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.

A skew true INAR(1) process with application

Integer-valued time series models have been a recurrent theme considered in many papers in the last three decades, but only a few of them have dealt with models on $mathbb Z$ (that is, including both negative and positive integers). Our aim in this paper is to introduce a first-order integer-valued autoregressive process on $mathbb Z$ with skew discrete Laplace marginals (Kozubowski and Inusah, 2006). For this, we define a new operator that acts on two independent latent processes, similarly as made by Freeland (2010). We derive some joint and conditional basic properties of the proposed process such as characteristic function, moments, higher-order moments and jumps. Estimators for the parameters of our model are proposed and their asymptotic normality are established. We run a Monte Carlo simulation to evaluate the finite-sample performance of these estimators. In order to illustrate the potentiality of our process, we apply it to a real data set about population increase rates.

A new lifetime model with decreasing failure rate

In this paper we introduce a new lifetime distribution by compounding exponential and Poisson-Lindley distributions, named exponential Poisson-Lindley distribution. Several properties are derived, such as density, failure rate, mean lifetime, moments, order statistics and Renyi entropy. Furthermore, estimation by maximum likelihood and inference for large sample are discussed. The paper is motivated by two applications to real data sets and we hope that this model be able to attract wider applicability in survival and reliability.