Rank procedures for some two-population multivariate extended classification problems

Given independent samples from three multivariate populations with cumulative distribution functions F⁽¹⁾(x), F⁽²⁾(x), and F⁽⁰⁾(x) = θF⁽¹⁾(x) + (1 ? θ)F⁽²⁾(x), where 0 ≤ θ ≤ 1 is unknown, the three-action problem involving decision as to whether the value of θ is high, low, or intermediate, is considered. A class of consistent procedures based on the relative spacing of three sample averages of linearly compounded rank scores is formulated. The asymptotic operating characteristics of the procedures when F⁽¹⁾ and F⁽²⁾ come close together are studied and the best choice of the compounding coefficients in terms of these considered. The consequence of using estimates of the best coefficients on the asymptotic operating characteristics is also examined.     

Multivariate data-driven k-NN function estimation

In Bhattacharya and Mack (Ann. Statist. 15 (1987), 976–994), it was shown (among other things) that adapting for the optimal choice of k in univariate k-nearest neighbor density and regression estimation is feasible using weak convergence techniques. We now show that the same holds true for the multivariate case. Our results parallel Krieger and Pickands (Ann. Statist. 9 (1981), 1066–1078) and Mack and Müller (J. Multivariate Anal. 23 (1987), 169–182) for adaptive multivariate kernel density, respectively, regression, estimation.     

Decreasing in transposition property of overlapping sums,and applications

In this paper a large class of multivariate densities and frequency functions, including the multivariate Poisson distribution and the compound multivariate Poisson distribution, are shown to have the decreasing in transposition property introduced by Hollander, Proschan, and Sethuraman (1977, Ann. Statist.5, 722–733). Sample applications in ecology and reliability are given; other applications to cumulation of damage and component down times are mentioned, but are not presented in detail.     

The properties of orthogonal matrix-valued wavelet packets in higher dimensions

In the paper matrix-valued multiresolution analysis and matrix-valued wavelet packets of spaceL ²(R ⁿ ,C ^{s x s}) are introduced. A procedure for constructing a class of matrix-valued wavelet packets in higher dimensions is proposed. The properties for the matrix-valued multivariate wavelet packets are investigated by using integral transform, algebra theory and operator theory. Finally, a new orthonormal basis ofL ²(R ⁿ ,C ^{s x s}) is derived from the orthogonal multivariate matrix-valued wavelet packets.     

Invariant scale matrix hypothesis tests under elliptical symmetry

The usual assumption in multivariate hypothesis testing is that the sample consists of n independent, identically distributed Gaussian m-vectors. In this paper this assumption is weakened by considering a class of distributions for which the vector observations are not necessarily either Gaussian or independent. This class contains the elliptically symmetric laws with densities of the form f(X(n × m)) = ψ[tr(X ? M)′ (X ? M)Σ^?1]. For testing the equality of k scale matrices and for the sphericity hypothesis it is shown, by using the structure of the underlying distribution rather than any specific form of the density, that the usual invariant normal-theory tests are exactly robust, for both the null and non-null cases, under this wider class.     

Asymptotic distributions of functions of the eigenvalues of sample covariance matrix and canonical correlation matrix in multivariate time series

Let S = (1/n) Σ_t=1ⁿ X(t) X(t)′, where X(1), …, X(n) are p × 1 random vectors with mean zero. When X(t) (t = 1, …, n) are independently and identically distributed (i.i.d.) as multivariate normal with mean vector 0 and covariance matrix Σ, many authors have investigated the asymptotic expansions for the distributions of various functions of the eigenvalues of S. In this paper, we will extend the above results to the case when {X(t)} is a Gaussian stationary process. Also we shall derive the asymptotic expansions for certain functions of the sample canonical correlations in multivariate time series. Applications of some of the results in signal processing are also discussed.     

Convergence properties of an empirical error criterion for multivariate density estimation

For the purpose of comparing different nonparametric density estimators, Wegman (J. Statist. Comput. Simulation 1 225–245) introduced an empirical error criterion. In a recent paper by Hall (Stochastic Process. Appl. 13 11–25) it is shown that this empirical error criterion converges to the mean integrated square error. Here, in the case of kernel estimation, the results of Hall are improved in several ways, most notably multivariate densities are treated and the range of allowable bandwidths is extended. The techniques used here are quite different from those of Hall, which demonstrates that the elegant Brownian Bridge approximation of Komlós, Major, and Tusnády (Z. Warsch. Verw. Gebrete 32 111–131) does not always give the strongest results possible.     

Maximizing the probability of correctly ordering random variables using linear predictors

Let (T₁, x₁), (T₂, x₂), …, (T_n, x_n) be a sample from a multivariate normal distribution where T_i are (unobservable) random variables and x_i are random vectors in R^k. If the sample is either independent and identically distributed or satisfies a multivariate components of variance model, then the probability of correctly ordering {T_i} is maximized by ranking according to the order of the best linear predictors {E(T_i|x_i)}. Furthermore, it orderings are chosen according to linear functions {b′x_i} then the conditional probability of correct order given (T_i = t₁; i = 1, …, n) is maximized when b′x_i is the best linear predictor. Examples are given to show that linear predictors may not be optimal and that using a linear combination other that the best linear predictor may give a greater probability of correctly ordering {T_i} if {(T_i, x_i)} are independent but not identically distributed, or if the distributions are not normal.     

Tractability and Strong Tractability of Linear Multivariate Problems

Linear multivariate problems are defined as the approximation of linear operators on functions of d variables. We study the complexity of computing an ϵ-approximation in different settings. We are particularly interested in large d and/or large ϵ⁻¹. Tractability means that the complexity is bounded by c(d) K(d, ϵ), where c(d) is the cost of one information operation and K(d, ϵ) is a polynomial in d and/or in ϵ⁻¹. Strong tractability means that K(d, ϵ) is a polynomial in ϵ⁻¹, independent of d. We provide necessary and sufficient conditions for linear multivariate problems to be tractable or strongly tractable in the worst case, average case, randomized, and probabilistic settings. This is done for the class Λ^all where an information operation is defined as the computation of any continuous linear functional. We also consider the class Λ^std where an information operation is defined as the computation of a function value. We show under mild assumptions that tractability in the class Λ^all is equivalent to tractability in the class Λ^std. The proof is, however, not constructive. Finally, we consider linear multivariate problems over reproducing kernel Hilbert spaces, showing that such problems are strongly tractable even in the worst case setting.     

MARM Processes Part I: General Theory

This paper introduces a new class of real vector-valued stochastic processes, called MARM (Multivariate Autoregressive Modular) processes, which generalizes the class of (univariate) ARM (Autoregressive Modular) processes. Like ARM processes, the key advantage of MARM processes is their ability to fit a strong statistical signature consisting of first-order and second-order statistics. More precisely, MARM processes exactly fit an arbitrary multi-dimensional marginal distribution and approximately fit a set of leading autocorrelations and cross-correlations. This capability appears to render the MARM modeling methodology unique in its ability to fit a multivariate model to such a class of strong statistical signatures. The paper describes the construction of two flavors of MARM processes, MARM ⁺ and MARM ^? , studies the statistics of MARM processes (transition structure and second order statistics), and devises MARM-based fitting and forecasting algorithms providing point estimators and confidence intervals. The efficacy of the MARM fitting and forecasting methodology will be illustrated on real-life data in a companion paper.     

On the robustness of least squares procedures in regression models

The criterion robustness of the standard likelihood ratio test (LRT) under the multivariate normal regression model and also the inference robustness of the same test under the univariate set up are established for certain nonnormal distributions of errors. Restricting attention to the normal distribution of errors in the context of univariate regression models, conditions on the design matrix are established under which the usual LRT of a linear hypothesis (under homoscedasticity of errors) remains valid if the errors have an intraclass covariance structure. The conditions hold in the case of some standard designs. The relevance of C. R. Rao's (1967 In Proceedings Fifth Berkeley Symposium on Math. Stat. and Prob., Vol. 1, pp. 355–372) and G. Zyskind's (1967, Ann. Math. Statist.38 1092–1110) conditions in this context is discussed.     

Binary Segmentation for Multivariate Polynomials

This paper develops a fast method of binary segmentation for multivariate integer polynomials and applies it to polynomial multiplication, pseudodivision, resultant and gcd calculations. In the univariate case, binary segmentation yields the fastest known method for multiplication of integer polynomials, even slightly faster than fast Fourier transform methods (however, the underlying fast integer multiplication method is based on fast Fourier transforms). Fischer and Paterson ("SIAM-AMS Proceedings, 1974," pp. 113-125) seem to be the first authors who suggest binary segmentation for polynomials with coefficients in {0, 1}. Schönhage (J. Complexity1 (1985), 118-137), as well as Bini and Pan (J. Complexity2 (1986), 179-203) have applied the binary segmentation to univariate polynomial division and gcd calculation. In the multivariate case, well- known methods are the iterated application of univariate binary segmentation and Kronecker′s map. Our method of binary segmentation to the contrary is based on a one-step algebra homomorphism mapping multivariate polynomials directly into integers. More precisely, the variables x₁, . . . , x_n of a multivariate polynomial are substituted by integers 2^ν₁, . . . , 2^ν_n, where ν₁ < · · · < ν_n are chosen as small as possible. If n is the number of variables, d bounds the total degrees, and l bounds the bit lengths of the input, we achieve the bit complexities O(ψ((2d)ⁿ (n log d + l))) for multivariate multiplication, O(d ψ(d²ⁿ(n log d + l))) for multivariate pseudo-division (provided n = O(d log d)), and O(d ψ((2d²)ⁿ (n log d + l))) for multivariate subresultant chain calculation. Here ψ(l) = l log l log log l denotes the complexity of integer multiplication.     

L p -Norms, Log-Barriers and Cramer Transform in Optimization

We show that the Laplace approximation of a supremum by L ^p -norms has interesting consequences in optimization. For instance, the logarithmic barrier functions (LBF) of a primal convex problem P and its dual P ^* appear naturally when using this simple approximation technique for the value function g of P or its Legendre–Fenchel conjugate g ^*. In addition, minimizing the LBF of the dual P ^* is just evaluating the Cramer transform of the Laplace approximation of g. Finally, this technique permits to sometimes define an explicit dual problem P ^* in cases when the Legendre–Fenchel conjugate g ^* cannot be derived explicitly from its definition.     

Some properties and characterizations for generalized multivariate Pareto distributions

In this paper, several distributional properties and characterization theorems of the generalized multivariate Pareto distributions are studied. It is found that the multivariate Pareto distributions have many mixture properties. They are mixed either by geometric, Weibull, or exponential variables. The multivariate Pareto, MP^(k)(I), MP^(k)(II), and MP^(k)(IV) families have closure property under finite sample minima. The MP^(k)(III) family is closed under both geometric minima and geometric maxima. Through the geometric minima procedure, one characterization theorem for MP^(k)(III) distribution is developed. Moreover, the MP^(k)(III) distribution is proved as the limit multivariate distribution under repeated geometric minimization. Also, a characterization theorem for the homogeneous MP^(k)(IV) distribution via the weighted minima among the ordered coordinates is developed. Finally, the MP^(k)(II) family is shown to have the truncation invariant property.     

Bivariate Gonarov polynomials and integer sequences

Univariate Gonarov polynomials arose from the Gonarov interpolation problem in numerical analysis.They provide a natural basis of polynomials for working with u-parking functions,which are integer sequences whose order statistics are bounded by a given sequence u.In this paper,we study multivariate Gonarov polynomials,which form a basis of solutions for multivariate Gonarov interpolation problem.We present algebraic and analytic properties of multivariate Gonarov polynomials and establish a combinatorial relation with integer sequences.Explicitly,we prove that multivariate Gonarov polynomials enumerate k-tuples of integers sequences whose order statistics are bounded by certain weights along lattice paths in Nk.It leads to a higher-dimensional generalization of parking functions,for which many enumerative results can be derived from the theory of multivariate Gonarov polynomials.     

Ergodic theorems for tensor products

An ergodic theorem is proved for tensor products of Banach spaces. As a special case, an ergodic theorem is proved for vector-valued L^p-spaces. This theorem generalizes results of Aribaud, J. Funct. Anal.5 (1970), 395–411, and Dinculeanu, J. Funct. Anal.12 (1973), 229–235.     

Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions

A function f(x) defined on $\text{X}$ = $\text{X}$₁ × $\text{X}$₂ × … × $\text{X}$_n where each $\text{X}$_i is totally ordered satisfying f(x ∨ y) f(x ∧ y) ≥ f(x) f(y), where the lattice operations ∨ and ∧ refer to the usual ordering on $\text{X}$, is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies ?DΣ^?1D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.     

Some remarks on multivariate stable distributions

This paper deals with multivariate stable distributions. Press has given an explicit algebraic representation of characteristic functions of such distributions [J. Multivariate Analysis2 (1972), 444–462]. We present counter-examples and correct proofs of some of the statements of Press. The properties of multivariate stable distributions, connected with the spectral measure Γ, present in the expression of the characteristic function, are studied.     

Limit theorems for the critical age-dependent branching process with infinite variance

Let Z(t) be the population at time t of a critical age-dependent branching process. Suppose that the offspring distribution has a generating function of the form f(s) = s + (1 ? s)^1+αL(1 ? s) where α ∈ (0, 1) and L(x) varies slowly as x → 0⁺. Then we find, as t → ∞, (P{Z(t)> 0})^αL(P{Z(t)>0})～ μ/αt where μ is the mean lifetime of each particle. Furthermore, if we condition the process on non-extinction at time t, the random variable P{Z(t)>0}Z(t) converges in law to a random variable with Laplace-Stieltjes transform 1 - u(1 + u^α)^?1/α for u ?/ 0. Moment conditions on the lifetime distribution required for the above results are discussed.     

On the observation closest to the origin

Out of n i.i.d. random vectors in $\text{R}$^d let X¹_n be the one closest to the origin. We show that X¹_n has a nondegenerate limit distribution if and only if the common probability distribution satisfies a condition of multidimensional regular variation. The result is then applied to a problem of density estimation.