Asymptotic Expansion of the Null Distribution of LR Statistic for Multivariate Linear Hypothesis when the Dimension is Large

Abstract In this paper we derive an asymptotic expansion of the null distribution of likelihood ratio statistic for multivariate linear hypothesis when the dimension is comparable to the sample size. Our asymptotic approximations are numerically compared with some other approximations including the large sample approximation due to Box [Box, G. E. P. (1949). A general distribution theory for a class of likelihood criteria. Biometrika 36:317–346]. It is shown that the approximations proposed in this paper are good when the dimension is large and close to the sample size, and further our approximations are similar to the Box's approximation even for most of the usual large sample cases.

[1]  G. S. Mudholkar,et al.  A Normal Approximation for the Multivariate Likelihood Ratio Statistics , 1981 .

[2]  C. R. Rao,et al.  An asymptotic expansion of the distribution of Wilk's criterion , 1951 .

[3]  Peter Hall,et al.  On the Removal of Skewness by Transformation , 1992 .

[4]  R. Butler,et al.  Saddlepoint approximations for the generalized variance and Wilks statistic , 1992 .

[5]  M. Srivastava,et al.  Saddlepoint method for obtaining tail probability of Wilks' likelihood ratio test , 1989 .

[6]  G. Box,et al.  A general distribution theory for a class of likelihood criteria. , 1949, Biometrika.

[7]  Z. Bai.  METHODOLOGIES IN SPECTRAL ANALYSIS OF LARGE DIMENSIONAL RANDOM MATRICES , A REVIEW , 1999 .

[8]  Yasunori Fujikoshi,et al.  Error Bounds for Asymptotic Approximations of the Linear Discriminant Function When the Sample Sizes and Dimensionality are Large , 2000 .

[9]  George Henry Dunteman,et al.  Introduction To Multivariate Analysis , 1984 .

[10]  G. S. Mudholkar,et al.  A normal approximation for the distribution of the likelihood ratio statistic in multivariate analysis of variance , 1980 .

[11]  Y. Fujikoshi,et al.  Asymptotic aproximations for EPMC’s of the linear and the quadratic discriminant functions when the sample sizes and the dimension are large , 1998 .

[12]  Yoong-Sin Lee,et al.  Some results on the distribution of Wilks's likelihood-ratio criterion , 1972 .

[13]  A. J. Collins,et al.  Introduction To Multivariate Analysis , 1981 .