Nonparametric Empirical Bayes Estimation of the Matrix Parameter of the Wishart Distribution

We consider independent pairs (X₁, Σ₁), (X₂, Σ₂), …, (X_n, Σ_n), where eachΣ_iis distributed according to some unknown density functiong(Σ) and, givenΣ_i=Σ,X_ihas conditional density functionq(xΣ) of the Wishart type. In each pair the first component is observable but the second is not. After the (n+1)th observationX_n+1is obtained, the objective is to estimateΣ_n+1corresponding toX_n+1. This estimator is called the empirical Bayes (EB) estimator ofΣ. An EB estimator ofΣis constructed without any parametric assumptions ong(Σ). Its posterior mean square risk is examined, and the estimator is demonstrated to be pointwise asymptotically optimal.     

Some properties of BLUE in a linear model and canonical correlations associated with linear transformations

Let (x, Xβ, V) be a linear model and let A′ = (A′₁, A′₂) be a p × p nonsingular matrix such that A₂X = 0, Rank A₂ = p − Rank X. We represent the BLUE and its covariance matrix in alternative forms under the conditions that the number of unit canonical correlations between y₁ ( = A₁x) and y₂ ( = A₂x) is zero. For the second problem, let x′ = (x′₁, x′₂) and let a g-inverse V⁻ of V be written as (V⁻)′ = (A′₁, A′₂). We investigate the reations (if any) between the nonzero canonical correlations {1 ₁ … ₁ > 0} due to y₁ ( = A₁x) and y₂ ( = A₂x), and the nonzero canonical correlations {1 λ₁ … λ_v+r > 0} due to x₁ and x₂. We answer some of the questions raised by Latour et al. (1987, in Proceedings, 2nd Int. Tampere Conf. Statist. (T. Pukkila and S. Puntanen, Eds.), Univ. of Tampere, Finland) in the case of the Moore-Penrose inverse V⁺ = (A′₁, A′₂) of V.     

On the conditional probability density functions of multivariate uniform random vectors and multivariate normal random vectors

It is shown that the conditional probability density function of Y₁ given (1/n) Σ_i=1ⁿ Y_i=1Y_i^t = Σ, where Y₁, Y₂,…, Y_n are i.i.d, p-variate uniform random vectors with mean 0 equals to that of Y₁ given (1/n) Σ_i=1ⁿ Y_iY_i^t,…, Y_n are i.i.d, p-variate normal random vectors with mean 0 and covariance matrix Σ.     

Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances

In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is y_t = Bz_t + v_t. The unobserved error sequence {v_t} is a sequence of martingale differences with conditional covariance matrices {Σ_t} and satisfying sup_{t=1,…, n} {v′_tv_tI(v′_tv_t>a) |z_t, v_t−1, z_t−1, …} 0 as a → ∞. The sample covariance of the independent variables z₁, …, z_n, is assumed to have a probability limit M, constant and nonsingular; max_t=1,…,nz′_tz_t/n 0. If (1/n)Σ_t=1ⁿΣ_t Σ, constant, then √nvec( _n−B) N(0,M⁻¹Σ) and _n Σ. The autoregression model is x_t = Bx_{t − 1} + v_t with the maximum absolute value of the characteristic roots of B less than one, the above conditions on {v_t}, and (1/n)Σ_t=max(r,s)+1(Σ_tv_t−1−rv′_t−1−s) δ_rs(ΣΣ), where δ_rs is the Kronecker delta. Then √nvec( _n−B) N(0,Γ⁻¹Σ), where Γ = Σ_{s = 0}^∞B^sΣ(B′)^s.     

Approximation algorithms for MAX–MIN tiling

The MAX–MIN tiling problem is as follows. We are given A[1,…,n][1,…,n], a two-dimensional array where each entry A[i][j] stores a non-negative number. Define a tile of A to be a subarray A[ℓ,…,r][t,…,b] of A, the weight of a tile to be the sum of all array entries in it, and a tiling of A to be a collection of tiles of A such that each entry A[i][j] is contained in exactly one tile. Given a weight bound W the goal of our MAX–MIN tiling problem is to find a tiling of A such that: (1) each tile is of weight at least W (the MIN condition), and (2) the number of tiles is maximized (the MAX condition). The MAX–MIN tiling problem is known to be NP-hard; here, we present first non-trivial approximations algorithms for solving it.     

Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions: II. The Heterogeneous Case

We consider the problem of discriminating between two independent multivariate normal populations, N_p(μ₁, Σ₁) and N_p(μ₂, Σ₂), having distinct mean vectors μ₁ and μ₂ and distinct covariance matrices Σ₁ and Σ₂. The parameters μ₁, μ₂, Σ₁, and Σ₂ are unknown and are estimated by means of independent random training samples from each population. We derive a stochastic representation for the exact distribution of the “plug-in” quadratic discriminant function for classifying a new observation between the two populations. The stochastic representation involves only the classical standard normal, chi-square, and F distributions and is easily implemented for simulation purposes. Using Monte Carlo simulation of the stochastic representation we provide applications to the estimation of misclassification probabilities for the well-known iris data studied by Fisher (Ann. Eugen.7 (1936), 179–188); a data set on corporate financial ratios provided by Johnson and Wichern (Applied Multivariate Statistical Analysis, 4th ed., Prentice–Hall, Englewood Cliffs, NJ, 1998); and a data set analyzed by Reaven and Miller (Diabetologia16 (1979), 17–24) in a classification of diabetic status.     

Majorants and Minorants for Elliptic Measures on

Let μ(· ; Σ, G₁) and μ(· ; Ω, G₂) be elliptically contoured measures on ^k centered at 0, having scale parameters (Σ, Ω) and radial cdf′s (G₁, G₂). Elliptical measures v_m(·) and v_M(·), depending on (Σ, Ω, G₁, G₂), are constructed such that V_m(C) ≤ {μ(C; Σ, G₁), μ(C; Ω, G₂)} for every symmetric convex set C ^k with equality for certain sets. These in turn rely on the construction of spectral lower and upper matrix bounds for (Σ, Ω). Extensions include bounds for certain ensembles and mixtures, including versions having star-shaped contours. The lindings specialize to give envelopes for some nonstandard distributions of quadratic forms, with applications to stochastic characteristics of ballistic systems.     

Decompositions of complete graphs and complete bipartite graphs into isomorphic supersubdivision graphs

A graph H is called a supersubdivison of a graph G if H is obtained from G by replacing every edge uv of G by a complete bipartite graph K_2,m (m may vary for each edge) by identifying u and v with the two vertices in K_2,m that form one of the two partite sets. We denote the set of all such supersubdivision graphs by SS(G). Then, we prove the following results.

1. Each non-trivial connected graph G and each supersubdivision graph HSS(G) admits an α-valuation. Consequently, due to the results of Rosa (in: Theory of Graphs, International Symposium, Rome, July 1966, Gordon and Breach, New York, Dunod, Paris, 1967, p. 349) and El-Zanati and Vanden Eynden (J. Combin. Designs 4 (1996) 51), it follows that complete graphs K_2cq+1 and complete bipartite graphs K_mq,nq can be decomposed into edge disjoined copies of HSS(G), for all positive integers m,n and c, where q=|E(H)|.

2. Each connected graph G and each supersubdivision graph in SS(G) is strongly n-elegant, where n=|V(G)| and felicitous.

3. Each supersubdivision graph in EASS(G), the set of all even arbitrary supersubdivision graphs of any graph G, is cordial.

Further, we discuss a related open problem.     

Automorphism groups of certain simple 2-(q,3,λ) designs constructed from finite fields

Let F be a finite field of characteristic not 2, and SF a subset with three elements. Consider the collection

S={S·a+b | a,bF, a≠0}.

Then (F,S) is a simple 2-design and the parameter λ of (F,S) is 1, 2, 3 or 6. We find in this paper the full automorphism group of (F,S). Namely, if we put U={r | {0,1,r}S} and K the subfield of F generated by U, then the automorphisms of (F,S) are the maps of the form xg(α(x))+b, xF, where bF, α : F→F is a field automorphism fixing U, and g is a linear transformation of F considered as a vector space over K.     

Nonbelief and the Desire-As-Belief Thesis

I show the incompatibility of two theses: (a) to desire the truth of p amounts to believing a certain proposition about the value of p’s truth; (b) one cannot be said to desire the truth of p if one believes that p is true. Thesis (a), the Desire-As-Belief Thesis, has received much attention since the late 1980s. Thesis (b) is an epistemic variant of Socrates’ remark in the Symposium that one cannot desire what one already has. It turns out that (a) and (b) cannot both be true if it is possible for there to exist an agent who has a desire initially, say the desire for the truth of p, and then expands the corpus of propositions she believes to include p. This result provides a new route to the denial of (a).

Fast pattern-matching on indeterminate strings

In a string x on an alphabet Σ, a position i is said to be indeterminate iff x[i] may be any one of a specified subset {λ₁,λ₂,…,λ_j} of Σ, 2j|Σ|. A string x containing indeterminate positions is therefore also said to be indeterminate. Indeterminate strings can arise in DNA and amino acid sequences as well as in cryptological applications and the analysis of musical texts. In this paper we describe fast algorithms for finding all occurrences of a pattern p=p[1..m] in a given text x=x[1..n], where either or both of p and x can be indeterminate. Our algorithms are based on the Sunday variant of the Boyer–Moore pattern-matching algorithm, one of the fastest exact pattern-matching algorithms known. The methodology we describe applies more generally to all variants of Boyer–Moore (such as Horspool's, for example) that depend only on calculation of the δ (“rightmost shift”) array: our method therefore assumes that Σ is indexed (essentially, an integer alphabet), a requirement normally satisfied in practice.     

Admissibility and complete class results for the multinomial estimation problem with entropy and squared error loss

Let X ≡ (X₁, …, X_t) have a multinomial distribution based on N trials with unknown vector of cell probabilities p ≡ (p₁, …, p_t). This paper derives admissibility and complete class results for the problem of simultaneously estimating p under entropy loss (EL) and squared error loss (SEL). Let and f(x¦p) denote the (t − 1)-dimensional simplex, the support of X and the probability mass function of X, respectively. First it is shown that δ is Bayes w.r.t. EL for prior P if and only if δ is Bayes w.r.t. SEL for P. The admissible rules under EL are proved to be Bayes, a result known for the case of SEL. Let Q denote the class of subsets of of the form T = _j=1^kF_j where k ≥ 1 and each F_j is a facet of which satisfies: F a facet of such that F naF_jF ncT. The minimal complete class of rules w.r.t. EL when N ≥ t − 1 is characterized as the class of Bayes rules with respect to priors P which satisfy P( ⁰) = 1, ξ(x) ≡ ∫ f(x¦p) P(dp) > 0 for all x in {x : sup ^₀ f(x¦p) > 0} for some ⁰ in Q containing all the vertices of . As an application, the maximum likelihood estimator is proved to be admissible w.r.t. EL when the estimation problem has parameter space Θ = but it is shown to be inadmissible for the problem with parameter space Θ = ( minus its vertices). This is a severe form of “tyranny of boundary.” Finally it is shown that when N ≥ t − 1 any estimator δ which satisfies δ(x) > 0 x is admissible under EL if and only if it is admissible under SEL. Examples are given of nonpositive estimators which are admissible under SEL but not under EL and vice versa.     

More on Ordinary Differential Equations Which Yield Periodic Solutions of Delay Differential Equations

We construct a Poincaré operator for the system where λ is a real parameter, x ³, x = (x₁, x₂, x₃), [formula], and ƒ is an odd C² function such that ƒ′(0) = 1, xƒ(x) > 0, for x ≠ 0. We also consider the case where ƒ is C¹. We will express F in linearized form, that is, F(x) = Ax + G(x), where A is the linearized part of F around zero and G(x) = o(|x|) near zero. Fixed points of the Poincaré operator correspond to periodic solutions of the functional differential equation

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where T is the period of x.     

Inequalities for predictive ratios and posterior variances in natural exponential families

The predictive ratio is considered as a measure of spread for the predictive distribution. It is shown that, in the exponential families, ordering according to the predictive ratio is equivalent to ordering according to the posterior covariance matrix of the parameters. This result generalizes an inequality due to Chaloner and Duncan who consider the predictive ratio for a beta-binomial distribution and compare it with a predictive ratio for the binomial distribution with a degenerate prior. The predictive ratio at x₁ and x₂ is defined to be p_g(x₁)p_g(x₂)/[p_g( )]² = h_g(x₁, x₂), where p_g(x₁) = ∫ ƒ(x₁θ) g(θ) dθ is the predictive distribution of x₁ with respect to the prior g. We prove that h_g(x₁, x₂) ≥ h_g^*(x₁, x₂) for all x₁ and x₂ if ƒ(xθ) is in the natural exponential family and Cov_gx(θ) ≥ Cov_g^*x(θ) in the Loewner sense, for all x on a straight line from x₁ to x₂. We then restrict the class of prior distributions to the conjugate class and ask whether the posterior covariance inequality obtains if g and g^* differ in that the “sample size”     

The structure of the algebraic eigenspace to the spectral radius of eventually compact, nonnegative integral operators

Let K be an eventually compact linear integral operator on L^p(Ω, μ), 1 p < ∞, with nonnegative kernel k(x, y), where the underlying measure μ is totally σ-finite on the domain set Ω when P = 1. This work extends the previous analysis of the author who characterized the distinguished eigenvalues of K and K^*, and the support sets for the eigenfunctions and generalized eigenfunctions belonging to the spectral radius of K or K^*. The characterizations of the support sets for the algebraic eigenspaces of K or K^* are phrased in terms of significant k-components which are maximal irreducible subsets of Ω and which yield a positive spectral radius for the integral operator defined by the restriction of k(x, y) to the Cartesian product of such sets. In this paper, we show that a basis for the functions, constituting the algebraic eigenspaces of K and K^* belonging to the spectral radius of K, can be chosen to consist of elements which are positive on their sets of support, except possibly on sets of measure less than some arbitrarily specified positive number. In addition, we present necessary and sufficient conditions, in terms of the significant k-components, for both K and K^* to possess a positive eigenfunction (a.e. μ) corresponding to the spectral radius, as well as necessary and sufficient conditions for the sequence γⁿKⁿg _p to converge whenever g 0, where − _p denotes the norm in L^p(Ω, μ), and γ₁ the smallest (in modulus) characteristic value of K. This analysis is made possible by introducing the concepts of chains, lengths of chains, height, and depth of a significant k-component as was done by U. Rothblum [Lin. Alg. Appl. 12 (1975), 281–292] for the matrix setting.     

Common Principal Components for Dependent Random Vectors

Let the kp-variate random vector X be partitioned into k subvectors X_i of dimension p each, and let the covariance matrix Ψ of X be partitioned analogously into submatrices Ψ_ij. The common principal component (CPC) model for dependent random vectors assumes the existence of an orthogonal p by p matrix β such that β^tΨ_ijβ is diagonal for all (i, j). After a formal definition of the model, normal theory maximum likelihood estimators are obtained. The asymptotic theory for the estimated orthogonal matrix is derived by a new technique of choosing proper subsets of functionally independent parameters.     

Sequential Estimation of the Mean Vector of a Multivariate Linear Process

Sequential procedures are proposed to estimate the unknown mean vector of a multivariate linear process of the form X_t − μ = ∑^∞_{j = 0}A_jZ_{t − j}, where the Z_t are i.i.d. (0, Σ) with unknown covariance matrix Σ. The proposed point estimation is asymptotically risk efficient in the sense of Starr (1966, Ann. Math. Statist.37 1173-1185). The fixed accuracy confidence set procedure is asymptotically efficient with prescribed coverage probability in the sense of Chow and Robbins (1965, Ann. Math. Statist.36 457-462). A random central limit theorem for this process, under a mild summability condition on the coefficient matrices A_j, is also obtained.     

On the Conditional Variance for Scale Mixtures of Normal Distributions

For a scale mixture of normal vector, X=A^1/2G, where X, G ⁿ and A is a positive variable, independent of the normal vector G, we obtain that the conditional variance covariance, Cov(X₂ X₁), is always finite a.s. for m2, where X₁ ⁿ and m<n, and remains a.s. finite even for m=1, if and only if the square root moment of the scale factor is finite. It is shown that the variance is not degenerate as in the Gaussian case, but depends upon a function S_A, m(·) for which various properties are derived. Application to a uniform and stable scale of normal distributions are also given.     

Canonical correlation analysis based on information theory

In this article, we propose a new canonical correlation method based on information theory. This method examines potential nonlinear relationships between p×1 vector Y-set and q×1 vector X-set. It finds canonical coefficient vectors a and b by maximizing a more general measure, the mutual information, between a^TX and b^TY. We use a permutation test to determine the pairs of the new canonical correlation variates, which requires no specific distributions for X and Y as long as one can estimate the densities of a^TX and b^TY nonparametrically. Examples illustrating the new method are presented.     

Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables

Consider Z₊^d (d2)—the positive d-dimensional lattice points with partial ordering , let {X_k,kZ₊^d} be i.i.d. random variables with mean 0, and set S_n=∑_knX_k, nZ₊^d. We establish precise asymptotics for ∑_n|n|^r/p−2P(|S_n||n|^1/p), and for

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, (0δ1) as 0, and for

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as .