Skewness-kurtosis adjusted confidence estimators and significance tests

TL;DR

This paper introduces skewness-kurtosis adjusted confidence intervals and significance tests that precisely control asymptotic error probabilities, with applications to exponential families.

Contribution

It develops modifications to confidence intervals and tests that asymptotically control error probabilities under skewness and kurtosis adjustments.

Findings

Asymptotic control of non-coverage and coverage probabilities achieved.

Modified confidence intervals and tests perform well under tail conditions.

Applications demonstrated for exponential family distributions.

Abstract

First and second kind modifications of usual confidence intervals for estimating the expectation and of usual local alternative parameter choices are introduced in a way such that the asymptotic behavior of the true non-covering probabilities and the covering probabilities under the modified local non-true parameter assumption can be asymptotically exactly controlled. The orders of convergence to zero of both types of probabilities are assumed to be suitably bounded below according to an Osipov-type condition and the sample distribution is assumed to satisfy a corresponding tail condition due to Linnik. Analogous considerations are presented for the power function when testing a hypothesis concerning the expectation both under the assumption of a true hypothesis as well as under a modified local alternative. Applications are given for exponential families.