Two testing problems relating the real and complex multivariate normal distributions

The first problem considered is that of testing for the reality of the covariance matrix of a p-dimensional complex normal distribution, while the second is that of testing that a 2p-dimensional real normal distribution has a p-dimensional complex structure. Both problems are reduced by invariance to their maximal invariant statistics, and the null and non-null distributions of these are obtained. Complete classes of unbiased, invariant tests are described for both problems, the locally most powerful invariant tests are obtained, and the admissibility of the likelihood ratio tests is established.     

On q-orthogonal polynomials over a collection of complex origin intervals related to little q-Jacobi polynomials

We construct the sequence of orthogonal polynomials with respect to an inner product which is defined by q-integrals over a collection of intervals in the complex plane. We prove that they are connected with little q-Jacobi polynomials. For such polynomials we discuss a few representations, a recurrence relation, a difference equation, a Rodrigues-type formula and a generating function. 2000 Mathematics Subject Classification Primary—33D45, 42C05     

The expected values of invariant polynomials with matrix argument of elliptical distributions

ThisresearchissupportedbytheChineseAcademyofSciences.1.IntroductionTherearemailypublicatiollsonmatrixvariatedistributions(ormatrixdistributionforsimplicity),inparticular,abollttheirexpectedvaluesofzonalpolynomialsoftheirquadraticforms(of.[12],[151,[171and…     

Nonnull distributions of some statistics associated with testing for the equality of two covariance matrices

The nonnull distribution of some statistics, used for testing Σ₁ = Σ₂ are obtained as mixtures of incomplete beta functions as well as mixtures of incomplete gamma functions. The introduction of the convergence factors and certain recurrence relations are useful in the computation of the power of the tests as well as computation of exact percentage points for tests of significance.     

A Determinant Representation for the Distribution of Quadratic Forms in Complex Normal Vectors

Let the column vectors of X:: M×N, M<N, be distributed as independent complex normal vectors with the same covariance matrix Σ. Then the usual quadratic form in the complex normal vectors is denoted by Z=XLX^H where L: N×N is a positive definite hermitian matrix. This paper deals with a representation for the density function of Z in terms of a ratio of determinants. This representation also yields a compact form for the distribution of the generalized variance |Z|.     

On linear functions of certain noncentral versions of independent gamma variables

A general Laplace transform and its inverse and their generalizations are considered in this article. This inverse contains the density functions of quadratic expressions in nonsingular as well as singular normal variables and a non-central version of linear functions of gamma variables, among others. Various representations of the inverse in power series, in gamma series, in Laguerre polynomials, in hypergeometric functions and in zonal polynomials are also discussed.     

Asymptotic distributions of the likelihood ratio test statistics for covariance structures of the complex multivariate normal distributions

In this paper, the authors derived asymptotic expressions for the null distributions of the likelihood ratio test statistics for multiple independence and multiple homogeneity of the covariance matrices when the underlying distributions are complex multivariate normal. Also, asymptotic expressions are obtained in the non-null cases for the likelihood ratio test statistics for independence of two sets of variables and the equality of two covariance matrices. The expressions obtained in this paper are in terms of beta series. In the null cases, the accuracy of the first terms alone is sufficient for many practical purposes.     

On confidence sequences for the mean vector of a multivariate normal distribution

Let X₁, X₂,… be idd random vectors with a multivariate normal distribution N(μ, Σ). A sequence of subsets {R_n(a₁, a₂,…, a_n), n ≥ m} of the space of μ is said to be a (1 − α)-level sequence of confidence sets for μ if P(μ R_n(X₁, X₂,…, X_n) for every n ≥ m) ≥ 1 − α. In this note we use the ideas of Robbins Ann. Math. Statist. 41 (1970) to construct confidence sequences for the mean vector μ when Σ is either known or unknown. The constructed sequence R_n(X₁, X₂, …, X_n) depends on Mahalanobis' or Hotelling's according as Σ is known or unknown. Confidence sequences for the vector-valued parameter in the general linear model are also given.     

On discrete distributions of orderk

Summary The class of discrete distributions of orderk is defined as the class of the generalized discrete distributions with generalizer a discrete distribution truncated at zero and from the right away fromk+1. The probability function and factorial moments of these distributions are expressed in terms of the (right) truncated Bell (partition) polynomials and several special cases are briefly examined. Finally a Poisson process of orderk, leading in particular to the Poisson distribution of orderk, is discussed.     

On a generalized mixture of standard normal and skew normal distributions

Here we propose a new class of distributions as a generalized mixture of standard normal and skew normal distributions (GMNSND) and study some of its properties by deriving its characteristic function, mean, variance, coefficient of skewness etc. Further, certain reliability aspects of GMNSND are studied and a location scale extension of GMNSND is considered. The estimation of the parameters of this extended GMNSND by the method of maximum likelihood is discussed.     

On a characterization of the normal distribution by means of identically distributed linear forms

Let X₁, X₂,…, be independent, identically distributed random variables. Suppose that the linear forms L₁ = Σ_j=1^∞a_jX_j and L₂ = Σ_j=1^∞b_jX_j exist with probability one and are identically distributed; necessary and sufficient conditions assuring that X₁ is normally distributed are presented. The result is an extension of a theorem of Linnik (Ukrainian Math. J.5 (1953), 207–243, 247–290) concerning the case that the linear forms L₁ and L₂ have a finite number of nonvanishing components. This proof only makes use of elementary properties of characteristic functions and of meromorphic functions.     

On a Ramanujan equation connected with the median of the gamma distribution

In this paper, we consider the sequence solving the Ramanujan equation

The three main achievements are the following. We introduce a continuous-time extension of and show its close connections with the medians of the distributions and the Charlier polynomials. We give upper and lower bounds for both and , in particular for , which are sharper than other known estimates. Finally, we show (and at the same time complete) two conjectures by Chen and Rubin referring to the sequence of medians .

     

On Schur's Q-functions and the Primitive Idempotents of a Commutative Hecke Algebra

Let B _n denote the centralizer of a fixed-point free involution in the symmetric group S _2n . Each of the four one-dimensional representations of B _n induces a multiplicity-free representation of S _2n , and thus the corresponding Hecke algebra is commutative in each case. We prove that in two of the cases, the primitive idempotents can be obtained from the power-sum expansion of Schur's Q-functions, from which follows the surprising corollary that the character tables of these two Hecke algebras are, aside from scalar multiples, the same as the nontrivial part of the character table of the spin representations of S _n.     

The bivariate normal copula function is regularly varying

We derive the rate of decay of the tail dependence of the bivariate normal distribution and establish its link with regularly varying functions. This result is an initial step in explaining the discrepancy between a zero asymptotic tail dependence coefficient and mass in the tail of a joint distribution.     

Trigonometric polynomials with many real zeros and a Littlewood-type problem

We examine the size of a real trigonometric polynomial of degree at most having at least zeros in (counting multiplicities). This result is then used to give a new proof of a theorem of Littlewood concerning flatness of unimodular trigonometric polynomials. Our proof is shorter and simpler than Littlewood's. Moreover our constant is explicit in contrast to Littlewood's approach, which is indirect.

     

On covariant functions and distributions under the action of a compact group

Let G be a compact subgroup of GLn(R) acting linearly on a finite dimensional complex vector space E. B. Malgrange has shown that the space C^∞G(Rn,E) of C^∞ and G-covariant functions is a finite module over the ring C^∞G(Rn) of C^∞ and G-invariant functions. First, we generalize this result for the Schwartz space SG(Rn,E) of G-covariant functions. Secondly, we prove that any G-covariant distribution can be decomposed into a sum of G-invariant distributions multiplied with a fixed family of G-covariant polynomials. This gives a generalization of an Oksak result proved in [4].     

On a test of equality of two-parameter exponential distributions

In this paper, an asymptotic expansion of the distribution of the statistic for testing the equality of p two-parameter exponential distributions is obtained upto the order n^?4 with the second term of the order n^?3 where n is the size of the sample drawn from the i th exponential population. The asymptotic expansion can therefore be used to obtain accurate approximations to the critical values of the test statistic even for comparatively small values of n. Also we have shown that F-tables can be used to test the hypothesis when the sample size is moderately large.     

On a new system of discrete distributions and characterizations of several discrete distributions by equality of distributions

This paper deals with a new system of discrete distributions. It also gives several characterizations of the Waring (and hence the Yule) distribution (and its truncated versions), the super-Poisson, the discrete uniform and other discrete distributions by using this system and other such systems existing in the literature, and linear regression. Continuous analogues of the above results are also briefly discussed.     

On characterization of power series distributions by a marginal distribution and a regression function

The conditional distribution of Y given X=x, where X and Y are non-negative integer-valued random variables, is characterized in terms of the regression function of X on Y and the marginal distribution of X which is assumed to be of a power series form. Characterizations are given for a binomial conditional distribution when X follows a Poisson, binomial or negative binomial, for a hypergeometric conditional distribution when X is binomial and for a negative hypergeometric conditional distribution when X follows a negative binomial.