High-Accuracy Confidence Regions for Distribution Parameters

Abstract

With the help of today’s computers, it is always relatively easy to find maximum-likelihood estimators of one or more parameters of any specific statistical distribution, and use these to construct the corresponding approximate confidence interval/region, facilitated by the well-known asymptotic properties of the likelihood function. The purpose of this article is to make this approximation substantially more accurate by extending the Taylor expansion of the corresponding probability density function to include quadratic and cubic terms in several centralized sample means, and thus finding the corresponding -proportional correction to the original algorithm. We then demonstrate the new procedure’s usage, both for constructing confidence regions and for testing hypotheses, emphasizing that incorporating this correction carries minimal computational and programming cost. In our final chapter, we present two examples to indicate how significantly the new approximation improves the procedure’s accuracy.