**4. Model Discrimination**

In this section we discuss some techniques to discriminate among the GTPN distribution and other models.

### *4.1. GTPN versus Submodels*

An interesting problem to solve is the discrimination between GTPN and the three submodels represented in Figure 1. In other words, we are interested in testing the following hypotheses:

• *H*(1) 0: *α* = 1 versus *H*(1) 1: *α* = 1 (TPN versus GTPN distribution).

• *H*(2) 0: *λ* = 0 versus *H*(2) 1: *λ* = 0 (GHN versus GTPN distribution).

• *H*(3) 0: (*<sup>α</sup>*, *λ*)=(1, 0) versus *H*(3) 1: (*<sup>α</sup>*, *λ*) = (1, 0) (HN versus GTPN distribution).

The three hypotheses can be tested considering the LRT, ST, and GT. Below we present the statistics for the three tests considered and for the three hypotheses of interest.

## 4.1.1. Likelihood Ratio Test

The statistic for the LRT (say SLR) to tests *H*(*j*) 0 , *j* = 1, 2, 3, is defined as

$$\text{SLR}\_{\hat{\jmath}} = 2\left[\ell\left(\hat{\sigma}, \hat{\lambda}, \hat{\kappa}\right) - \ell\left(\hat{\sigma}\_{0j}, \hat{\lambda}\_{0j}, \hat{\kappa}\_{0j}\right)\right],$$

where *<sup>σ</sup>*0*j*, *<sup>λ</sup>*0*j* and *<sup>α</sup>*0*j* denote the ML estimators for *σ*, *λ* and *α* restricted to *H*(*j*) 0 , *j* = 1, 2, 3. Under *H*(*j*) 0 , *j* = 1, 2, SLR*j* ∼ *<sup>χ</sup>*<sup>2</sup>(1) and under *H*(3) 0 , SLR3 ∼ *<sup>χ</sup>*<sup>2</sup>(2), where *<sup>χ</sup>*<sup>2</sup>(*p*) denotes the Chi-squared distribution with *p* degrees of freedom. For *H*(3) 0 , we obtain

$$
\hat{\sigma}\_{03} = \sqrt{\frac{1}{n} \sum\_{i=1}^{n} z\_i^2}, \quad \hat{\lambda}\_{03} = 0 \quad \text{and} \quad \hat{a}\_{03} = 1,
$$

whereas to test *H*(1) 0 and *H*(2) 0 the ML estimators under the null hypotheses need to be computed numerically. However, in both cases the problem is reduced to a unidimensional maximization. For details see Cooray and Ananda [9] and Gómez et al. [13], respectively.
