Noncentrality Matrix Research Articles

Approximation for the distribution function of non-central complex Wilks statistic

In this article, we use the saddlepoint approximation method to compute the cumulative distribution function of non-central complex Wilks statistic. This approximation does not require conditions on the dimension, degrees of freedom, and non-centrality matrix. We also compare our results with known approximations to show the accuracy of our approximation. For applications, we compute the power functions of the Wilks statistic in the complex multivariate analysis of variance and the probability of detection of the Wilks detector in signal detection in noise model.

Estimation of a covariance matrix in multivariate skew-normal distribution

This article addresses the problem of estimating a covariance matrix in a multivariate skew-normal distribution relative to two different losses. The estimation problem can be reduced to that of a scale matrix of a noncentral Wishart distribution. The noncentrality parameter matrix, which is a nuisance parameter, brings about non optimality of the best triangular invariant estimators which are minimax under normality. Some improving techniques under normality are proven to remain robust under the multivariate skew-normal distribution.

The Generalized Conditional Autoregressive Wishart Model for Multivariate Realized Volatility

It is well known that in finance variances and covariances of asset returns move together over time. Recently, much interest has been aroused by an approach involving the use of the realized covariance (RCOV) matrix constructed from high-frequency returns as the ex-post realization of the covariance matrix of low-frequency returns. For the analysis of dynamics of RCOV matrices, we propose the generalized conditional autoregressive Wishart (GCAW) model. Both the noncentrality matrix and scale matrix of the Wishart distribution are driven by the lagged values of RCOV matrices, and represent two different sources of dynamics, respectively. The GCAW is a generalization of the existing models, and accounts for symmetry and positive definiteness of RCOV matrices without imposing any parametric restriction. Some important properties such as conditional moments, unconditional moments, and stationarity are discussed. Empirical examples including sequences of daily RCOV matrices from the New York Stock Exchange illustrate that our model outperforms the existing models in terms of model fitting and forecasting.

NON-CENTRAL MULTIVARIATE CHI-SQUARE AND GAMMA DISTRIBUTIONS

A (p-1)-variate integral representation is given for the cumulative distribution function of the general p-variate non-central gamma distribution with a non-centrality matrix of any admissible rank. The real part of products of well known analytical functions is integrated over arguments from (-pi,pi). To facilitate the computation, these formulas are given more detailed for p=2 and p=3. These (p-1)-variate integrals are also derived for the diagonal of a non-central complex Wishart Matrix. Furthermore, some alternative formulas are given for the cases with an associated one-factorial (pxp)-correlation matrix R, i.e. R differs from a suitable diagonal matrix only by a matrix of rank 1, which holds in particular for all (3x3)-R with no vanishing correlation.

High-dimensional consistency of rank estimation criteria in multivariate linear model

This paper is concerned with consistency properties of rank estimation criteria in a multivariate linear model, based on the model selection criteria AIC , BIC and C p . The consistency properties of these criteria are studied under a high-dimensional framework with two different assumptions on the noncentrality matrix such that the number of response variables and the sample size tend to infinity. In general, it is known that under a large-sample asymptotic framework, the criteria based on AIC and C p are not consistent, but the criterion based on BIC is consistent. However, we note that there are cases that the criteria based on AIC and C p are consistent, but the criterion based on BIC is not consistent. Such consistency properties are also shown for the generalized criteria with a tuning parameter. Further, the modified criteria with a ridge-type estimator are also examined. Through a Monte Carlo simulation experiment, our results are checked numerically, and the estimation criteria are compared.

Chi-square mixture representations for the distribution of the scalar Schur complement in a noncentral Wishart matrix

We show that the distribution of the scalar Schur complement in a noncentral Wishart matrix is a mixture of central chi-square distributions with different degrees of freedom. For the case of a rank-1 noncentrality matrix, we reveal that the mixture weights arise from a noncentral beta mixture of Poisson distributions.

Consistency properties of AIC, BIC, Cp and their modifications in the growth curve model under a large-(q,n) framework

This paper is concerned with consistency properties of some criteria for selecting row vectors of a k×p design matrix within individuals in the growth curve model, based on a sample of size n. Recently Enomoto, Sakurai and Fujikoshi (2013) showed that AIC and its modification have a consistency property for selecting hierarchical models of the row vectors under a condition on the order of the noncentrality matrix, assuming a large-(q,n) asymptotic framework such that q/n→d∈[0,1). We extend the result to a family of log-likelihood-based information criteria including AIC and BIC, and Cp. Further, their consistency properties are also obtained under a new condition on the order of the noncentrality matrix. Our results are checked numerically by conducting a Mote Carlo simulation.

Consistency of high-dimensional AIC-type and [formula omitted]-type criteria in multivariate linear regression

The AIC, the multivariate Cp and their modifications have been proposed for multivariate linear regression models under a large-sample framework when the sample size n is large, but the dimension p of the response variables is fixed. In this paper, first we propose a high-dimensional AIC (denoted by HAIC) which is an asymptotic unbiased estimator of the risk function defined by the expected log-predictive likelihood or equivalently the Kullback–Leibler information under a high-dimensional framework p/n→c∈[0,1). It is noted that our new criterion provides better approximations to the risk function in a wide range of p and n. Recently Yanagihara et al. (2012) [17] noted that AIC has a consistency property under Ω=O(np) when p/n→c∈[0,1), where Ω is a noncentrality matrix. In this paper we show that several criteria including HAIC and Cp have also a consistency property under Ω=O(n) as well as Ω=O(np) when p/n→c∈[0,1). Our results are checked numerically by conducting a Monte Carlo simulation.

Consistency of AIC and its modification in the growth curve model under a large-(q,n) framework

The AIC and its modifications have been proposed for selecting the degree in a polynomial growth curve model under a large-sample framework and a high-dimensional framework by Satoh, Kobayashi and Fujikoshi [9] and Fujikoshi, Enomoto and Sakurai [4], respectively. They note that the AIC and its modifications have no consistency property. In this paper we consider asymptotic properties of the AIC and its modification when the number q of groups or explanatory variables and the sample size n are large. First we show that the AIC has a consistency property under a large-(q,n) framework such that q/n→d∈[0,1), under a condition on the noncentrality matrix, but the dimension p is fixed. Next we propose a modification of the AIC (denoted by MAIC) which is an asymptotic unbiased estimator of the risk under the asymptotic framework. It is shown that MAIC has a consistency property under a condition on the noncentrality matrix. Our results are checked numerically by conducting a Mote Carlo simulation.

Estimation of the eigenvalues of noncentrality parameter matrix in noncentral Wishart distribution

We consider the problem of estimating the eigenvalues of noncentrality parameter matrix in noncentral Wishart distribution when the scale parameter is known. A decision theoretic approach is taken with squared error as the loss function. We propose two new estimators and show their superior performance to an usual estimator theoretically and numerically.

Estimation of the eigenvalues of noncentrality parameter in matrix variate noncentral beta distribution

We consider the problem of estimating the eigenvalues of noncentrality parameter matrix in a matrix variate noncentral beta distribution, also known as multivariate noncentral F distribution. A decision theoretic approach is taken with square error as the loss function. We propose two types of new estimators and show their superior performance theoretically as well as numerically.

An identity for the noncentral wishart distribution with application

This paper generalizes an identity for the Wishart distribution (derived independently by C. Stein and L. Haff) to the noncentral Wishart distribution. As an application of this noncentral Wishart identity, we consider the problem of estimating the noncentrality matrix of a noncentral Wishart distribution. This noncentral Wishart identity is used to develop a class of orthogonally invariant estimators which dominate the usual unbiased estimator.

A comparison of three invariant tests of additivity in two-way classifications with no replications

Tusell (1990) proposed an innovative test of additivity in non-replicated two-way classifications. Tusell's procedure consists of employing the well-known likelihood ratio (LR) test of sphericity as a test of additivity. Ordinarily, additivity is assumed when testing sphericity. Tusell deduced that if sphericity is assumed, then application of the LR criterion results in a test of additivity. This article gives the limiting distribution of the LR sphericity criterion under nonadditivity. Simulation studies that compare Tusell's test with Johnson and Graybill's (1972) LR test against a rank-1 alternative and with Boik's (1990) locally best invariant test of addivity are reported. The LR sphericity test is the most powerful test (among the three) when the noncentrality matrix has multiple non-zero eigenvalues which are comparable in magnitude. For most other conditions, the locally best invariant test is the most powerful test. In practice, prior information about the structure of the noncentrality matrix is generally not available. For these cases, the locally best invariant test is recommended.

Canonical variate analysis with unequal covariance matrices: Generalizations of the usual solution

Canonical variate analysis is extended for use when the covariance matrices are not equal. Linear combinations of variates are derived by generalizing either a weighted between-groups approach or the likelihood-ratio test and the associated noncentrality matrix. The usual solution and the two generalizations are compared via generated data for a few typical configurations of means in a situation in which the covariance matrices are in fact equal. The MSE of the canonical variate coefficients and group means for the generalizations are approximately three times those for the usual solution, due to corresponding changes in the variances. Two examples are discussed.

Optimal bounds for the distributions of some test criteria for tests of dimensionality

Optimal upper bounds are obtained for the distributions of two functions of the m r smallest latent roots of HE1, where E and H have Wishart distributions with identical covariance matrices; E has a central distribution while H has a noncentral distribution with unknown noncentrality matrix A of rank r. These bounds are then used to investigate the chi-squared approximation for some test criteria used in tests of dimensionality.

DIRECTION AND COLLINEARITY FACTORS OF WILK'S A ASSOCIATED WITH THE GROWTH CURVE MODEL

SummaryKhatri (1966) has derived a Wilks' s A test of a general linear hypothesis in the growth curve model. In this paper we give the direction and collinearity factors and their null distributions when the hypothesis is not true but the noncentrality matrix is of rank one. Interpretation of these tests and their usefulness in discrimination in growth curve models are discussed.

Influence functions for certain parameters in multivariate analysis

The influence function introduced by Hampe1 (1968, 1973, 1974) is a tool that can be used for outlier detection. Campbell (1978) has obtained influence function for Mahalanobis’s distance between two populations which can be used for detecting outliers in discrim-inant analysis. In this paper influence functions for a variety of parametric functions in multivariate analysis are obtained. Influence functions for the generalized variance, the matrix of regression coefficients, the noncentrality matrix Σ-1 δ in multivariate analysis of variance and its eigen values, the matrix L, which is a generalization of 1-R2 , canonical correlations, principal components and parameters that correspond to Pillai’s statistic (1955), Hotelling’s (1951) generalized To2 and Wilk’s Λ (1932), which can be used for outlier detection in multivariate analysis, are obtained. Delvin, Ginanadesikan and Kettenring (1975) have obtained influence function for the population correlation co-efficient in the bivariate case. It is shown in this paper that influence functions for parameters corresponding to r2, R2, and Mahalanobis D2 can be obtained as particular cases.

Asymptotic Representations of the Densities of Canonical Correlations and Latent Roots in MANOVA When the Population Parameters Have Arbitrary Multiplicity

Asymptotic representations of the joint densities of the canonical correlation coefficients, calculated from a sample from a multivariate normal population, and of the latent roots of $B(B + W)^{-1},$ where $B$ is $W_p(n_1, \Sigma, \Omega)$ and $W$ is $W_p(n_2, \Sigma),$ are obtained by deriving asymptotic representations of the hypergeometric functions in the joint densities. The results hold in the first case for large sample size and arbitrary values of the population canonical correlations and in the second case for large $n_2$ and $\Omega = n_2\Theta,$ where the latent roots of the noncentrality matrix $\Omega$ are arbitrary.

A Canonical Representation for the Noncentral Wishart Distribution Useful for Simulation

Abstract This article provides a simple distributional representation for the noncentral Wishart distribution most useful when the noncentrality matrix is of less than full rank. It is shown that the representation leads to a method for simulating a noncentral Wishart matrix that has advantages over a procedure currently used.

A Canonical Representation for the Noncentral Wishart Distribution Useful for Simulation

Abstract This article provides a simple distributional representation for the noncentral Wishart distribution most useful when the noncentrality matrix is of less than full rank. It is shown that the representation leads to a method for simulating a noncentral Wishart matrix that has advantages over a procedure currently used.