Moments of generalized quadratic forms and distributions of certain multivariate test statistics

Summary Moments and distributions of quadratic forms or quadratic expressions in normal variables are available in literature. Such quadratic expressions are shown to be equivalent to a linear function of independent central or noncentral chi-square variables. Some results on linear functions of generalized quadratic forms are also available in literature. Here we consider an arbitrary linear function of matrix-variate gamma variables. Moments of the determinant of such a linear function are evaluated when the matrix-variate gammas are independently distributed. By using these results, arbitrary non-null moments as well as the non-null distribution of the likelihood ratio criterion for testing the hypothesis of equality of covariance matrices in independent multivariate normal populations are derived. As a related result, the distribution of a linear function of independent matrix-variate gamma random variables, which includes linear functions of independent Wishart matrices, is also obtained. Some properties of generalized special functions of several matrix arguments are used in deriving these results.     

Estimation of noncentrality parameters

We propose some estimators of noncentrality parameters which improve upon usual unbiased estimators under quadratic loss. The distributions we consider are the noncentral chi-square and the noncentral F. However, we give more general results for the family of elliptically contoured distributions and propose a robust dominating estimator.     

On approximating distribution of the quadratic discriminant function

The quadratic discriminant function (QDF) with known parameters has been represented in terms of a weighted sum of independent noncentral chi-square variables. To approximate the density function of the QDF as m-dimensional exponential family, its moments in each order have been calculated. This is done using the recursive formula for the moments via the Stein's identity in the exponential family. We validate the performance of our method using simulation study and compare with other methods in the literature based on the real data. The finding results reveal better estimation of misclassification probabilities, and less computation time with our method.     

Bivariate distributions of some ratios of independent noncentral chi-square random variables

The bivariate distributions of three pairs of ratios of in¬dependent noncentral chi-square random variables are considered. These ratios arise in the problem of computing the joint power function of simultaneous F-tests in balanced ANOVA and ANCOVA. The distributions obtained are generalizations to the noncentral case of existing results in the literature. Of particular note is the bivariate noncentral F distribution, which generalizes a special case of Krishnaiah*s (1964,1965) bivariate central F distribution. Explicit formulae for the cdf's of these distribu¬tions are given, along with computational procedures     

Expectation of quadratic forms in normal and nonnormal variables with applications

We derive some new results on the expectation of quadratic forms in normal and nonnormal variables. Using a nonstochastic operator, we show that the expectation of the product of an arbitrary number of quadratic forms in noncentral normal variables follows a recurrence formula. This formula includes the existing result for central normal variables as a special case. For nonnormal variables, while the existing results are available only for quadratic forms of limited order (up to 3), we derive analytical results to a higher order 4. We use the nonnormal results to study the effects of nonnormality on the finite sample mean squared error of the OLS estimator in an AR(1) model and the QMLE in an MA(1) model.     

Evaluation of probabilities for the noncentral distributions and the difference of two T-variables with a desk calculator

It is demonstrated that integrals of the noncentral chi-square, noncentral F and noncentral T distributions can be evaluated on desk calculators. The same procedure can be used to compute probabilities for the distribution of the difference of two T-variables with equal degrees of freedom. The proposed method of computation can be used with any computer which yields probabilities for the chi-square and F distributions.     

The noncentral bivariate chisquared distribution and extensions

The distribution of certain correlated noncentral chisquared variates P, Q, is termed the noncentral bivariate chisquared distribution. Moment generating functions of the distributions of (P, Q), (P+Q) and other quadratic forms have been obtained. A relationship to the linear case of the noncentral Wishart distribution is indicated. Convolution properties and applications are presented.     

A simple approximation for the doubly noncentral t-distribution

A simple approximation for the doubly noncentral t-distribution, based upon the Fieller-Geary Theorem (1930) and approximate normality of the square root of the noncentral chi-square variable observed by Patnaik (1949), is developed, This approximation and an Edgeworth series expansion associated with it are evaluated. The simple approximation is seen to be reasonably accurate for most practical purposes.     

Modified Normal-based Approximation to the Percentiles of Linear Combination of Independent Random Variables with Applications

A modified normal-based approximation for calculating the percentiles of a linear combination of independent random variables is proposed. This approximation is applicable in situations where expectations and percentiles of the individual random variables can be readily obtained. The merits of the approximation are evaluated for the chi-square and beta distributions using Monte Carlo simulation. An approximation to the percentiles of the ratio of two independent random variables is also given. Solutions based on the approximations are given for some classical problems such as interval estimation of the normal coefficient of variation, survival probability, the difference between or the ratio of two binomial proportions, and for some other problems. Furthermore, approximation to the percentiles of a doubly noncentral F distribution is also given. For all the problems considered, the approximation provides simple satisfactory solutions. Two examples are given to show applications of the approximation.     

On the distribution of quadratic forms and applications in the analysis of repeated measurements

In this paper a unified approach is given to the distribution of scalar quadratic forms for dependent variables. Necessary and sufficient conditions are found for the sums of squares of the various hierarchical layers in ANOVA to be distributed as multiples of chi-square variables. Results concerning the usual univariate F-tests in ANOVA of repeated measurements are derived as a special case.     

Further remarks on estimating the parameter of a noncentral chi-square distribution

Perlman and Rasmussen (1975) have found estimators of the non-centrality parameter of a noncentral chi-square distribution which have lower mean square error than the maximum likelihood estimator. This paper studies some extensions of their estimators and some related problems.     

New representations of the noncentral chi-square density and cumulative

Representations of noncentral chi-square cumulative distribution function and probability density function are reviewed and new repre¬sentations are given. One representation of the cdf in terms of an integral is easily computed on any machine which has an accurate algorithm for computing the normal cdf.     

The Distribution of a Ratio of Quadratic Forms in Noncentral Normal Variables

Abstract

An expression for the exact cumulative distribution function of a ratio of quadratic forms in noncentral normal variable is derived in terms of infinite series of top order invariant polynomials.     

Power of analysis of covariance tests under conditional specification

The procedure-wise power functions of two strategies for balanced single-factor analysis of covariance in the presence of possibly unequal regression slopes are evaluated and illustrated. The strategies differ in the action to be taken following a re-^ jection by the preliminary test for equal slopes. The first strategy simply discards the covariate and respecifies the model as the one-way ANOVA model for testing factor effects. The second leaves the unequal slopes covariance model intact, but respecifies the factor effects hypothesis to address the factor level means adjusted to the sample average of the covariate. One additional strategy, that of testing factor effects only if the preliminary slopes test does not reject, is included for comparison purposes. Computation of the power functions requires extensive use of the results obtained in Hawkins and Han (1986) concerning the bivariate distributions of certain ratios of independent noncentral chi-square random variables.     

ROBUST TESTS BASED ON THE SAMPLE CHARACTERISTIC FUNCTION

A new method is described for robust analysis of variance in the balanced fixed effects case. The method uses the empirical characteristic function of the treatment samples, and has an interpretation in terms of S-estimators. The test statistic, under the null hypothesis, asymptotically follows a central chi-square distribution, and under contiguous alternatives a noncentral chi-square distribution. A Monte Carlo study suggests that, for finite samples, this is reasonably well approximated by the usual F distribution used in analysis of variance. The test statistic has a bounded influence function. The new procedure competes well with Huber's and a Wald-type procedure except in very heavy-tailed cases.     

SOME SIMPLE APPROXIMATIONS FOR THE DOUBLY NONCENTRAL z DISTRIBUTION

We give two simple approximations for evaluating the cumulative probabilities of the doubly noncentral z distribution. These can easily be used for evaluating the cumulative probabilities of the doubly noncentral F distribution as well. We compare our results with those obtained by Tiku (1965) using series expansion. An industrial situation where a quality characteristic of interest follows the doubly noncentral z distribution is also cited. However, in this case the exact probabilities could be calculated using results on the ratio of two normal variables.     

Multivariate hypothesis testing using generalized and {2}-inverses – with applications

The use of generalized inverses in Wald's-type quadratic forms of test statistics having singular normal limiting distributions does not guarantee to obtain chi-square limiting distributions. In this article, the use of {2} -inverses for that problem is investigated. Alternatively, Imhof-based test statistics can also be defined, which converge in distribution to weighted sum of chi-square variables. The asymptotic distributions of these test statistics under the null and alternative hypotheses are discussed. Under fixed and local alternatives, the asymptotic powers are compared theoretically. Simulation studies are also performed to compare the exact powers of the test statistics in finite samples. A data analysis on the temperature and precipitation variability in the European Alps illustrates the proposed methods.     

On approximating the central and noncentral multivariate gamma distributions

In this paper a finite series approximation involving Laguerre polynomials is derived for central and noncentral multivariate gamma distributions. It is shown that if one approximates the density of any k nonnegative continuous random variables by a finite series of Laguerre polynomials up to the (n₁, …, n_k)th degree, then all the mixed moments up to the order (n₁, …, n_k) of the approximated distribution equal to the mixed moments up to the same order of the random variables. Some numerical results are given for the bivariate central and noncentral multivariate gamma distributions to indicate the usefulness of the approximations.     

Approximations for the doubly noncentral-F distribution

The approximate normality of the cube root of the noncentral chi-square observed by Aty (1954) and an Edgeworth-series expansion are used to derive an approximation for the doubly noncentral-F distribution. Another approximation in terms of a noncentral-F distribution is also proposed. Both these approximations are seen to compare favorably with some earlier approximations due to Das Gupta (1968) and Tiku (1972). The problem of approximating the cumulants of the doubly noncentral-F variable, which is pivotal in Tiku’s approximation, is examined and use of a noncentral-F distribution is seen to provide a good solution for it. A FORTRAN routine for the Edgeworth-series approximation is given.     

The probability to select the correct model using likelihood-ratio based criteria in choosing between two nested models of which the more extended one is true

The probability to select the correct model is calculated for likelihood-ratio-based criteria to compare two nested models. If the more extended of the two models is true, the difference between twice the maximised log-likelihoods is approximately noncentral chi-square distributed with d.f. the difference in the number of parameters. The noncentrality parameter of this noncentral chi-square distribution can be approximated by twice the minimum Kullback–Leibler divergence (MKLD) of the best-fitting simple model to the true version of the extended model.The MKLD, and therefore the probability to select the correct model increases approximately proportionally to the number of observations if all observations are performed under the same conditions. If a new set of observations can only be performed under different conditions, the model parameters may depend on the conditions, and therefore have to be estimated for each set of observations separately. An increase in observations will then go together with an increase in the number of model parameters. In this case, the power of the likelihood-ratio test will increase with an increasing number of observations. However, the probability to choose the correct model with the AIC will only increase if for each set of observations the MKLD is more than 0.5. If the MKLD is less than 0.5, that probability will decrease. The probability to choose the correct model with the BIC will always decrease, sometimes after an initial increase for a small number of observation sets. The results are illustrated by a simulation study with a set of five nested nonlinear models for binary data.