Testing the equality of matrix distributions
- PDF / 689,926 Bytes
- 19 Pages / 439.37 x 666.142 pts Page_size
- 105 Downloads / 202 Views
Testing the equality of matrix distributions Lingzhe Guo1 · Reza Modarres1 Accepted: 2 June 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract While matrices are usually used as the basic data structure for experiments with repeated measurements or longitudinal data, testing methods for the equality of two matrix distributions have not been fully discussed in the literature. In this article, we propose three methods to test the equality of two matrix distributions: the likelihood ratio test, the Frobenius norm methods and triangle tests. We present a simulation to compare their performance under the matrix normal distribution. We apply the testing methods to compare the US economy, as measured by closing prices of five market indices, before and after the US stock market crash of 2008. Keywords Matrix distribution · Matrix normal · Homogeneity · Frobenius norm Mathematics Subject Classification 62H10 · 62H15 · 62G20
1 Introduction The purpose of this paper is to construct tests for the equality of matrix distributions. Suppose A = {X1 , . . . , Xn 1 } are i.i.d. d × m random matrices drawn from distribution function F, and B = {Y1 , . . . , Yn 2 } are i.i.d. d × m random matrices drawn from distribution function G. Our aim is to test the null hypothesis H0 :F = G versus the general alternative H1 :F = G. With applications in physics, economics, psychology, biomedicine and epidemiology, matrix distributions are widely used in the experiments with repeated measurements or longitudinal data where vectors of attributes from independent subjects are measured over time, in different occasions or geographic locations. Observation of various attributes measured on a set of subjects in different spatio-temporal conditions is often referred to as a three-way dataset (Carroll and Arabie 1980) and represented by a random matrix. In multiple ranking
B 1
Reza Modarres [email protected] Department of Statistics, The George Washington University, Washington, DC 20052, USA
123
L. Guo, R. Modarres
and selection procedures, matrix observations arise when some objects are ranked on multiple attributes by multiple judges (Vermunt 2007). Gupta and Nagar (1999) present a systematic discussion of the matrix distributions, including normal, Wishart, beta, Dirichlet, spherical and elliptical, among others. Dawid (1981) considers theory and a Bayesian application of matrix distributions. Naik and Rao (2001) considered a dataset of repeated measurements on the individuals from different groups and analyzed the time and group effects in the experiment. Lovison (2006) proposes a matrix Bernoulli distribution and discusses extensions to categorical matrix data. Banerjee et al. (2015) transform vectors into well-separated matrices in order to study the change in the correlation structure. Harrar and Gupta (2008) introduce matrix variate skew-normal distributions to model data involving skewness. Gallaugher and McNicholas (2017) extend this work to a matrix variate skew-t distribution. Lu and Zimmerman (2005), Mitchell
Data Loading...
