Jornal de Biometria e Bioestatística

Relative Likelihood Differences to Examine Asymptotic Convergen ce: A Bootstrap Simulation Approach

Abstract

Milan Bimali and Michael Brimacombe

Maximum likelihood estimators (mle) and their large sample properties are extensively used in descriptive as well as inferential statistics. In the framework of large sample distribution of mle, it is important to know the relationship between the sample size and asymptotic convergence i.e. for what sample size does the mle behave satisfactorily attaining asymptotic normality. Previous works have discussed the undesirable impacts of using large sample approximations of the mles when such approximations do not hold. It has been argued that relative likelihood functions must be examined before making inferences based on mle. It was also demonstrated that transformation of mle can help achieve asymptotic normality with smaller sample sizes. Little has been explored regarding the appropriate sample size that would allow the mle achieve asymptotic normality from relative likelihood perspective directly. Our work proposes bootstrap/simulation based approach in examining the relationship between sample size and asymptotic behaviors of mle. We propose two measures of the convergence of observed relative likelihood function to the asymptotic relative likelihood functions namely: differences in areas and dissimilarity in shape between the two relative likelihood functions. These two measures were applied to datasets from literatures as well as simulated datasets.