"Tests for Multivariate Analysis of Variance in High Dimension Under Non-Normality"

In this article, we consider the problem of testing the equality of mean vectors of dimension ρ of several groups with a common unknown non-singular covariance matrix Σ, based on N independent observation vectors where N may be less than the dimension ρ. This problem, known in the literature as the Multivariate Analysis of variance (MANOVA) in high-dimension has recently been considered in the statistical literature by Srivastava and Fujikoshi[7], Srivastava [5] and Schott[3]. All these tests are not invariant under the change of units of measurements. On the lines of Srivastava and Du[8] and Srivastava[6], we propose a test that has the above invariance property. The null and the non-null distributions are derived under the assumption that ( N, ρ) → ∞ and N may be less than ρ and the observation vectors follow a general non-normal model.

"Selection of Variables in Multivariate Regression Models for Large Dimensions"

The Akaike information criterion, AIC, and Mallows' Cp statistic have been proposed for selecting a smaller number of regressor variables in the multivariate regression models with fully unknown covariance matrix. All these criteria are, however, based on the implicit assumption that the sample size is substantially larger than the dimension of the covariance matrix. To obtain a stable estimator of the covariance matrix, it is required that the dimension of the covariance matrix be much smaller than the sample size. When the dimension is close to the sample size, it is necessary to use ridge type of estimators for the covariance matrix. In this paper, we use a ridge type of estimators for the covariance matrix and obtain the modified AIC and modified Cp statistic under the asymptotic theory that both the sample size and the dimension go to infinity. It is numerically shown that these modified procedures perform very well in the sense of selecting the true model in large dimensional cases.

On testing the equality of mean vectors in high dimension

In this article, we review various tests that have been proposed in the literature for testing the equality of several mean vectors. In particular, it includes testing the equality of two mean vectors, the so-called two-sample problem as well as that of testing the equality of several mean vectors, the so-called multivariate analysis of variance or MANOVA problem. The total sample size, however, may be less than the dimension of the mean vectors, and so usual tests cannot be used. Powers of these tests are compared using simulation

Minimum distance classification rules for high dimensional data

AbstractIn this article, the problem of classifying a new observation vector into one of the two known groups Πi,i=1,2, distributed as multivariate normal with common covariance matrix is considered. The total number of observation vectors from the two groups is, however, less than the dimension of the observation vectors. A sample-squared distance between the two groups, using Moore–Penrose inverse, is introduced. A classification rule based on the minimum distance is proposed to classify an observation vector into two or several groups. An expression for the error of misclassification when there are only two groups is derived for large p and n=O(pδ),0<δ<1

Asia-Pacific Regional Integration Index: Construction, Interpretation, and Comparison

We develop an index to measure the degree of regional integration in Asia and the Pacific (48 economies in six subregions). The index comprises 26 indicators in six dimensions of regional integration, i.e., trade and investment, money and finance, regional value chains, infrastructure and connectivity, free movement of people, and institutional and social integration. We use principal component analysis to apportion a weight to each dimension and indicator to construct composite indexes. The resulting indexes help assess the state of regional integration on diverse socioeconomic dimensions, evaluate progress against goals, identify strengths and weaknesses, and track progress. Cross-country, cross-regional comparisons also allow policy makers to prioritize areas for further efforts

Some tests criteria for the covariance matrix with fewer observations than the dimension

We consider testing certain hypotheses concerning the covariance matrix&nbsp;Σ&nbsp;when the number of observations&nbsp;N=n+1&nbsp;on the&nbsp;p-dimensional random vector&nbsp;x, distributed as normal, is less than&nbsp;p,&nbsp;n&lt;p, and&nbsp;n/p&nbsp;goes to zero. Specifically, we consider testing&nbsp;Σ=σ2Ip,&nbsp; Σ=Ip,&nbsp; Σ=Λ, a diagonal matrix, and&nbsp;Σ=σ2[(1−ρ)Ip+ρ1p1′p], an intraclass correlation structure, where&nbsp;1′p=(1,1,…,1), is a&nbsp;p-row vector of ones, and&nbsp;Ip&nbsp;is the&nbsp;p×p&nbsp;identity matrix. The first two tests are the adapted versions of the likelihood ratio tests when&nbsp;n&gt;p,&nbsp;p-fixed, and&nbsp;p/n&nbsp;goes to zero, to the case when&nbsp;n&lt;p,&nbsp;n-fixed, and&nbsp;n/p&nbsp;goes to zero. The third test is the normalized version of Fisher’s&nbsp;z-transformation which is shown to be asymptotically normally distributed as&nbsp;n&nbsp;and&nbsp;p&nbsp;go to infinity (irrespective of the manner). A test for the fourth hypothesis is constructed using the spherecity test for a&nbsp;(p−1)-dimensional vector but this test can only reject the hypothesis, that is, if the hypothesis is not rejected, it may not imply that the hypothesis is true. The first three tests are compared with some recently proposed tests

Minimum distance classification rules for high dimensional data

In this article, the problem of classifying a new observation vector into one of the two known groups [Pi]i,i=1,2, distributed as multivariate normal with common covariance matrix is considered. The total number of observation vectors from the two groups is, however, less than the dimension of the observation vectors. A sample-squared distance between the two groups, using Moore-Penrose inverse, is introduced. A classification rule based on the minimum distance is proposed to classify an observation vector into two or several groups. An expression for the error of misclassification when there are only two groups is derived for large p and n=O(p[delta]),0Fisher discriminant rule Misclassification error Moore-Penrose inverse Multivariate normal Singular Wishart