Distribuição qui-quadrado inf: uma nova abordagem para o aperfeiçoamento do teste da razão de verossimilhanças

Abstract

One of the main objectives of statistics is to make inference in a population or phenomenon from a subset of data from this, called a sample. One way to make inference is to perform hypothesis testing. In general, we can say that it is from a sample of the population that we will establish a decision rule according to which we will reject or not reject the proposed hypothesis, called null hypothesis. A general procedure that produces reasonable tests is the Likelihood Ratio test (TRV). To apply the TRV, we need to know the true distribution of the likelihood ratio λ∗(x) which is generally not easy to obtain. However, it is known in the literature that the RV = −2 log(λ∗(x)) statistic follows an approximate chi-square distribution when the test is based on a large sample size. However, using the chi-square distribution as an approximation to the true distribution of the RV statistic can lead to inaccurate inferences when the sample size is small. The objective of this research is to improve the TRV when the test is based on a small or moderate sample. For this, we make use of new families of distributions, denoted sup and inf. Based on some properties of the inf distribution family, we propose a new approach for the improvement of TRV. Specifically, we used the chi-square inf distribution as a chi-square correction distribution to obtain the quantile that determines the critical region of the test. In addition, this work creates the computational package LikRatioTest, written in the R language, with the objective of evaluating the performance of the improved TRV (TRV∗) and comparing it with the classic asymptotic TRV. This package allows you to do Monte Carlo simulations, imposing various scenarios, and thus, calculate the rejection rate of the null hypothesis using the two approaches (classical and improved). This package is open source and is available on GitHub for installation on R. To illustrate the usefulness of our proposal, we show two numerical examples using real data sets.