Beta and Kumaraswamy distributions as non-nested hypotheses in the modeling of continuous bounded data

Nowadays, beta and Kumaraswamy distributions are the most popular models to fit continuous bounded data. These models present some characteristics in common and to select one of them in a practical situation can be of great interest. With this in mind, in this paper we propose a method of selection between the beta and Kumaraswamy distributions. We use the logarithm of the likelihood ratio statistic (denoted by $T_n$, where $n$ is the sample size) and obtain its asymptotic distribution under the hypotheses $H_{\mathcal B}$ and $H_{\mathcal K}$, where $H_{\mathcal B}$ ($H_{\mathcal K}$) denotes that the data come from the beta (Kumaraswamy) distribution. Since both models has the same number of parameters, based on the Akaike criterion, we choose the model that has the greater log-likelihood value. We here propose to use the probability of correct selection (given by $P(T_n>0)$ or $P(T_n<0)$ depending on the null hypothesis) instead of only to observe the maximized log-likelihood values. We obtain an approximation for the probability of correct selection under the hypotheses $H_{\mathcal B}$ and $H_{\mathcal K}$ and select the model that maximizes it. A simulation study is presented in order to evaluate the accuracy of the approximated probabilities of correct selection. We illustrate our method of selection in two applications to real data sets involving proportions.