Hierarchical Missing Data and Multivariate Behrens–Fisher Problem

Jianqi Yu,Ljubisa Kocinac

doi:10.1155/2021/8837044

Abstract

This article firstly defines hierarchical data missing pattern, which is a generalization of monotone data missing pattern. Then multivariate Behrens–Fisher problem with hierarchical missing data is considered to illustrate that how ideas in dealing with monotone missing data can be extended to deal with hierarchical missing pattern. A pivotal quantity similar to the Hotelling T 2 is presented, and the moment matching method is used to derive its approximate distribution which is for testing and interval estimation. The precision of the approximation is illustrated through Monte Carlo data simulation. The results indicate that the approximate method is very satisfactory even for moderately small samples.

Highlights

Inferences with incomplete data have aroused lots of interest among statisticians in the past as well as present. e causes for missing data could be various which will not be discussed in this article
Ere are a few missing patterns considered in the literatures, but the incomplete data with monotone pattern (see display (1) and (2)) occur frequently in practice and it allows the exact calculation of the maximum likelihood estimators (MLEs) and the likelihood ratio statistics and relevant distributions if multivariate normality is assumed
We define hierarchical data missing pattern and point out that the strategy in many papers dealing with monotone missing data can be extended to deal with hierarchical missing data

Summary

Introduction

Inferences with incomplete data have aroused lots of interest among statisticians in the past as well as present. e causes for missing data could be various which will not be discussed in this article. Ere are a few missing patterns considered in the literatures, but the incomplete data with monotone pattern (see display (1) and (2)) occur frequently in practice and it allows the exact calculation of the maximum likelihood estimators (MLEs) and the likelihood ratio statistics and relevant distributions if multivariate normality is assumed. Yu et al [10] considered the problem of testing equality of two normal covariance matrices with monotone missing data. Batsidis [11,12,13] extends the inferences on monotone missing data to the assumption of elliptically contoured distributions of which the multivariate normal is a special case. We consider the multivariate Behrens–Fisher problem with hierarchical missing data.

Hierarchical Data Missing Pattern

Accuracy of the Approximations

An Illustrative Example

Concluding Remarks