ANewTestforaNormalCovarianceMatrix

禹建奇

【Abstract】The problem of testing the normal covariance matrix equal to a specified matrix is considered.A new Chi-Square test statistic is derived for multivariate normal population.Unlike the likelihood ratio test，the new test is an exact one.

【Key words】exact test； Chi-Square test； likelihood ratio test

1.Introduction of the Problem

Suppose we have a sampleand we want to test the problemvs，whereis a known matrix.It is easy to see that the problem is invariant under the changeswhere T is nonsingular.Therefore，without loss of generality，we can assume that，the identity matrix.

2.The Test Procedure

Letbe the sample means and sample covariance，and the -th component of S，where.The null becomes .let .It is easy to see that H0 is equivalent to.

For testing，there is a well known test statistics nS11and under .

For testing，we regressonand get

where

Letthenandunder H0，and they can be used for testing respectively.Moreover，the three statistics above are independent under H0，so we combine them together and get a new statistics：

Let ，and then under H0，.

Finally，we get an exact test which is totally different from the ones we have already：

If or，reject the null H0 at level .

3.The Generalization

At the end，we point out that the test can be easily extended to the p-variate normal cases where p>2.but the notation will become very complicated.So，we just point out this and do not show the results.

4.Concluding Remarks

In this article，we have developed a Chi-Square test procedure for testing a normal covariance matrix equal to a given matrix.Unlike the usual likelihood ratio test，the Chi-Square test is an exact one and easy to apply.Moreover，the powers of the Chi-Square test are always higher than are those of LRT，especially for small samples.In a word，the Chi-Square test is superior to the LRT.

References：

[1]Das Gupta，S.，1969.Properties of power functions of some tests concerning dispersion matrices of multivariate normal distributions.Ann.Math.Statist.40，697-701.

[2]Muirhead，R.J.，1982.Aspects of Multivariate Statistical Theory.Wiley，New York.

[3]Seber，G.A.F.，1984.Multivariate Observations，Wiley，New York.

【资助项目】本文获“桂林理工大学博士科研启动基金（2014）”及“桂林理工大学校级教改项目（2014B06）”支持。