**4. Application**

In this section, a numerical illustration based on a real dataset is presented. The goal of this application is to show empirical evidence that the MPN yields a better fit to data than the PN , SN , and t-student (T S) with *α* degrees of freedom distributions. For that reason, we consider a set of 3848 observations of the variable "density" included in the dataset verb "POLLEN5.DA" available at http://lib.stat.cmu.edu/datasets/pollen.data. This variable measures a geometric characteristic of a specific type of pollen. This dataset was previously used by Pewsey et al. [9] to compare the PN and SN distributions. A summary of some descriptive statistics are displayed in Table 4 below.


**Table 4.** Summary of descriptive statistics for the pollen density dataset.

By using the results derived in Proposition 4, we have computed the moment estimates for the parameters (*μ*, *σ*, *α*) of the MPN distribution, obtaining (−5.609, 4.576, 11.857). Then, by taking these numbers as initial values, the ML estimates are derived. In Table 5, the ML estimates for the parameters of the MPN , PN , SN , and T S distributions. The figures between brackets are the asymptotic standard errors of the estimates obtained by inverting the Fisher's information matrices for the three models evaluated at their respective ML estimates. Additionally, for each model, the values of the maximum of the log-likelihood function (*max*) are reported. The MPN distribution attains the largest value, and consequently provides a better fit to data.

**Table 5.** Parameter estimates; standard errors (SE); and maximum of the log-likelihood function, *max*, for the T S, SN , PN , and MPN corresponding to the pollen density dataset.


To compare the fit achieved by each distribution, the values of several measures of model selection, i.e., Akaike's information criterion (AIC) (see Akaike [22]) and Bayesian information criterion (BIC) (see Schwarz [23]) are reported in Table 6. A model with lower numbers in these measures of model selection is preferable. It can be seen that the MPN is preferable in terms of these two measures of model validation. In addition, the Kolmogorov–Smirnov test statistics and the corresponding *p*-values has been included in this table for all the models considered. It can be observed that none of the models is rejected at the usual significance levels. However, the MPN distribution has a higher *p*-value and is rejected later than the other two models. Alternative methods of model selection to the Kolmogorov–Smirnov test that can be applied here can be found in Ja¨ntschi and Bolboaca˘ [24] and Ja¨ntschi [25]. Furthermore, the histogram associated to the empirical distribution of the variable "density" in the pollen dataset is illustrated in the left hand side of Figure 5. In addition, the densities of T S, SN , PN , and MPN , by using the maximum likelihood estimates of their parameters, have been superimposed. Similarly, on the right hand side of Figure 5, the fit in both tails is shown. It is observable that, for this dataset, the MPN has thicker tails than the other three distributions. Finally, the QQ-plots for each distribution considered have been illustrated in Figure 6. Here, note that the MPN distribution exhibits an almost perfect alignment with the 45◦ line, and therefore it provides a better fit for extreme quantiles. Finally, Figure 7 displays the profile log-likelihood of *μ*, *σ*, and *α* of the MPN distribution. It is noticeable that the estimates are unique.

**Table 6.** Akaike's information criterion (AIC), Bayesian information criterion (BIC), Kolmogorov– Smirnov (KSS) test, and the corresponding *p*-values for all the models considered.


**Figure 5. Left** panel: Histogram of the empirical distribution and fitted densities by ML superimposed for pollen dataset. **Right** panel: Plots of the tails for the four models.

**Figure 6.** QQ-plots: (**a**) MPN model; (**b**) PN model; (**c**) SN model; (**d**) T S model.
