Asymptotic expansions for a class of tests for a general covariance structure under a local alternative

AbstractLet S be a p×p random matrix having a Wishart distribution Wp(n,n−1Σ). For testing a general covariance structure Σ=Σ(ξ), we consider a class of test statistics Th=nρh(S,Σ(ξˆ)), where ρh(Σ1,Σ2)=∑i=1ph(λi) is a distance measure from Σ1 to Σ2, λi’s are the eigenvalues of Σ1Σ2−1, and h is a given function with certain properties. Wakaki, Eguchi and Fujikoshi (1990) suggested this class and gave an asymptotic expansion of the null distribution of Th. This paper gives an asymptotic expansion of the non-null distribution of Th under a sequence of alternatives. By using results, we derive the power, and compare the power asymptotically in the class. In particular, we investigate the power of the sphericity tests

A class of tests for a general covariance structure

AbstractLet S be a p × p random matrix having a Wishart distribution Wp(n,n−1Σ). For testing a general covariance structure Σ = Σ(ξ), we consider a class of test statistics Th = n inf ϱh(S, Σ(ξ)), where ϱh(Σ1, Σ2) = Σj = 1ph(λj) is a distance measure from Σ1 to Σ2, λi's are the eigenvalues of Σ1Σ2−1, and h is a given function with certain properties. This paper gives an asymptotic expansion of the null distribution of Th up to the order n−1. Using the general asymptotic formula, we give a condition for Th to have a Bartlett adjustment factor. Two special cases are considered in detail when Σ is a linear combination or Σ−1 is a linear combination of given matrices

Contributions to multivariate analysis by Professor Yasunori Fujikoshi

The purpose of this article is to review the findings of Professor Fujikoshi which are primarily in multivariate analysis. He derived many asymptotic expansions for multivariate statistics which include MANOVA tests, dimensionality tests and latent roots under normality and nonnormality. He has made a large contribution in the study on theoretical accuracy for asymptotic expansions by deriving explicit error bounds. A large contribution has been also made in an important problem involving the selection of variables with introducing "no additional information hypotheses" in some multivariate models and the application of model selection criteria. Recently he is challenging to a high-dimensional statistical problem. He has been involved in other topics in multivariate analysis, such as power comparison of a class of tests, monotone transformations with improved approximations, etc.Asymptotic expansions MANOVA Dimensionality Latent roots Growth curve model Selection of variables Error bounds Monotone transformations

Asymptotic expansions for a class of tests for a general covariance structure under a local alternative

Let be a pxp random matrix having a Wishart distribution . For testing a general covariance structure , we consider a class of test statistics , where is a distance measure from to , [lambda]i's are the eigenvalues of , and h is a given function with certain properties. Wakaki, Eguchi and Fujikoshi (1990) suggested this class and gave an asymptotic expansion of the null distribution of Th. This paper gives an asymptotic expansion of the non-null distribution of Th under a sequence of alternatives. By using results, we derive the power, and compare the power asymptotically in the class. In particular, we investigate the power of the sphericity tests.Asymptotic expansion Class of test statistics General covariance structure Non-null distribution Local alternative Power comparison Linear structure Sphericity test

Optimal discriminant functions for normal populations

A class of discriminant rules which includes Fisher's linear discriminant function and the likelihood ratio criterion is defined. Using asymptotic expansions of the distributions of the discriminant functions in this class, we derive a formula for cut-off points which satisfy some conditions on misclassification probabilities, and derive the optimal rules for some criteria. Some numerical experiments are carried out to examine the performance of the optimal rules for finite numbers of samples.62H30 62H20 Linear discriminant function W-rule Z-rule Asymptotic expansion

High-dimensional Edgeworth expansion of a test statistic on independence and its error bound

In this paper, we calculate Edgeworth expansion of a test statistic on independence when some of the parameters are large, and simulate the goodness of fit of its approximation. We also calculate an error bound for Edgeworth expansion. Some tables of the error bound are given, which show that the derived bound is sufficiently small for practical use.Edgeworth expansion Error bound High dimension Likelihood ratio

A class of tests for a general covariance structure

Let S be a p - p random matrix having a Wishart distribution Wp(n,n-1[Sigma]). For testing a general covariance structure [Sigma] = [Sigma]([xi]), we consider a class of test statistics Th = n inf [varrho]h(S, [Sigma]([xi])), where [varrho]h([Sigma]1, [Sigma]2) = [Sigma]j = 1ph([lambda]j) is a distance measure from [Sigma]1 to [Sigma]2, [lambda]i's are the eigenvalues of [Sigma]1[Sigma]2-1, and h is a given function with certain properties. This paper gives an asymptotic expansion of the null distribution of Th up to the order n-1. Using the general asymptotic formula, we give a condition for Th to have a Bartlett adjustment factor. Two special cases are considered in detail when [Sigma] is a linear combination or [Sigma]-1 is a linear combination of given matrices.asymptotic expansion class of test statistics general covariance structure linear structure null distribution