Asymptotic Distribution of Probabilities of Misclassification for Edgeworth Series Distribution (ESD)

Awogbemi Clement Adeyeye

doi:10.11648/j.engmath.20200401.11

Abstract

The exact distribution of the test statistics in multivariate case is quite complicated in many situations, even when the underlying distribution is multivariate normal. This is due to the complex nature of the expression and therefore, there is a need to derive the asymptotic expression for the distribution. In this study, the asymptotic distribution of errors of misclassification for Edgeworth Series is derived by using Taylor’s expansion. The error of misclassification for the conditional probability of misclassification was expanded around the means emanating from populations one and two using approximated mean and variance of the errors of misclassification. The distribution of error of misclassification of the conditional probability of misclassification for ESD is approximately normal with mean zero and variance one.

Highlights

Edgeworth Series Distribution (ESD) constitutes an expansion which is a series that approximates a probability distribution in terms of its cumulants and the Hermite polynomials
The use of ESD is expedient because approximations to distribution of sample statistics of higher order than
Finite sample approximations for the distribution functions of Generalized Empirical Likelihood (GEL) were derived, and the analytical results obtained were applied to estimators which serve as alternatives to Generalized Method of Moment (GMM) [7]

Summary

Introduction

Edgeworth Series Distribution (ESD) constitutes an expansion which is a series that approximates a probability distribution in terms of its cumulants and the Hermite polynomials. The distribution of the conditional probability of misclassification of Edgeworth Series Distribution (ESD) is intractable due to the complex nature of the expression [16] It comprises of the normal density function, the cumulative normal distribution function and the Chebyshev’s Hermite polynomial. Edgeworth series expansion and the saddle point method were investigated [13] This is to estimate the distribution function for the standardized mean of independent and identically distributed random variables. Finite sample approximations for the distribution functions of Generalized Empirical Likelihood (GEL) were derived, and the analytical results obtained were applied to estimators which serve as alternatives to Generalized Method of Moment (GMM) [7]. By considering the standardized sum of n independent and identically distributed random variables, Edgeworth Series is obtained by collecting terms in equation (1) according to the power of n [8]. Λ3 is the skewness factor [2]

Methodology

Approximate Mean and Variance of Conditional Probability of Misclassification

Conclusion