TY - JOUR

T1 - A general near-exact distribution theory for the most common likelihood ratio test statistics used in Multivariate Analysis

AU - Coelho, Carlos Manuel Agra

AU - Marques, Filipe José Gonçalves Pereira

PY - 2011/1/1

Y1 - 2011/1/1

N2 - In this paper we first show how the exact distributions of the most common likelihood ratio test (l.r.t.) statistics, that is, the ones used to test the independence of several sets of variables, the equality of several variance-covariance matrices, sphericity, and the equality of several mean vectors, may be expressed as the distribution of the product of independent Beta random variables or the product of a given number of independent random variables whose logarithm has a Gamma distribution times a given number of independent Beta random variables. Then, we show how near-exact distributions for the logarithms of these statistics may be expressed as Generalized Near-Integer Gamma distributions or mixtures of these distributions, whose rate parameters associated with the integer shape parameters, for samples of size n, all have the form (n-j)/n for j=2,aEuro broken vertical bar,p, where for three of the statistics, p is the number of variables involved, while for the other one, it is the sum of the number of variables involved and the number of mean vectors being tested. What is interesting is that the similarities exhibited by these statistics are even more striking in terms of near-exact distributions than in terms of exact distributions. Moreover all the l.r.t. statistics that may be built as products of these basic statistics also inherit a similar structure for their near-exact distributions. To illustrate this fact, an application is made to the l.r.t. statistic to test the equality of several multivariate Normal distributions.

AB - In this paper we first show how the exact distributions of the most common likelihood ratio test (l.r.t.) statistics, that is, the ones used to test the independence of several sets of variables, the equality of several variance-covariance matrices, sphericity, and the equality of several mean vectors, may be expressed as the distribution of the product of independent Beta random variables or the product of a given number of independent random variables whose logarithm has a Gamma distribution times a given number of independent Beta random variables. Then, we show how near-exact distributions for the logarithms of these statistics may be expressed as Generalized Near-Integer Gamma distributions or mixtures of these distributions, whose rate parameters associated with the integer shape parameters, for samples of size n, all have the form (n-j)/n for j=2,aEuro broken vertical bar,p, where for three of the statistics, p is the number of variables involved, while for the other one, it is the sum of the number of variables involved and the number of mean vectors being tested. What is interesting is that the similarities exhibited by these statistics are even more striking in terms of near-exact distributions than in terms of exact distributions. Moreover all the l.r.t. statistics that may be built as products of these basic statistics also inherit a similar structure for their near-exact distributions. To illustrate this fact, an application is made to the l.r.t. statistic to test the equality of several multivariate Normal distributions.

KW - Wilks lambda statistic

KW - Mixtures

KW - Generalized Near-Integer Gamma distribution

KW - Generalized Integer Gamma distribution

KW - Independence test

KW - Sphericity test

U2 - 10.1007/s11749-010-0193-3

DO - 10.1007/s11749-010-0193-3

M3 - Article

SN - 1133-0686

VL - 20

SP - 180

EP - 203

JO - Test

JF - Test

IS - 1

ER -