*6.2. Application 2*

The data set to be investigated was taken from Nierenberg et al. [23], and is related to a study on plasma retinol and betacarotene levels from a sample of *n* = 315 subjects. More specifically, the response variable observed is grams of cholesterol consumed per day. Descriptive statistics for the variable are provided in Table 4.

Initially, we calculate the estimators based on the centiles, naive, and moments of the GTPN distribution, which are *θcent* = (2.326, 2.741, 2.376), *θnaive* = (275.960, 0, <sup>1</sup>), and *θmom* = (0.003, −86.024, 1.854). We used these estimations as initial values in computing the ML estimators for the GTPN model. Table 5 summarizes the fit for this data set. As in last application, we noted the high estimated standard error for the *γ* parameter in the GL model. Again, based on the AIC criteria and BIC criteria, the preferred model is the GTPN. Figure 8 shows the estimated density for each model in the cholesterol data set. The GTPN model appears to provide a better fit. For this data set, we also present the hypothesis tests for the three particular models of the GTPN distribution discussed in Section 4.1. Specifically, for *H*(1) 0 : *α* = 1 versus *H*(1) 1 : *α* = 1 we obtained *p* − *values* < 0.0001, 0.0157, and < 0.0001 for the SLR, SR, and ST tests, respectively. Therefore, with a significance of 5% we preferred the GTPN over the TPN model. For *H*(2) 0 : *λ* = 0 versus *H*(2) 1 : *λ* = 0 we obtained *p* − *values* < 0.0001, 0.0008 and < 0.0001 for the SLR, SR and ST tests, respectively. Hence, with a significance of 5% we preferred the GTPN over the GHN model. Finally, *H*(3) 0 : (*<sup>α</sup>*, *λ*)=(1, 0) versus *H*(3) 1 : (*<sup>α</sup>*, *λ*) = (1, 0) we obtained *p* − *values* < 0.0001 for the three tests. Therefore, with a significance of 5% we preferred the GTPN over the HN model. Finally, Figure 9 also presents the qq-plot for the QR in the fitted models and the *p* − *values* for the three normality tests. Results sugges<sup>t</sup> that the GTPN model is appropriate for this data set while the rest of the models are not.

**Table 4.** Descriptive statistics for the cholesterol data set.


**Table 5.** Estimated parameters and their standard errors (in parentheses) for the GTPN, TPN, WEI, and GL models for the *cholesterol* data set. The AIC and BIC criteria are also presented.


**Figure 8.** Fit of the distributions for the cholesterol data set.

**Figure 9.** RQ for the fitted models in the cholesterol data set. The *p* − *values* for the AD, CVM, and SW normality tests are also presented to check if the QR came from the standard normal distribution.
