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      ANewTestforaNormalCovarianceMatrix

      2015-05-19 13:13:27禹建奇
      校園英語·中旬 2015年2期
      關(guān)鍵詞:校級(jí)理工大學(xué)桂林

      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.

      【資助項(xiàng)目】本文獲“桂林理工大學(xué)博士科研啟動(dòng)基金(2014)”及“桂林理工大學(xué)校級(jí)教改項(xiàng)目(2014B06)”支持。

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