P.R. Krishnaiah

On detection of the number of signals in presence of white noise

In this paper, the authors propose procedures for detection of the number of signals in presence of Gaussian white noise under an additive model. This problem is related to the problem of finding the multiplicity of the smallest eigenvalue of the covariance matrix of the observation vector. The methods used in this paper fall within the framework of the model selection procedures using information theoretic criteria. The strong consistency of the estimates of the number of signals, under different situations, is established. Extensions of the results are also discussed when the noise is not necessarily Gaussian. Also, certain information-theoretic criteria are investigated for determination of the multiplicities of various eigenvalues.

On the limit of the largest eigenvalue of the large dimensional sample covariance matrix

SummaryIn this paper the authors show that the largest eigenvalue of the sample covariance matrix tends to a limit under certain conditions when both the number of variables and the sample size tend to infinity. The above result is proved under the mild restriction that the fourth moment of the elements of the sample sums of squares and cross products (SP) matrix exist.

On detection of the number of signals when the noise covariance matrix is arbitrary

In this paper, the authors proposed model selection methods for determination of the number of signals in presence of noise with arbitrary covariance matrix. This problem is related to finding the multiplicity of the smallest eigenvalue of [Sigma]2[Sigma]1-1, where [Sigma]2 = [Gamma] + [lambda][Sigma]1, [Sigma]1 and [Sigma]2 are covariance matrices, [lambda] is a scalar, and [Gamma] is non-negative definite matrix and is not of full rank. Also, the authors proposed methods for determination of the multiplicities of various eigenvalues of [Sigma]2[Sigma]1-1. The methods used in these procedures are based upon certain information theoretic criteria. The strong consistency of these criteria is established in this paper.

On some nonparametric methods for detection of the number of signals

A simpler, more elegant, and more flexible method for estimating the number of signals is proposed. Strong consistency is proved. Bounds for error probabilities are given. This method depends on choosing a function with a parameter r. The previous methods correspond to the case r = 1. It seems the larger the r, the faster the convergence. We can choose r as any positive integer, even ∞. Applications are discussed, including the coherent cases.

Complex elliptically symmetric distributions

In this paper, the authors introduce a class of distributions known as complex elliptically symmetric distributions. The complex multivariate normal and complex multivariate t distributions are members of this class. Various properties of the complex elliptically symmetric distributions are studied. Finally, the robustness of certain test procedures are discussed when the assumption of complex multivariate normality is violated but the underlying distribution still belongs to the class of elliptically symmetric distributions.

On rates of convergence of efficient detection criteria in signal processing with white noise

L.C. Zhao et al. (1986) proposed certain information-theoretic criteria for detection of the number of signals under an additive model with white noise when the noise variance is known or unknown. It was shown that these criteria are strongly consistent even when the underlying distribution is not necessarily Gaussian. Upper bounds on the probabilities of error detection are obtained here. >

Remarks on certain criteria for detection of number of signals

In this paper, we derive the asymptotic distribution of the logarithm of the likelihood ratio statistic for testing the hypothesis that the number of signals is equal to q against the alternative that it is equal to k (specified) for a special case. This distribution is not chi-square. The above statistic also rises in studying consistency property of the MDL criterion and AIC for detection of the number of signals.

Asymptotic distributions of functions of the eigenvalues of some random matrices for nonnormal populations

The authors investigated the asymptotic joint distributions of certain functions of the eigenvalues of the sample covariance matrix, correlation matrix, and canonical correlation matrix in nonnull situations when the population eigenvalues have multiplicities. These results are derived without assuming that the underlying distribution is multivariate normal. In obtaining these expressions, Edgeworth type expansions were used.

16 Likelihood ratio tests for mean vectors and covariance matrices

Publisher Summary This chapter describes various likelihood ratio tests on mean vectors and covariance matrices and discusses computations of the critical values associated with these tests. Likelihood ratio tests play an important role in testing various hypotheses under the univariate analysis of variance (ANOVA) and multivariate analysis of variance (MANOVA) models. The chapter discusses the likelihood ratio test for testing for the equality of mean vectors of several multivariate normal populations, the test specifying the mean vector, the distribution of the determinant of the multivariate beta matrix, likelihood ratio test for multiple independence of several sets of variables, the likelihood ratio tests for sphericity and for the multiple homogeneity of the covariance matrices. It also provides likelihood ratio procedure specifying the mean vector and covariance matrix simultaneously, likelihood ratio test for the equality of the mean vectors and the equality of covariance matrices simultaneously, likelihood ratio tests for certain linear structures on the covariance matrices, and the applications of the tests on linear structures in the area of the components of variance.

On the asymptotic joint distributions of certain functions of the eigenvalues of four random matrices

In this paper, the authors obtained asymptotic expressions for the joint distributions of certain functions of the eigenvalues of the Wishart matrix, correlation matrix, MANOVA matrix and canonical correlation matrix when the population roots have multiplicity.