*4.2. Non-Nested Models*

The comparison of non-nested models can be performed based on the AIC criteria (Akaike [17]), where the model with a lower AIC is preferred. However, in practice we can have a set of inappropriate models for a certain data set. For this reason, we also need to perform a goodness-of-fit validation. This can be performed, for instance, based on the quantile residuals (QR). For more details see Dunn and Smyth [18]. These residuals are defined as

$$\mathcal{QR}\_{i} = G(z\_{i}; \widehat{\psi}), \quad i = 1, \ldots, n\_{r}$$

where *<sup>G</sup>*(; *ψ*) is the cdf of the specified distribution evaluated in the estimator for *ψ*. If the model is correctly specified, such residuals are a random sample from the standard normal distribution. This can be assessed using, for instance, the Anderson–Darling (AD), Cramer-Von-Mises (CVM) and Shapiro–Wilks (SW) tests. A discussion of these tests can be seen in Yazici and Yocalan [19].
