Likelihood Ratio Tests in Multivariate Linear Model

The aim of this chapter is to review likelihood ratio test procedures in multivariate linear models, focusing on projection matrices. It is noted that the projection matrices to the spaces spanned by mean vectors in hypothesis and alternatives play an important role. Some basic properties are given for projection matrices. The models treated include multivariate regression model, discriminant analysis model, and growth curve model. The hypotheses treated involve a generalized linear hypothesis and no additional information hypothesis, in addition to a usual liner hypothesis. The test statistics are expressed in terms of both projection matrices and sums of squares and products matrices

An Asymptotic Expansion for the Distribution of Hotelling'sT2-Statistic under Nonnormality

AbstractIn this paper we obtain an asymptotic expansion for the distribution of Hotelling'sT2-statisticT2under nonnormality when the sample size is large. In the derivation we find an explicit Edgeworth expansion of the multivariatet-statistic. Our method is to use the Edgeworth expansion and to expand the characteristic function ofT2

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

Asymptotics for testing hypothesis in some multivariate variance components model under non-normality

AbstractWe consider the problem of deriving the asymptotic distribution of the three commonly used multivariate test statistics, namely likelihood ratio, Lawley–Hotelling and Bartlett–Nanda–Pillai statistics, for testing hypotheses on the various effects (main, nested or interaction) in multivariate mixed models. We derive the distributions of these statistics, both in the null as well as non-null cases, as the number of levels of one of the main effects (random or fixed) goes to infinity. The robustness of these statistics against departure from normality will be assessed.Essentially, in the asymptotic spirit of this paper, both the hypothesis and error degrees of freedom tend to infinity at a fixed rate. It is intuitively appealing to consider asymptotics of this type because, for example, in random or mixed effects models, the levels of the main random factors are assumed to be a random sample from a large population of levels.For the asymptotic results of this paper to hold, we do not require any distributional assumption on the errors. That means the results can be used in real-life applications where normality assumption is not tenable.As it happens, the asymptotic distributions of the three statistics are normal. The statistics have been found to be asymptotically null robust against the departure from normality in the balanced designs. The expressions for the asymptotic means and variances are fairly simple. That makes the results an attractive alternative to the standard asymptotic results. These statements are favorably supported by the numerical results

Selection for linear structure models with different variances

This paper is concerned with selection of some linear structure models with different variances. The models are based on the study of relationships between two fracture toughness testing methods of dental luting cemments. The measurements are made by using several kinds of materials. They are assumed to have different variances depending on the materials, but they have the same variance between two testing methods. For such data, we consider three types of structures between two methods: (1) proportionality, (2) linearity, and (3) no structure. We give Akaike information criterion, AIC, to evaluate these models. Then, we derive corrected AIC (CAIC) which is useful for small samples. By simulation experiments, we find that CAIC is more effective than AIC in the case of small samples. Our results are applied to a real data of dental luting cements.【査読有