Distributions of orientations on Stiefel manifolds

The Riemann space whose elements are m × k (m k) matrices X, i.e., orientations, such that X′X = I_k is called the Stiefel manifold V_k,m. The matrix Langevin (or von Mises-Fisher) and matrix Bingham distributions have been suggested as distributions on V_k,m. In this paper, we present some distributional results on V_k,m. Two kinds of decomposition are given of the differential form for the invariant measure on V_k,m, and they are utilized to derive distributions on the component Stiefel manifolds and subspaces of V_k,m for the above-mentioned two distributions. The singular value decomposition of the sum of a random sample from the matrix Langevin distribution gives the maximum likelihood estimators of the population orientations and modal orientation. We derive sampling distributions of matrix statistics including these sample estimators. Furthermore, representations in terms of the Hankel transform and multi-sample distribution theory are briefly discussed.     

High dimensional limit theorems and matrix decompositions on the Stiefel manifold

The main purpose of this paper is to investigate high dimensional limiting behaviors, as m becomes infinite (m → ∞), of matrix statistics on the Stiefel manifold V_{k, m}, which consists of m × k (m ≥ k) matrices X such that X′X = I_k. The results extend those of Watson. Let X be a random matrix on V_{k, m}. We present a matrix decomposition of X as the sum of mutually orthogonal singular value decompositions of the projections P X and P X, where and are each a subspace of R^m of dimension p and their orthogonal compliment, respectively (p ≥ k and m ≥ k + p). Based on this decomposition of X, the invariant measure on V_{k, m} is expressed as the product of the measures on the component subspaces. Some distributions related to these decompositions are obtained for some population distributions on V_{k, m}. We show the limiting normalities, as m → ∞, of some matrix statistics derived from the uniform distribution and the distributions having densities of the general forms f(P X) and f(m^1/2P X) on V_{k, m}. Subsequently, applications of these high dimensional limit theorems are considered in some testing problems.     

Concentrated matrix Langevin distributions

This paper concerns the matrix Langevin distributions, exponential-type distributions defined on the two manifolds of our interest, the Stiefel manifold V_k,m and the manifold P_k,m−k of m×m orthogonal projection matrices idempotent of rank k which is equivalent to the Grassmann manifold G_k,m−k. Asymptotic theorems are derived when the concentration parameters of the distributions are large. We investigate the asymptotic behavior of distributions of some (matrix) statistics constructed based on the sample mean matrices in connection with testing hypotheses of the orientation parameters, and obtain asymptotic results in the estimation of large concentration parameters and in the classification of the matrix Langevin distributions.     

MU-Estimation and Smoothing

In the M-estimation theory developed by Huber (1964, Ann. Math. Statist.43, 1449–1458), the parameter under estimation is the value of θ which minimizes the expectation of what is called a discrepancy measure (DM) δ(X, θ) which is a function of θ and the underlying random variable X. Such a setting does not cover the estimation of parameters such as the multivariate median defined by Oja (1983) and Liu (1990), as the value of θ which minimizes the expectation of a DM of the type δ(X₁, …, X_m, θ) where X₁, …, X_m are independent copies of the underlying random variable X. Arcones et al. (1994, Ann. Statist.22, 1460–1477) studied the estimation of such parameters. We call such an M-type MU-estimation (or μ-estimation for convenience). When a DM is not a differentiable function of θ, some complexities arise in studying the properties of estimators as well as in their computation. In such a case, we introduce a new method of smoothing the DM with a kernel function and using it in estimation. It is seen that smoothing allows us to develop an elegant approach to the study of asymptotic properties and possibly apply the Newton–Raphson procedure in the computation of estimators.     

A Geometric Approach to Maximum Likelihood Estimation of the Functional Principal Components From Sparse Longitudinal Data

In this article, we consider the problem of estimating the eigenvalues and eigenfunctions of the covariance kernel (i.e., the functional principal components) from sparse and irregularly observed longitudinal data. We exploit the smoothness of the eigenfunctions to reduce dimensionality by restricting them to a lower dimensional space of smooth functions. We then approach this problem through a restricted maximum likelihood method. The estimation scheme is based on a Newton–Raphson procedure on the Stiefel manifold using the fact that the basis coefficient matrix for representing the eigenfunctions has orthonormal columns. We also address the selection of the number of basis functions, as well as that of the dimension of the covariance kernel by a second-order approximation to the leave-one-curve-out cross-validation score that is computationally very efficient. The effectiveness of our procedure is demonstrated by simulation studies and an application to a CD4+ counts dataset. In the simulation studies, our method performs well on both estimation and model selection. It also outperforms two existing approaches: one based on a local polynomial smoothing, and another using an EM algorithm. Supplementary materials including technical details, the R package fpca, and data analyzed by this article are available online.     

Elementary Symmetric Polynomials in Random Variables

The subject of the paper is the probability-theoretic properties of elementary symmetric polynomials σ _k of arbitrary degree k in random variables X _i (i=1,2,…,m) defined on special subsets of commutative rings ℛ _m with identity of finite characteristic m. It is shown that the probability distributions of the random elements σ _k (X ₁,…,X _m ) tend to a limit when m→∞ if X ₁,…,X _m form a Markov chain of finite degree μ over a finite set of states V, V⊂ℛ _m , with positive conditional probabilities. Moreover, if all the conditional probabilities exceed a prescribed positive number α, the limit distributions do not depend on the choice of the chain.      

Asymptotically optimal choice of the smoothing parameter in a nonparametric kernel estimator for a probability density

This paper presents a method of estimation of an “optimal” smoothing parameter (window width) in kernel estimators for a probability density. The obtained estimator is calculated directly from observations. By “optimal” smoothing parameters we mean those parameters which minimize the mean integral square error (MISE) or the integral square error (ISE) of approximation of an unknown density by the kernel estimator. It is shown that the asymptotic “optimality” properties of the proposed estimator correspond (with respect to the order) to those of the well-known cross-validation procedure [1, 2]. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 67–80, Perm, 1990.     

Generalized Hodges-Lehmann estimators for the analysis of variance

For X_i, …, X_n a random sample and K(·, ·) a symmetric kernel this paper considers large sample properties of location estimator satisfying , . Asymptotic normality of is obtained and two forms of interval estimators for parameter θ satisfying EK(X₁ − θ, X₂ − θ) = 0, are discussed. Consistent estimation of the variance parameters is obtained which permits the construction of asymptotically distribution free procedures. The p-variate and multigroup extension is accomplished to provide generalized one-way MANOVA. Monte Carlo results are included.     

On the Expected Number of Shadow Vertices of the Convex Hull of Random Points

Leta₁, . . . ,a_mbe independent random points in ⁿthat are independent and identically distributed spherically symmetrical in ⁿ. Moreover, letXbe the random polytope generated as the convex hull ofa₁, . . . ,a_mand letL_kbe an arbitraryk-dimensional subspace of ⁿwith 2 ≤k≤n− 1. LetX_kbe the orthogonal projection image ofXinL_k. We call those vertices ofXwhose projection images inL_kare vertices ofX_kshadow vertices ofXwith respect to the subspaceL_k. We derive a distribution independent sharp upper bound for the expected number of shadow vertices ofXinL_k.     

A Low Complexity Lie Group Method on the Stiefel Manifold

A low complexity Lie group method for numerical integration of ordinary differential equations on the orthogonal Stiefel manifold is presented. Based on the quotient space representation of the Stiefel manifold we provide a representation of the tangent space suitable for Lie group methods. According to this representation a special type of generalized polar coordinates (GPC) is defined and used as a coordinate map. The GPC maps prove to adapt well to the Stiefel manifold. For the n×k matrix representation of the Stiefel manifold the arithmetic complexity of the method presented is of order nk ², and for nk this leads to huge savings in computation time compared to ordinary Lie group methods. Numerical experiments compare the method to a standard Lie group method using the matrix exponential, and conclude that on the examples presented, the methods perform equally on both accuracy and maintaining orthogonality.     

On the Lusternik-Schnirelmann category of Stiefel manifolds

We determine the Lusternik-Schnirelmann category of real Stiefel manifolds Vn,k and quaternionic Stiefel manifolds Xn,k for n?2k which is equal to the cup-length of the mod 2 cohomology of Vn,k and the integer cohomology of Xn,k, respectively.     

Nonparametric density estimation for linear processes with infinite variance

We consider nonparametric estimation of marginal density functions of linear processes by using kernel density estimators. We assume that the innovation processes are i.i.d. and have infinite-variance. We present the asymptotic distributions of the kernel density estimators with the order of bandwidths fixed as h =  cn ^−1/5, where n is the sample size. The asymptotic distributions depend on both the coefficients of linear processes and the tail behavior of the innovations. In some cases, the kernel estimators have the same asymptotic distributions as for i.i.d. observations. In other cases, the normalized kernel density estimators converge in distribution to stable distributions. A simulation study is also carried out to examine small sample properties.     

On the estimation of entropy

Motivated by recent work of Joe (1989,Ann. Inst. Statist. Math.,41, 683–697), we introduce estimators of entropy and describe their properties. We study the effects of tail behaviour, distribution smoothness and dimensionality on convergence properties. In particular, we argue that root-n consistency of entropy estimation requires appropriate assumptions about each of these three features. Our estimators are different from Joe's, and may be computed without numerical integration, but it can be shown that the same interaction of tail behaviour, smoothness and dimensionality also determines the convergence rate of Joe's estimator. We study both histogram and kernel estimators of entropy, and in each case suggest empirical methods for choosing the smoothing parameter.     

Modifiable low‐rank approximation to a matrix

A truncated ULV decomposition (TULVD) of an m×n matrix X of rank k is a decomposition of the form X = ULV^T+E, where U and V are left orthogonal matrices, L is a k×k non‐singular lower triangular matrix, and E is an error matrix. Only U,V, L, and ∥E∥_F are stored, but E is not stored. We propose algorithms for updating and downdating the TULVD. To construct these modification algorithms, we also use a refinement algorithm based upon that in (SIAM J. Matrix Anal. Appl. 2005; 27 (1):198–211) that reduces ∥E∥_F, detects rank degeneracy, corrects it, and sharpens the approximation. Copyright © 2009 John Wiley & Sons, Ltd.     

Resolvable even cycle decompositions of the tensor product of complete graphs

In this paper, we consider resolvable k-cycle decompositions (for short, k-RCD) of K_m×K_n, where × denotes the tensor product of graphs. It has been proved that the standard necessary conditions for the existence of a k-RCD of K_m×K_n are sufficient when k is even.     

Laws of large numbers for classes of functions

Let (X, , P) be a probability space and _n, n ≥ 1, a sequence of classes of measurable complex-valued functions on (X, , P). Under a weak metric entropy condition on _n and sup {g_∞: g _n}, Glivenko-Cantelli theorems are established for the classes _n with respect to the probability measure P; i.e., lim_{n → ∞} sup_{g n} ∫ g(dP_n − dP) = 0 a.s. The result is applied to kernel density estimation and a law of the logarithm is derived for the maximal deviation between a kernel density estimator and its expected value, improving upon and generalizing the recent results of W. Stute (Ann. Probab. 10 (1982), 414–422). This result is also used to derive improved rates of uniform convergence for the empirical characteristic function.     

Non–parametric probability density function estimation on a bounded support: Applications to shape classification and speech coding

In statistics, it is usually difficult to estimate the probability density function from N independent samples X₁,X₂, …?, X_N identically distributed. A lot of work has been done in the statistical literature on the problem of probability density estimation (e.g. Cencov, 1962; Devroye and Gyorfi, 1981; Hall, 1980 and 1982; Hominal, 1979; Izenman, 1991; Kronmal and Tarter, 1968; Parzen, 1962; Rosenblatt, 1956). In this paper, we consider random variables on bounded support. Orthogonal series estimators, studied in detail by Kronmal and Tarter (1968), by Hall (1982) and by Cencov (1962), show that there is a disadvantage related to the Gibbs phenomenon on the bias of these estimators. We suggest a new method for the non–parametric probability density function estimation based on the kernel method using an appropriately chosen regular change of variable. The new method can be used for several problems of signal processing applications (scalar or vector quantization, speech or image processing, pattern recognition, etc.). Applications to shape classification and speech coding are given.     

1 -convex manifolds are p- k?hler

We study here K?hler-type properties of 1-convex manifolds, using the duality between forms and compactly supported currents, and some properties of the Aeppli groups of (q-convex manifolds. We prove that, when the exceptional setS of the l-convex manifoldX has dimensionk, X is p-K?hler for everyp > k, and isk-K?hler if and only if “the fundamental class” ofS does not vanish. There are classical examples whereX is notk-K?hler even with a smoothS, but we prove that this cannot happen if2k ≥n = dimX, nor for suitable neighborhoods of S; in particular,X is always balanced (i.e.,(n - 1)-Kahler). Partially supported by MIUR research funds.     

Strong convergence rates for the estimation of a covariance operator for associated samples

Let X _n , n ≥ 1, be a strictly stationary associated sequence of random variables, with common continuous distribution function F. Using histogram type estimators we consider the estimation of the two-dimensional distribution function of (X ₁,X _k+1) as well as the estimation of the covariance function of the limit empirical process induced by the sequence X _n , n ≥ 1. Assuming a convenient decrease rate of the covariances Cov(X ₁,X _n+1), n ≥ 1, we derive uniform strong convergence rates for these estimators. The condition on the covariance structure of the variables is satisfied either if Cov(X ₁,X _n+1) decreases polynomially or if it decreases geometrically, but as we could expect, under the latter condition we are able to establish faster convergence rates. For the two-dimensional distribution function the rate of convergence derived under a geometrical decrease of the covariances is close to the optimal rate for independent samples.      

Simultaneous estimation of location parameters of the distribution with finite support

Summary LetX _i ,i=1,..., p be theith component of thep×1 vectorX=(X ₁,X ₂,...,X _p )′. Suppose thatX ₁,X ₂,...,X _p are independent and thatX _i has a probability density which is positive on a finite interval, is symmetric about θ _i and has the same variance. In estimation of the location vector θ=(θ₁, θ₂,...,θ _p )′ under the squared error loss function explicit estimators which dominateX are obtained by using integration by parts to evaluate the risk function. Further, explicit dominating estimators are given when the distributions ofX _i ′s are mixture of two uniform distributions. For the loss function such an estimator is also given when the distributions ofX _i ′s are uniform distributions.