A general method of inference for two-parameter continuous distributions

Abstract

This article presents a general method of inference of the parameters of a continuous distribution with two unknown parameters. Except in a few distributions such as the normal distribution, the classical approach fails in this context to provide accurate inferences with small samples.Therefore, by taking the generalized approach to inference (cf. Weerahandi, 1995 Weerahandi, S. (1995). Exact Statistical Methods for Data Analysis. New York: Springer Verlag. Google Scholar]), in this article we present a general method of inference to tackle practically useful two-parameter distributions such as the gamma distribution as well as distributions of theoretical interest such as the two-parameter uniform distribution. The proposed methods are exact in the sense that they are based on exact probability statements and exact expected values. The advantage of taking the generalized approach over the classical approximate inferences is shown via simulation studies.

This article has the potential to motivate much needed further research in non normal regressions, multiparameter problems, and multivariate problems for which basically there are only large sample inferences available. The approach that we take should pave the way for researchers to solve a variety of non normal problems, including ANOVA and MANOVA problems, where even the Bayesian approach fails. In the context of testing of hypotheses, the proposed method provides a superior alternative to the classical generalized likelihood ratio method.