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.     

A new statistical model for random unit vectors

This paper proposes a new statistical model for symmetric axial directional data in dimension p. This proposal is an alternative to the Bingham distribution and to the angular central Gaussian family. The statistical properties for this model are presented. An explicit form for its normalizing constant is given and some moments and limiting distributions are derived. The proposed density is shown to apply to the modeling of 3×3 rotation matrices by representing them as quaternions, which are unit vectors in . The moment estimators of the parameters of the new model are calculated; explicit expressions for their sampling variances are given. The analysis of data measuring the posture of the right arm of subjects performing a drilling task illustrates the application of the proposed model.     

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.     

From Archimedean to Liouville copulas

We use a recent characterization of the d-dimensional Archimedean copulas as the survival copulas of d-dimensional simplex distributions (McNeil and Nešlehová (2009) [1]) to construct new Archimedean copula families, and to examine the relationship between their dependence properties and the radial parts of the corresponding simplex distributions. In particular, a new formula for Kendall’s tau is derived and a new dependence ordering for non-negative random variables is introduced which generalises the Laplace transform order. We then generalise the Archimedean copulas to obtain Liouville copulas, which are the survival copulas of Liouville distributions and which are non-exchangeable in general. We derive a formula for Kendall’s tau of Liouville copulas in terms of the radial parts of the corresponding Liouville distributions.     

Asymptotic power comparison of three tests in GMANOVA when the number of observed points is large

This paper is concerned with the testing problem of generalized multivariate linear hypothesis for the mean in the growth curve model(GMANOVA). Our interest is the case in which the number of the observed points p is relatively large compared to the sample size N. Asymptotic expansions of the non-null distributions of the likelihood ratio criterion, Lawley-Hotelling’s trace criterion and Bartlett-Nanda-Pillai’s trace criterion are derived under the asymptotic framework that N and p go to infinity together, while p/N→c∈(0,1). It also can be confirmed that Rothenberg’s condition on the magnitude of the asymptotic powers of the three tests is valid when p is relatively large, theoretically and numerically.     

Wishart ratios with dependent structure: New members of the bimatrix beta type IV

In multivariate statistics under normality, the problems of interest are random covariance matrices (known as Wishart matrices) and “ratios” of Wishart matrices that arise in multivariate analysis of variance (MANOVA) (see 24). The bimatrix variate beta type IV distribution (also known in the literature as bimatrix variate generalised beta; matrix variate generalization of a bivariate beta type I) arises from “ratios” of Wishart matrices. In this paper, we add a further independent Wishart random variate to the “denominator” of one of the ratios; this results in deriving the exact expression for the density function of the bimatrix variate extended beta type IV distribution. The latter leads to the proposal of the bimatrix variate extended F distribution. Some interesting characteristics of these newly introduced bimatrix distributions are explored. Lastly, we focus on the bivariate extended beta type IV distribution (that is an extension of bivariate Jones’ beta) with emphasis on P(X₁<X₂) where X₁ is the random stress variate and X₂ is the random strength variate.     

Densities for random balanced sampling

A random balanced sample (RBS) is a multivariate distribution with n components X_k, each uniformly distributed on [-1,1], such that the sum of these components is precisely 0. The corresponding vectors lie in an (n-1)-dimensional polytope M(n). We present new methods for the construction of such RBS via densities over M(n) and these apply for arbitrary n. While simple densities had been known previously for small values of n (namely 2,3, and 4), for larger n the known distributions with large support were fractal distributions (with fractal dimension asymptotic to n as n→∞). Applications of RBS distributions include sampling with antithetic coupling to reduce variance, and the isolation of nonlinearities. We also show that the previously known densities (for n?4) are in fact the only solutions in a natural and very large class of potential RBS densities. This finding clarifies the need for new methods, such as those presented here.     

Exact distribution of the generalized Wilks’s statistic and applications

We establish the exact expression of the density of Wilks’s statistic Λ(n,p,q), and also those of the densities of the product and ratio of two independent such statistics, in terms of Meijer functions, and provide applications with numerical illustrations in various domains of Multivariate Analysis.     

On Mardia's and Song's measures of kurtosis in elliptical distributions

The main objective of this paper is the calculation and the comparative study of two general measures of multivariate kurtosis, namely Mardia's measure β_2,p and Song's measure S(f). In this context, general formulas for the said measures are derived for the broad family of the elliptically contoured symmetric distributions and also for specific members of this family, like the multivariate t-distribution, the multivariate Pearson type II, the multivariate Pearson type VII, the multivariate symmetric Kotz type distribution and the uniform distribution in the unit sphere. Analytic expressions for computing Shannon and Rényi entropies are obtained under the elliptic family. The behaviour of Mardia's and Song's measures, their similarities and differences, possible interpretations and uses in practice are investigated by comparing them in specific members of the elliptic family of multivariate distributions. An empirical estimator of Song's measure is moreover proposed and its asymptotic distribution is investigated under the elliptic family of multivariate distributions.     

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.     

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≠?.     

The distribution of the characteristic roots ofS 1 S 2 −1 under violations in the complex case and power comparisons of four tests

Summary The joint density function of the latent roots ofS ₁ S ₂ ⁻¹ under violations is obtained whereS ₁ has a complex non-central Wishart distributionW _c (p,n ₁,Σ ₁,Ω) andS ₂, an independent complex central Wishart,W _c (p,n ₂,Σ ₂, 0). The density and moments of Hotelling's trace are also derived under violations. Further, the non-null distributions of the following four criteria in the two-roots case are studied for tests of three hypotheses: Hotelling's trace, Pillai's trace, Wilks' criterion and Roy's largest root. In addition, tabulations of powers are carried out and power comparisons for tests of each of three hypotheses based on the four criteria are made in the complex case extending such work of Pillai and Jayachandran in the classical Gaussian case. The findings in the complex Gaussian are generally similar to those in the classical.     

Characterizations of Arnold and Strauss’ and related bivariate exponential models

Characterizations of probability distributions is a topic of great popularity in applied probability and reliability literature for over last 30 years. Beside the intrinsic mathematical interest (often related to functional equations) the results in this area are helpful for probabilistic and statistical modelling, especially in engineering and biostatistical problems. A substantial number of characterizations has been devoted to a legion of variants of exponential distributions. The main reliability measures associated with a random vector X are the conditional moment function defined by m_φ(x)=E(φ(X)|X?x) (which is equivalent to the mean residual life function e(x)=m_φ(x)-x when φ(x)=x) and the hazard gradient function h(x)=-∇logR(x), where R(x) is the reliability (survival) function, R(x)=Pr(X?x), and ∇ is the operator . In this paper we study the consequences of a linear relationship between the hazard gradient and the conditional moment functions for continuous bivariate and multivariate distributions. We obtain a general characterization result which is the applied to characterize Arnold and Strauss’ bivariate exponential distribution and some related models.     

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).     

On a new multivariate two-sample test

In this paper we propose a new test for the multivariate two-sample problem. The test statistic is the difference of the sum of all the Euclidean interpoint distances between the random variables from the two different samples and one-half of the two corresponding sums of distances of the variables within the same sample. The asymptotic null distribution of the test statistic is derived using the projection method and shown to be the limit of the bootstrap distribution. A simulation study includes the comparison of univariate and multivariate normal distributions for location and dispersion alternatives. For normal location alternatives the new test is shown to have power similar to that of the t- and T²-Test.     

Estimation of regression contour clusters—an application of the excess mass approach to regression

The paper shows that the technique known as excess mass can be translated to non-parametric regression with random design in d-dimensional Euclidean space, where the regression function m is given by m(x)=E(Y∣X=x),x∈R^d. The approach is applied to estimating regression contour clusters, which are sets where m exceeds a certain threshold value. This is accomplished without prior estimation of the regression function. Consistency of the resulting estimators is studied, and a functional central limit theorem for the excess mass is derived in the regression context.     

Maximum entropy characterizations of the multivariate Liouville distributions

A random vector X=(X₁,X₂,…,X_n) with positive components has a Liouville distribution with parameter θ=(θ₁,θ₂,…,θ_n) if its joint probability density function is proportional to , θ_i>0 [R.D. Gupta, D.S.P. Richards, Multivariate Liouville distributions, J. Multivariate Anal. 23 (1987) 233-256]. Examples include correlated gamma variables, Dirichlet and inverted Dirichlet distributions. We derive appropriate constraints which establish the maximum entropy characterization of the Liouville distributions among all multivariate distributions. Matrix analogs of the Liouville distributions are considered. Some interesting results related to I-projection from a Liouville distribution are presented.     

Application of fast spherical Fourier transform to density estimation

This paper is on density estimation on the 2-sphere, S², using the orthogonal series estimator corresponding to spherical harmonics. In the standard approach of truncating the Fourier series of the empirical density, the Fourier transform is replaced with a version of the discrete fast spherical Fourier transform, as developed by Driscoll and Healy. The fast transform only applies to quantitative data on a regular grid. We will apply a kernel operator to the empirical density, to produce a function whose values at the vertices of such a grid will be the basis for the density estimation. The proposed estimation procedure also contains a deconvolution step, in order to reduce the bias introduced by the initial kernel operator. The main issue is to find necessary conditions on the involved discretization and the bandwidth of the kernel operator, to preserve the rate of convergence that can be achieved by the usual computationally intensive Fourier transform. Density estimation is considered in L²(S²) and more generally in Sobolev spaces H_v(S²), any v?0, with the regularity assumption that the probability density to be estimated belongs to H_s(S²) for some s>v. The proposed technique to estimate the Fourier transform of an unknown density keeps computing cost down to order O(n), where n denotes the sample size.     

Nonparametric tests of independence between random vectors

A nonparametric test of the mutual independence between many numerical random vectors is proposed. This test is based on a characterization of mutual independence defined from probabilities of half-spaces in a combinatorial formula of Möbius. As such, it is a natural generalization of tests of independence between univariate random variables using the empirical distribution function. If the number of vectors is p and there are n observations, the test is defined from a collection of processes R_n,A, where A is a subset of {1,…,p} of cardinality |A|>1, which are asymptotically independent and Gaussian. Without the assumption that each vector is one-dimensional with a continuous cumulative distribution function, any test of independence cannot be distribution free. The critical values of the proposed test are thus computed with the bootstrap which is shown to be consistent. Another similar test, with the same asymptotic properties, for the serial independence of a multivariate stationary sequence is also proposed. The proposed test works when some or all of the marginal distributions are singular with respect to Lebesgue measure. Moreover, in singular cases described in Section 4, the test inherits useful invariance properties from the general affine invariance property.     

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.