*6.1. Application 1*

The data set was taken from Laslett [21], and consisted of *n* = 115 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 the surface roughness of the rollers. A statistical summary of the data set is presented in Table 2.

**Table 2.** Descriptive statistics of the Laslett data set.


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* = (3.557, 0, <sup>1</sup>), and *θmom* = (3.075, 1.572, 3.203) . We used these estimations as initial values in computing the ML estimators for the GTPN model. Results are presented in Table 3. Note the high estimated standard error for the *γ* parameter in the GL model. In addition, note that, based on the AIC criteria and BIC criteria [22], the GTPN model is preferred (among the fitted models) for this data set. Figure 6 shows the estimated density for each model in this data set, where the GTPN model appears to provide a better fit. Finally, Figure 7 also presents the qq-plot for the QR in the same models and the *p* − *values* for the three normality tests discussed in Section 4.2. Results sugges<sup>t</sup> that the GTPN model is an appropriate model for this data set while the rest of models are not.

**Table 3.** Estimation of the parameters and their standard errors (in parentheses) for the GTPN, TPN, WEI, and GL models for the data set. The AIC and BIC criteria are also included.


**Figure 6.** Fit of the distributions for the Laslett data set.

**Figure 7.** QR for the fitted models in the Laslett data set. The *p* − *values* for the Anderson–Darling (AD), Cramer-Von-Mises (CVM) and Shapiro–Wilks (SW) normality tests are also presented to check if the RQ came from the standard normal distribution.
