Modern likelihood inference for the maximum/minimum of a bivariate normal vector

V Mameli,A R Brazzale

doi:10.1080/00949655.2015.1089872

V Mameli, A R Brazzale

Open Access

https://doi.org/10.1080/00949655.2015.1089872

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Abstract

ABSTRACTWe consider the use of modern likelihood asymptotics in the construction of confidence intervals for the parameter which determines the skewness of the distribution of the maximum/minimum of an exchangeable bivariate normal random vector. Simulation studies were conducted to investigate the accuracy of the proposed methods and to compare them to available alternatives. Accuracy is evaluated in terms of both coverage probability and expected length of the interval. We furthermore illustrate the suitability of our proposals by means of two data sets, consisting of, respectively, measurements taken on the brains of 10 mono-zygotic twins and measurements of mineral content of bones in the dominant and non-dominant arms for 25 elderly women.

Highlights

Small sample sizes are rather common to researchers in fields such as biology, genetics, medical sciences and psychology
In this paper we investigate the behavior of modern likelihood-based small-sample procedures to compute confidence intervals for the parameter of skewness which characterizes the distribution of the maximum/minimum of a bivariate Normal and exchangeable random vector
Extensive numerical investigation revealed that the higher order frequentist pivot r∗ is highly accurate, especially for the rather small sample sizes which may be encountered, and for the challenging situation where ρ is close to −1

Summary

Introduction

Small sample sizes are rather common to researchers in fields such as biology, genetics, medical sciences and psychology. Inference based on the classical first order Normal and χ2 approximations may be unreliable. The last four decades have seen the development of so-called higher order likelihood approximations, which require little more effort than is needed for their first order counterparts while providing highly accurate inferences in small samples. The aim of this paper is to encourage the use of these modern likelihood-based solutions for the analysis of bivariate normally distributed continuous data when interest focuses on the maximum (or minimum) value of correlated observations. A great variety of studies, especially in the medical area, are based on the analysis of extreme observations. The extremes coming from the left and right sides of the body are commonplace in comparative studies; see [2] for an illustration about the visual acuity of fellow eyes, or [3], [4] and reference therein, for other typical cases

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