Characterization of Wishart-Laplace distributions via Jordan algebra homomorphisms

For a real, Hermitian, or quaternion normal random matrix Y with mean zero, necessary and sufficient conditions for a quadratic form Q(Y) to have a Wishart-Laplace distribution (the distribution of the difference of two independent central Wishart W_p(m_i,Σ) random matrices) are given in terms of a certain Jordan algebra homomorphism ρ. Further, it is shown that {Q_k(Y)} is independent Laplace-Wishart if and only if in addition to the aforementioned conditions, the images ρ_k(Σ⁺) of the Moore-Penrose inverse Σ⁺ of Σ are mutually orthogonal: ρ_k(Σ⁺)ρ_?(Σ⁺)=0 for k≠?.     

A test of location for exchangeable multivariate normal data with unknown correlation

We consider the problem of testing whether the common mean of a single n-vector of multivariate normal random variables with known variance and unknown common correlation ρ is zero. We derive the standardized likelihood ratio test for known ρ and explore different ways of proceeding with ρ unknown. We evaluate the performance of the standardized statistic where ρ is replaced with an estimate of ρ and determine the critical value c_n that controls the type I error rate for the least favorable ρ in [0,1]. The constant c_n increases with n and this procedure has pathological behavior if ρ depends on n and ρ_n converges to zero at a certain rate. As an alternate approach, we replace ρ with the upper limit of a (1−β_n) confidence interval chosen so that c_n=c for all n. We determine β_n so that the type I error rate is exactly controlled for all ρ in [0,1]. We also investigate a simpler approach where we bound the type I error rate. The former method performs well for all n while the less powerful bound method may be a useful in some settings as a simple approach. The proposed tests can be used in different applications, including within-cluster resampling and combining exchangeable p-values.     

Robustness of A-optimal designs

Suppose that Y=(Y_i) is a normal random vector with mean Xb and covariance σ²I_n, where b is a p-dimensional vector (b_j),X=(X_ij) is an n×p matrix. A-optimal designs X are chosen from the traditional set D of A-optimal designs for ρ=0 such that X is still A-optimal in D when the components Y_i are dependent, i.e., for i≠i′, the covariance of Y_i,Y_i^′ is ρ with ρ≠0. Such designs depend on the sign of ρ. The general results are applied to X=(X_ij), where X_ij∈{-1,1}; this corresponds to a factorial design with -1,1 representing low level or high level respectively, or corresponds to a weighing design with -1,1 representing an object j with weight b_j being weighed on the left and right of a chemical balance respectively.     

Wishart-Laplace distributions associated with matrix quadratic forms

For a normal random matrix Y with mean zero, necessary and sufficient conditions are obtained for Y^′W_kY to be Wishart-Laplace distributed and {Y^′W_kY} to be independent, where each W_k is assumed to be symmetric rather than nonnegative definite.     

Error bounds for asymptotic expansions of Wilks’ lambda distribution

Let Λ=|S_e|/|S_e+S_h|, where S_h and S_e are independently distributed as Wishart distributions W_p(q,Σ) and W_p(n,Σ), respectively. Then Λ has Wilks’ lambda distribution Λ_p,q,n which appears as the distributions of various multivariate likelihood ratio tests. This paper is concerned with theoretical accuracy for asymptotic expansions of the distribution of T=-nlogΛ. We derive error bounds for the approximations. It is necessary to underline that our error bounds are given in explicit and computable forms.     

A characterization of distributions of independent random vectors by the independence of transformed vectors

If W and Z are independent random vectors and Y₁, Y₂, …, Y_n are the result of a transformation satisfying certain general conditions then W and Z are distributed according to a certain class of densities if and only if for suitable q, (Y₁, …, Y_q) and (Y_q+1, …, Y_n) are independent.     

The restricted EM algorithm under inequality restrictions on the parameters

One of the most powerful algorithms for maximum likelihood estimation for many incomplete-data problems is the EM algorithm. The restricted EM algorithm for maximum likelihood estimation under linear restrictions on the parameters has been handled by Kim and Taylor (J. Amer. Statist. Assoc. 430 (1995) 708-716). This paper proposes an EM algorithm for maximum likelihood estimation under inequality restrictions A₀β?0, where β is the parameter vector in a linear model W=Xβ+ε and ε is an error variable distributed normally with mean zero and a known or unknown variance matrix Σ>0. Some convergence properties of the EM sequence are discussed. Furthermore, we consider the consistency of the restricted EM estimator and a related testing problem.     

Equivalent conditions for noncentral generalized Laplacianness and independence of matrix quadratic forms

Let Y be an n×p multivariate normal random matrix with general covariance Σ_Y and W be a symmetric matrix. In the present article, the property that a matrix quadratic form Y^′WY is distributed as a difference of two independent (noncentral) Wishart random matrices is called the (noncentral) generalized Laplacianness (GL). Then a set of algebraic results are obtained which will give the necessary and sufficient conditions for the (noncentral) GL of a matrix quadratic form. Further, two extensions of Cochran’s theorem concerning the (noncentral) GL and independence of a family of matrix quadratic forms are developed.     

Wishart distributions associated with matrix quadratic forms

For a normally distributed random matrix Y with mean zero and general covariance matrix Σ_Y and for a symmetric matrix W, necessary and sufficient conditions are derived for the Wishartness of Y′WY.     

Empirical likelihood confidence region for parameter in the errors-in-variables models

This paper proposes a constrained empirical likelihood confidence region for a parameter β₀ in the linear errors-in-variables model: Y_i=x_i^τβ₀+ε_i,X_i=x_i+u_i,(1?i?n), which is constructed by combining the score function corresponding to the squared orthogonal distance with a constrained region of β₀. It is shown that the coverage error of the confidence region is of order n⁻¹, and Bartlett corrections can reduce the coverage errors to n⁻². An empirical Bartlett correction is given for practical implementation. Simulations show that the proposed confidence region has satisfactory coverage not only for large samples, but also for small to medium samples.     

On the dependence between the extreme order statistics in the proportional hazards model

Let X₁,…,X_n be a random sample from an absolutely continuous distribution with non-negative support, and let Y₁,…,Y_n be mutually independent lifetimes with proportional hazard rates. Let also X₍₁₎<?<X_(n) and Y₍₁₎<?<Y_(n) be their associated order statistics. It is shown that the pair (X₍₁₎,X_(n)) is then more dependent than the pair (Y₍₁₎,Y_(n)), in the sense of the right-tail increasing ordering of Avérous and Dortet-Bernadet [LTD and RTI dependence orderings, Canad. J. Statist. 28 (2000) 151-157]. Elementary consequences of this fact are highlighted.     

Asymptotic expansions for distributions of latent roots in multivariate analysis

Asymptotic expansions are given for the distributions of latent roots of matrices in three multivariate situations. The distribution of the roots of the matrix S₁(S₁ + S₂)^?1, where S₁ is $\text{W}{}_{\mathrm{<mtext>m</mtext>}}\text{(n}{}_1\text{, Σ, Ω)}$ and S₂ is W_m(n₂, Σ), is studied in detail and asymptotic series for the distribution are obtained which are valid for some or all of the roots of the noncentrality matrix Ω large. These expansions are obtained using partial-differential equations satisfied by the distribution. Asymptotic series are also obtained for the distributions of the roots of n^?1S, where S in W_m(n, Σ), for large n, and S₁S₂^?1, where S₁ is W_m(n₁, Σ) and S₂ is W_m(n₂, Σ), for large n₁ + n₂.     

The estimation of a multivariate linear relation

A multivariate linear relation η_n = β₀ξ_n is considered, in which ξ_n and η_n are observed subject to white noise errors, with covariance matrices σ₀, ω₀ respectively. If their elements lie in the null space of a suitable vector function, β₀, σ₀, ω₀ may be uniquely defined by second-order functions of the data. The asymptotic properties of estimates of β₀, σ₀, ω₀ are established under relatively mild conditions. We explore the possibility that explicit formulas for consistent estimates of β₀, σ₀, ω₀ may be available.     

On the range of heterogeneous samples

Let R_n be the range of a random sample X₁,…,X_n of exponential random variables with hazard rate λ. Let S_n be the range of another collection Y₁,…,Y_n of mutually independent exponential random variables with hazard rates λ₁,…,λ_n whose average is λ. Finally, let r and s denote the reversed hazard rates of R_n and S_n, respectively. It is shown here that the mapping t?s(t)/r(t) is increasing on (0,∞) and that as a result, R_n=X_(n)−X₍₁₎ is smaller than S_n=Y_(n)−Y₍₁₎ in the likelihood ratio ordering as well as in the dispersive ordering. As a further consequence of this fact, X_(n) is seen to be more stochastically increasing in X₍₁₎ than Y_(n) is in Y₍₁₎. In other words, the pair (X₍₁₎,X_(n)) is more dependent than the pair (Y₍₁₎,Y_(n)) in the monotone regression dependence ordering. The latter finding extends readily to the more general context where X₁,…,X_n form a random sample from a continuous distribution while Y₁,…,Y_n are mutually independent lifetimes with proportional hazard rates.     

Wishartness and independence of matrix quadratic forms in a normal random matrix

Let Y be an n×p multivariate normal random matrix with general covariance Σ_Y. The general covariance Σ_Y of Y means that the collection of all np elements in Y has an arbitrary np×np covariance matrix. A set of general, succinct and verifiable necessary and sufficient conditions is established for matrix quadratic forms Y^′W_iY's with the symmetric W_i's to be an independent family of random matrices distributed as Wishart distributions. Moreover, a set of general necessary and sufficient conditions is obtained for matrix quadratic forms Y^′W_iY's to be an independent family of random matrices distributed as noncentral Wishart distributions. Some usual versions of Cochran's theorem are presented as the special cases of these results.     

Polynomial expansions of density and distribution functions of scale mixtures

Let Y be an absolutely continuous random variable and W a nonnegative variable independent of Y. It is to be expected that when W is close to 1 in some sense, the distribution of the scale mixture YW will be close to Y. This notion has been investigated by a number of workers, who have provided bounds on the difference between the distribution functions of Y and YW. In this paper we examine the deeper problem of finding asymptotic expansions of the form P(YW ≤ x) = P(Y ≤ x) + Σ_n=1^∞E(W^r ? 1)ⁿG_n(x), where r > 0 and the functions G_n do not depend on W. We approach the problem very generally, and then consider the normal and gamma distributions in greater detail. Our results are applied to obtain better uniform and nonuniform estimates of the difference between the distribution functions of Y and YW.     

A universal strong law of large numbers for conditional expectations via nearest neighbors

For k_n-nearest neighbor estimates of a regression Y on X (d-dimensional random vector X, integrable real random variable Y) based on observed independent copies of (X,Y), strong universal pointwise consistency is shown, i.e., strong consistency P_X-almost everywhere for general distribution of (X,Y). With tie-breaking by indices, this means validity of a universal strong law of large numbers for conditional expectations E(Y|X=x).     

Asymptotic expansions of test statistics for dimensionality and additional information in canonical correlation analysis when the dimension is large

This paper examines asymptotic expansions of test statistics for dimensionality and additional information in canonical correlation analysis based on a sample of size N=n+1 on two sets of variables, i.e.,  and . These problems are related to dimension reduction. The asymptotic approximations of the statistics have been studied extensively when dimensions p₁ and p₂ are fixed and the sample size N tends to infinity. However, the approximations worsen as p₁ and p₂ increase. This paper derives asymptotic expansions of the test statistics when both the sample size and dimension are large, assuming that and have a joint (p₁+p₂)-variate normal distribution. Numerical simulations revealed that this approximation is more accurate than the classical approximation as the dimension increases.     

Nonparametric methods for unbalanced multivariate data and many factor levels

We propose different nonparametric tests for multivariate data and derive their asymptotic distribution for unbalanced designs in which the number of factor levels tends to infinity (large a, small n_i case). Quasi gratis, some new parametric multivariate tests suitable for the large a asymptotic case are also obtained. Finite sample performances are investigated and compared in a simulation study. The nonparametric tests are based on separate rankings for the different variables. In the presence of outliers, the proposed nonparametric methods have better power than their parametric counterparts. Application of the new tests is demonstrated using data from plant pathology.     

High-dimensional asymptotic expansions for the distributions of canonical correlations

This paper examines asymptotic distributions of the canonical correlations between and with q≤p, based on a sample of size of N=n+1. The asymptotic distributions of the canonical correlations have been studied extensively when the dimensions q and p are fixed and the sample size N tends toward infinity. However, these approximations worsen when q or p is large in comparison to N. To overcome this weakness, this paper first derives asymptotic distributions of the canonical correlations under a high-dimensional framework such that q is fixed, m=n−p→∞ and c=p/n→c₀∈[0,1), assuming that and have a joint (q+p)-variate normal distribution. An extended Fisher’s z-transformation is proposed. Then, the asymptotic distributions are improved further by deriving their asymptotic expansions. Numerical simulations revealed that our approximations are more accurate than the classical approximations for a large range of p,q, and n and the population canonical correlations.