Modelos Skellam generalizados

Abstract

In applied statistics, counting data are often observed in different areas of study. Due to the great diversity of problems that result in these types of data, it is necessary to propose new models. In this work, we propose generalizations of the Skellam distribution, whose support consists of the set formed by integers (positive and negative), aiming to also explore in the context of regression models. For the process of estimating and inferring model parameters, the classical (maximum likelihood method) and Bayesian (Markov Chain Monte Carlo) approaches were considered for comparison purposes. Specifically regarding the Bayesian approach, which was more efficient in the proposals presented here, we used a variant of the Hamiltonian Monte Carlo algorithm, which consists of reformulating Hamilton’s equations by introducing a stochastic component into the gradient equation, deriving the Stochastic Gradient Hamiltonian Monte Carlo algorithm. To illustrate the proposed models, we present the analyzes of data sets referring to two real problems (3 datasets in total): In the first problem, a set of data corresponding to observations of the weekly variation of the Ibovespa score was considered, that is, the price difference, measured in ticks (cents) of the current day in relation to the previous day, in the period between January 2000 and December 2022. The estimated values for the parameter p characterized the data set as inflated with observations -2 (ticks); In the second problem, two sets of data were considered, corresponding to the values of the differences between games won and games lost by teams in the 2022-2023 regular season of the National Basketball Association, in each conference (East and West). The selection criteria indicated the k-MS model with k = −12 as the best adjusted for the Eastern conference, while for the Western conference, the indicated value was k = −38. Given the good results, both the k-MS and the k-IS models proved to be good alternatives to explain the behavior of data with integer values.