"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.

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

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

Promoting Socio-Economic Equity in South Asia: Challenges and Prospects

For the past decade or so, questions related to equity have been front loaded on the global agenda of economic growth and development. Equity has an umbilical but extremely uneasy relationship with growth. It is almost meaningless to talk of equity if there is no growth, and growth, in the long run, may be seriously imperilled and jeopardised if it thwarts equity. The vision of “Growth with Equity”, so assiduously projected during the Sixties and the Seventies sadly stands shaken if not shattered. The illusion of “trickle-down” mythology has been exposed. Pope Francis, in a paper written in November 2013, described the “trickle-down theories” as “never being confirmed by facts”, adding that these theories express “a crude and naïve trust in the goodness of those wielding economic power and in the sacralised workings of the prevailing economic system.” A few months later, in April 2014, the Pontiff again twitted: “Inequality is the root of social evils.” Reading this tweet, my memory was thrown back to the early seventies when Indira Gandhi of India called poverty the greatest polluter of human society. DOI: 10.5281/zenodo.336690

Controllability of linear impulsive systems – an eigenvalue approach

summary:This article considers a class of finite-dimensional linear impulsive time-varying systems for which various sufficient and necessary algebraic criteria for complete controllability, including matrix rank conditions are established. The obtained controllability results are further synthesised for the time-invariant case, and under some special conditions on the system parameters, we obtain a Popov-Belevitch-Hautus (PBH)-type rank condition which employs eigenvalues of the system matrix for the investigation of their controllability. Numerical examples are provided that demonstrate--for the linear impulsive systems, null controllability need not imply their complete controllability, unlike for the non-impulsive linear systems