (9) and their respective 95% confidence intervals. Note that body mass index and lean body mass are significant in explaining all the quantiles modeled.

Figure 6 shows the profile density for the *q*-th quantile of body fat percentage for *q* = 0.1 and *q* = 0.75. Note that the distribution of the 0.1 quantile is unimodal, and the distribution of the 0.9 quantile is bimodal.

**Figure 5.** Estimates for regression coefficients (and 95% confidence interval)s for variables bmi (left panel) and lbm (right panel) in different quantile regression models with quantiles equal to 0.1, 0.25, 0.5, 0.75, and 0.9 and response variable Bfat.

**Figure 6.** Distribution for 0.1 and 0.75 quantiles of body fat percentage considering body mass index and lean body mass equal to 22.96 and 64.87, respectively. Curves in black, red, and green represent the density functions estimated by the GSC, SKL, and SKT models, respectively.
