Characterization of the p-generalized normal distribution

It is a well known fact that invariance under the orthogonal group and marginal independence uniquely characterizes the isotropic normal distribution. Here, a similar characterization is provided for the more general class of differentiable bounded L_p-spherically symmetric distributions: Every factorial distribution in this class is necessarily p-generalized normal.     

On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

The ratio of the largest eigenvalue divided by the trace of a p×p random Wishart matrix with n degrees of freedom and an identity covariance matrix plays an important role in various hypothesis testing problems, both in statistics and in signal processing. In this paper we derive an approximate explicit expression for the distribution of this ratio, by considering the joint limit as both p,n→∞ with p/n→c. Our analysis reveals that even though asymptotically in this limit the ratio follows a Tracy-Widom (TW) distribution, one of the leading error terms depends on the second derivative of the TW distribution, and is non-negligible for practical values of p, in particular for determining tail probabilities. We thus propose to explicitly include this term in the approximate distribution for the ratio. We illustrate empirically using simulations that adding this term to the TW distribution yields a quite accurate expression to the empirical distribution of the ratio, even for small values of p,n.     

A multivariate nonparametric test of independence

A new nonparametric approach to the problem of testing the joint independence of two or more random vectors in arbitrary dimension is developed based on a measure of association determined by interpoint distances. The population independence coefficient takes values between 0 and 1, and equals zero if and only if the vectors are independent. We show that the corresponding statistic has a finite limit distribution if and only if the two random vectors are independent; thus we have a consistent test for independence. The coefficient is an increasing function of the absolute value of product moment correlation in the bivariate normal case, and coincides with the absolute value of correlation in the Bernoulli case. A simple modification of the statistic is affine invariant. The independence coefficient and the proposed statistic both have a natural extension to testing the independence of several random vectors. Empirical performance of the test is illustrated via a comparative Monte Carlo study.     

The asymptotic distribution of a goodness of fit statistic for factorial invariance

Suppose that random factor models with k factors are assumed to hold for m, p-variate populations. A model for factorial invariance has been proposed wherein the covariance or correlation matrices can be written as Σ_i = LC_iL′ + σ_i²I, where C_i is the covariance matrix of factor variables and L is a common factor loading matrix, i = 1,…, m. Also a goodness of fit statistic has been proposed for this model. The asymptotic distribution of this statistic is shown to be that of a quadratic form in normal variables. An approximation to this distribution is given and thus a test for goodness of fit is derived. The problem of dimension is considered and a numerical example is given to illustrate the results.     

Multivariate quadratic forms of random vectors

We obtain the distribution of the sum of n random vectors and the distribution of their quadratic forms: their densities are expanded in series of Hermite and Laguerre polynomials. We do not suppose that these vectors are independent. In particular, we apply these results to multivariate quadratic forms of Gaussian vectors. We obtain also their densities expanded in Mac Laurin series or in the form of an integral. By this last result, we introduce a new method of computation which can be much simpler than the previously known techniques. In particular, we introduce a new method in the very classical univariate case. We remark that we do not assume the independence of normal variables.     

Random volumes under a general matrix-variate model

The convex hull generated by p linearly independent points in Euclidean n-space, n?p will almost surely determine a p-simplex and the corresponding p-parallelotope. The volume of this p-parallelotope is where the rows of the p×n,n?p matrix of rank p represent the p linearly independent points. If the points are random points in some sense then v becomes a random volume. The distribution of this random volume v when the matrix X has a very general real rectangular matrix-variate density is the topic of this paper. The complicated classical procedures based on integral geometry techniques for dealing with such problems are replaced by a simpler procedure based on Jacobians of matrix transformations and functions of matrix argument. Apart from the distribution of v under this general model, arbitrary moments of v, connection to the likelihood ratio statistic or λ-criterion for testing hypotheses on the parameters of multivariate normal distributions, connections to Mellin-Barnes integrals and Meijer’s G-function, connection to the concept of generalized variance, various structural decompositions of v and special cases are also examined here.     

Minimax and admissible minimax estimators of the mean of a multivariate normal distribution for unknown covariance matrix

Let X be a p-variate (p ≥ 3) vector normally distributed with mean μ and covariance Σ, and let A be a p × p random matrix distributed independent of X, according to the Wishart distribution W(n, Σ). For estimating μ, we consider estimators of the form δ = δ(X, A). We obtain families of Bayes, minimax and admissible minimax estimators with respect to the quadratic loss function (δ ? μ)′ Σ^?1(δ ? μ) where Σ is unknown. This paper extends previous results of the author [1], given for the case in which the covariance matrix of the distribution is of the form σ²I, where σ is known.     

A test for the mean vector with fewer observations than the dimension

In this paper, we consider a test for the mean vector of independent and identically distributed multivariate normal random vectors where the dimension p is larger than or equal to the number of observations N. This test is invariant under scalar transformations of each component of the random vector. Theories and simulation results show that the proposed test is superior to other two tests available in the literature. Interest in such significance test for high-dimensional data is motivated by DNA microarrays. However, the methodology is valid for any application which involves high-dimensional data.     

Minimum distance classification rules for high dimensional data

In this article, the problem of classifying a new observation vector into one of the two known groups Π_i,i=1,2, distributed as multivariate normal with common covariance matrix is considered. The total number of observation vectors from the two groups is, however, less than the dimension of the observation vectors. A sample-squared distance between the two groups, using Moore-Penrose inverse, is introduced. A classification rule based on the minimum distance is proposed to classify an observation vector into two or several groups. An expression for the error of misclassification when there are only two groups is derived for large p and n=O(p^δ),0<δ<1.     

Progressive Type II censored order statistics for multivariate observations

For a sequence of independent and identically distributed random vectors , i=1,2,…,n, we consider the conditional ordering of these random vectors with respect to the magnitudes of , where N is a p-variate continuous function defined on the support set of X₁ and satisfying certain regularity conditions. We also consider the Progressive Type II right censoring for multivariate observations using conditional ordering. The need for the conditional ordering of random vectors exists for example, in reliability analysis when a system has n independent components each consisting of p arbitrarily dependent and parallel connected elements. Let the vector of life lengths for the ith component of the system be , where denotes the life length of the jth element of the ith component. Then the first failure in the system occurs at time , and for this case . In this paper we introduce the conditionally ordered and Progressive Type II right-censored conditionally ordered statistics for multivariate observations and to study their distributional properties.     

Generalized p-values and generalized confidence regions for the multivariate Behrens-Fisher problem and MANOVA

For two multivariate normal populations with unequal covariance matrices, a procedure is developed for testing the equality of the mean vectors based on the concept of generalized p-values. The generalized p-values we have developed are functions of the sufficient statistics. The computation of the generalized p-values is discussed and illustrated with an example. Numerical results show that one of our generalized p-value test has a type I error probability not exceeding the nominal level. A formula involving only a finite number of chi-square random variables is provided for computing this generalized p-value. The formula is useful in a Bayesian solution as well. The problem of constructing a confidence region for the difference between the mean vectors is also addressed using the concept of generalized confidence regions. Finally, using the generalized p-value approach, a solution is developed for the heteroscedastic MANOVA problem.     

Bounds for the uniform deviation of empirical measures

If X₁,…,X_n are independent identically distributed R^d-valued random vectors with probability measure μ and empirical probability measure μ_n, and if $\text{a}$ is a subset of the Borel sets on R^d, then we show that P{sup_{A∈$\text{a}$}|μ_n(A)?μ(A)|≥ε} ≤ cs($\text{a}$, n²)e^?2n∈2, where c is an explicitly given constant, and s($\text{a}$, n) is the maximum over all (x₁,…,x_n) ∈ R^dn of the number of different sets in {{x₁…,x_n}∩A|A ∈$\text{a}$}. The bound strengthens a result due to Vapnik and Chervonenkis.     

Inference for the invariance of canonical analysis under linear transformations

Using a recent result about the invariance problem in linear canonical analysis (LCA), we introduce a criterion by means of which one can see if this invariance holds when the related random vectors are transformed by linear maps. Then, the estimation of this criterion is considered as well as the problem of testing for invariance of LCA. Particularly, a new test for additional information in canonical analysis is proposed and simulations are used to gain understanding of the finite sample performance of this test and to compare it with the likelihood ratio test.     

A characterization of the multivariate normal distribution

Let X_j (j = 1,…,n) be i.i.d. random variables, and let Y′ = (Y₁,…,Y_m) and X′ = (X₁,…,X_n) be independently distributed, and A = (a_jk) be an n × n random coefficient matrix with a_jk = a_jk(Y) for j, k = 1,…,n. Consider the equation U = AX, Kingman and Graybill [Ann. Math. Statist.41 (1970)] have shown U ～ N(O,I) if and only if X ～ N(O,I). provided that certain conditions defined in terms of the a_jk are satisfied. The task of this paper is to delete the identical assumption on X₁,…,X_n and then generalize the results to the vector case. Furthermore, the condition of independence on the random components within each vector is relaxed, and also the question raised by the above authors is answered.     

Monotonicity and independence axioms

Given σ, a family ofchoice problems, subsets ofR ⁿ representing the payoff vectors (measured in von Neumann-Morgenstern utility scales) attainable by a group ofn players, asolution f on σ associates to everyS in σ a unique elementf (S) ofS. Amonotonicity axiom specifies how the solution outcome should change when the choice problem is subjected to certain geometric transformations while anindependence axiom requires, in similar circumstances, the invariance of the solution outcome. A number of such axioms are here formulated and the logical relationships among them are established.Strong monotonicity is shown to be the strongest axiom and strongly monotonic solutions are characterized.     

Asymptotic expansions for conditional distributions

It is shown that—under appropriate regularity conditions—the conditional distribution of the first p components of a normalized sum of i.i.d. m-dimensional random vectors, given the complementary subvector, admits a Chebyshev-Cramér asymptotic expansion of order $\text{o(n}{}^{\mathrm{<mtext>?(s?2)</mtext><mtext>2</mtext>}}\text{)}$, uniformly over all Borelsets in $\text{R}$^p and uniformly in a region of the conditioning subvector that includes moderate deviations.     

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.     

Nonparametric estimation of mixed partial derivatives of a multivariate density

On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators f_n^(p) of f^(p), where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on R^m, p = (p₁,…,p_m), p_j ≥ 0, fixed integers, and for x = (x₁,…,x_m) in R^m, f^(p)(x) = ?^p₁+…+p_m f(x)/(?^p₁x₁ … ?^p_mx_m). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of f_n^(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of f_n^(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.     

Nonparametric tests for conditional independence in two-way contingency tables

Testing for the independence between two categorical variables R and S forming a contingency table is a well-known problem: the classical chi-square and likelihood ratio tests are used. Suppose now that for each individual a set of p characteristics is also observed. Those explanatory variables, likely to be associated with R and S, can play a major role in their possible association, and it can therefore be interesting to test the independence between R and S conditionally on them. In this paper, we propose two nonparametric tests which generalise the chi-square and the likelihood ratio ideas to this case. The procedure is based on a kernel estimator of the conditional probabilities. The asymptotic law of the proposed test statistics under the conditional independence hypothesis is derived; the finite sample behaviour of the procedure is analysed through some Monte Carlo experiments and the approach is illustrated with a real data example.     

Some tests for the covariance matrix with fewer observations than the dimension under non-normality

This article analyzes whether some existing tests for the p×p covariance matrix Σ of the N independent identically distributed observation vectors work under non-normality. We focus on three hypotheses testing problems: (1) testing for sphericity, that is, the covariance matrix Σ is proportional to an identity matrix I_p; (2) the covariance matrix Σ is an identity matrix I_p; and (3) the covariance matrix is a diagonal matrix. It is shown that the tests proposed by Srivastava (2005) for the above three problems are robust under the non-normality assumption made in this article irrespective of whether N≤p or N≥p, but (N,p)→∞, and N/p may go to zero or infinity. Results are asymptotic and it may be noted that they may not hold for finite (N,p).