Muni S. Srivastava

Multivariate Theory for Analyzing High Dimensional Data

In this article, we develop a multivariate theory for analyzing multivariate datasets that have fewer observations than dimensions. More specifically, we consider the problem of testing the hypothesis that the mean vector µ of a p-dimensional random vector x is a zero vector where N, the number of independent observations on x, is less than the dimension p. It is assumed that x is normally distributed with mean vector µ and unknown nonsingular covariance matrix Σ. We propose the

A measure of skewness and kurtosis and a graphical method for assessing multivariate normality

Using principal components, a measure of skewness and kurtosis is developed for multivariate populations. The sample analogues of these measures are proposed as tests of multivariate normality. Also, a graphical method is presented for assessing multivariate normality.

Likelihood ratio tests for a change in the multivariate normal mean

Abstract A sequence of independent multivariate normal vectors with equal but possibly unknown variance matrices are hypothesized to have equal mean vectors, and we wish to test that the mean vectors have changed after an unknown point in the sequence. The likelihood ratio test is based on the maximum Hotelling T 2 for the sequences before and after the change point. The main result is a conservative approximation for its null distribution based on an improved Bonferroni inequality. If the change is judged significant, then further changes are estimated by splitting the two subsequences formed by the first change point. The methods can also be used to test for a change in row probabilities of a contingency table, allowing for extramultinomial variation. The results are used to find changes in a set of geological data previously analyzed by Chernoff (1973) by the “faces” method and to find changes in the frequencies of pronouns in the plays of Shakespeare.

Models with a Kronecker Product Covariance Structure: Estimation and Testing

In this article we consider a pq-dimensional random vector x distributed normally with mean vector θ and covariance matrix Λ assumed to be positive definite. On the basis of N independent observations on the random vector x, we want to estimate parameters and test the hypothesis H: Λ = Ψ ⊗ Σ, where Ψ = (ψij): q × q, ψqq = 1, and Σ = (σij): p × p, and Λ = (ψijΣ), the Kronecker product of Ψ and Σ. That is instead of 1/2pq(pq + 1) parameters, it has only 1/2p(p + 1) + 1/2q(q + 1) − 1 parameters. A test based on the likelihood ratio is given to check if this model holds. And, when this model holds, we test the hypothesis that Ψ is a matrix with intraclass correlation structure. The maximum likelihood estimators (MLE) are obtained under the hypothesis as well as under the alternatives. Using these estimators the likelihood ratio tests (LRT) are obtained. One of the main objects of the paper is to show that the likelihood equations provide unique estimators.

On assessing multivariate normality based on shapiro-wilk W statistic

Shapiro and Wilks (1965) W statistic has been found to be the best omnibus test for detecting departures from univariate normality. Royston (1983) extends the application of W to testing multivariate normality but the procedure involves a certain approximation which needs to be justified. The procedures proposed in the present paper do not need such an approximation. The asymptotic null distributions are also given. Finally, a numerical example is used to illustrate the procedures.

Some One-Sided Tests for Change in Level

We consider tests based on one observation on each of N ≥ 2 random variables X l, …, XN to decide if the means μ of the xi s are all equal against the one-sided alternative that a shift has occurred at some unknown point γ, (i.e. μ1, = μ2 = … = μ r < μ r+1 = … = μ N ). The x i s are considered to be normally distributed with a common unknown variance. Bayesian tests as well as a test based on the maximum likelihood estimate of γ are considered and their powers are compared by Monte Carlo methods. The exact distribution of a Bayesian test statistic is derived. A simple application using traffic accident data is presented.

Multivariate data with missing observations

In this paper, multivariate data with missing observations, where missing values could be by chance or by design, are considered for various models including the growth curve model. The likelihood equations are derived and the consistency of the estimates established. The likelihood ratio tests are explicity derived.

Stein estimation under elliptical distributions

In a subclass of elliptical distributions, Stein estimators are robust in estimating the mean vector and the regression parameters in a linear regression model. Unbiased estimates of bias and risk are also given for the regression model.

Profile analysis of several groups

In this paper, profile analysis of several groups is considered. The maximum likelihood estimate, along with its covariance matrix, is given for the “level differences” between the grops. Likelihood ratio test procedures, with their distributions, are given for the three hypothesis known in the literature as “parallelism”, “level hypothesis” and “ no condition variation”. In the literature, even for the two groups case, some procedures are not likelihood procedures.

A second order approximation to taguchps online control procedure

In this paper, a second order approximation of the average loss function for on-line control procedure is developed under a Brownian motion model. The approximation depends on whether the ratio of the adjustment cost to the inspection cost is small or large. Both cases are dealt with. Simple formulae for the optimal parameters are given for both cases. These approximations are shown to be better than the ones given by Taguchi et al.(1989) and Adams and WoodaU(1989).