Minimax estimation of the mean of spherically symmetric distributions under general quadratic loss

For X one observation on a p-dimensional (p ≥ 4) spherically symmetric (s.s.) distribution about θ, minimax estimators whose risks dominate the risk of X (the best invariant procedure) are found with respect to general quadratic loss, L(δ, θ) = (δ − θ)′ D(δ − θ) where D is a known p × p positive definite matrix. For C a p × p known positive definite matrix, conditions are given under which estimators of the form δ_a,r,C,D(X) = (I − (ar(|X|²)) D^−1/2CD^1/2 |X|⁻²)X are minimax with smaller risk than X. For the problem of estimating the mean when n observations X₁, X₂, …, X_n are taken on a p-dimensional s.s. distribution about θ, any spherically symmetric translation invariant estimator, δ(X₁, X₂, …, X_n), with have a s.s. distribution about θ. Among the estimators which have these properties are best invariant estimators, sample means and maximum likelihood estimators. Moreover, under certain conditions, improved robust estimators can be found.     

On improved estimators of the generalized variance

Treated in this paper is the problem of estimating with squared error loss the generalized variance | Σ | from a Wishart random matrix S: p × p W_p(n, Σ) and an independent normal random matrix X: p × k N(ξ, Σ I_k) with ξ(p × k) unknown. Denote the columns of X by X₍₁₎ ,…, X_(k) and set ψ⁽⁰⁾(S, X) = {(n − p + 2)!/(n + 2)!} | S |, ψ⁽ⁱ⁾(X, X) = min[ψ⁽ⁱ⁻¹⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!} | S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |] and Ψ⁽ⁱ⁾(S, X) = min[ψ⁽⁰⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!}| S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |], i = 1,…,k. Our result is that the minimax, best affine equivariant estimator ψ⁽⁰⁾(S, X) is dominated by each of Ψ⁽ⁱ⁾(S, X), i = 1,…,k and for every i, ψ⁽ⁱ⁾(S, X) is better than ψ⁽ⁱ⁻¹⁾(S, X). In particular, ψ^(k)(S, X) = min[{(n − p + 2)!/(n + 2)!} | S |, {(n − p + 2)!/(n + 2)!} | S + X₍₁₎X′₍₁₎|,…,| {(n − p + k + 2)!/(n + k + 2)!} | S + X₍₁₎X′₍₁₎ + + X_(k)X′_(k)|] dominates all other ψ's. It is obtained by considering a multivariate extension of Stein's result (Ann. Inst. Statist. Math. 16, 155–160 (1964)) on the estimation of the normal variance.     

On confidence sequences for the mean vector of a multivariate normal distribution

Let X₁, X₂,… be idd random vectors with a multivariate normal distribution N(μ, Σ). A sequence of subsets {R_n(a₁, a₂,…, a_n), n ≥ m} of the space of μ is said to be a (1 − α)-level sequence of confidence sets for μ if P(μ R_n(X₁, X₂,…, X_n) for every n ≥ m) ≥ 1 − α. In this note we use the ideas of Robbins Ann. Math. Statist. 41 (1970) to construct confidence sequences for the mean vector μ when Σ is either known or unknown. The constructed sequence R_n(X₁, X₂, …, X_n) depends on Mahalanobis' or Hotelling's according as Σ is known or unknown. Confidence sequences for the vector-valued parameter in the general linear model are also given.     

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.     

Best approximation in L(μ, X), II

The object of this paper is to prove the following theorem: If Y is a closed subspace of the Banach space X, then L¹(μ, Y) is proximinal in L¹(μ, X) if and only if L^p(μ, Y) is proximinal in L^p(μ, Y) for every p, 1 < p < ∞. As an application of this result we prove that if Y is either reflexive or Y is a separable proximinal dual space, then L¹(μ, Y) is proximinal in L¹(μ, X).     

Best simultaneous approximation in

Let X be a Banach space, (Ω,Σ,μ) a finite measure space, and L₁(μ,X) the Banach space of X-valued Bochner μ-integrable functions defined on Ω endowed with its usual norm. Let us suppose that Σ₀ is a sub-σ-algebra of Σ, and let μ₀ be the restriction of μ to Σ₀. Given a natural number n, let N be a monotonous norm in . It is shown that if X is reflexive then L₁(μ₀,X) is N-simultaneously proximinal in L₁(μ,X) in the sense of Fathi et al. [Best simultaneous approximation in L^p(I,E), J. Approx. Theory 116 (2002), 369–379]. Some examples and remarks related with N-simultaneous proximinality are also given.     

Existence of maximal solutions

Let Ω be a plane bounded region. Let U = {U_μ(P):μ(P)εL∞(Ω), u_μ ε H²_{2, 0}(Ω) and a(P, μ(P))u_μ,xx + 2b(P, μ(P))u_μ,xy + c(P, μ(P))u_μ,vv = ƒ(P) for P ε Ω; here we are given a(P, X), b(P, X), c(P, X) ε L_∞(Ω × E¹), ƒ(P) ε L_p(Ω) with p > 2, and our partial differential equation is uniformly elliptic. The functions μ(P) are called profiles. We establish sufficient conditions—which when they apply are constructive—that there exist a μ₀ ε L_∞(Ω) such that u_μ0 (P) u_μ(P) for all P ε Ω and for each μ ε L_∞(Ω). Similar results are obtained for a difference equation and convergence is proved.     

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.     

Commutators for Fourier multipliers on Besov Spaces

If T is any bounded linear operator on Besov spaces B_p^σ_j,q_j(Rⁿ)(j=0,1, and 0<σ₁<σ<σ₀), it is proved that the commutator [T,T_μ]=TT_μ−T_μT is bounded on B_p^σ,q(Rⁿ), if T_μ is a Fourier multiplier such that μ is any (possibly unbounded) symbol with uniformly bounded variation on dyadic coronas.     

Likelihood-Based Local Polynomial Fitting for Single-Index Models

The parametric generalized linear model assumes that the conditional distribution of a response Y given a d-dimensional covariate X belongs to an exponential family and that a known transformation of the regression function is linear in X. In this paper we relax the latter assumption by considering a nonparametric function of the linear combination β^TX, say η₀(β^TX). To estimate the coefficient vector β and the nonparametric component η₀ we consider local polynomial fits based on kernel weighted conditional likelihoods. We then obtain an estimator of the regression function by simply replacing β and η₀ in η₀(β^TX) by these estimators. We derive the asymptotic distributions of these estimators and give the results of some numerical experiments.     

A discrete Korovkin theorem

In this paper we give a sufficient condition for the pointwise Korovkin property on B(X), the space of bounded real valued functions on an arbitrary countable set X = {x_l,…, x_j,…}. Our theorem follows from its L_p(X, μ) analogue (and conversely); here 1 p < ∞ and μ is a positive finite measure on X such that μ({x_j}) > 0 for all j.     

Limit distributions and one-parameter groups of linear operators on Banach spaces

Let = {U_t: t > 0} be a strongly continuous one-parameter group of operators on a Banach space X and Q be any subset of a set (X) of all probability measures on X. By (Q; ) we denote the class of all limit measures of {U_tn(μ₁ * μ₂*…*μ_n)*δ_xn}, where {μ_n}Q, {x_n}X and measures U_tnμ_j (j=1, 2,…, n; N=1, 2,…) form an infinitesimal triangular array. We define classes L_m( ) as follows: L₀( )= ( (X); ), L_m( )= (L_m−1( ); ) for m=1, 2,… and L_∞( )=_m=0^∞L_m( ). These classes are analogous to those defined earlier by Urbanik on the real line. Probability distributions from L_m( ), m=0, 1, 2,…, ∞, are described in terms of their characteristic functionals and their generalized Poisson exponents and Gaussian covariance operators.     

Pointwise Best Approximation in the Space of Strongly Measurable Functions with Applications to Best Approximation in L(μ,X)

The object of this paper is to prove the following theorem: Let Y be a closed subspace of the Banach space X, (S,Σ,μ) a σ-finite measure space, L(S,Y) (respectively, L(S, X)) the space of all strongly measurable functions from S to Y (respectively, X), and p a positive number. Then L(S,Y) is pointwise proximinal in L(S,X) if and only if L^p(μ,Y) is proximinal in L^p(μ,X). As an application of the theorem stated above, we prove that if Y is a separable closed subspace of the Banach space X, p is a positive number, then L^p(μ,Y) is proximinal in L^p(μ,X) if and only if Y is proximinal in X. Finally, several other interesting results on pointwise best approximation are also obtained.     

Inequalities for a Polynomial and Its Derivative II

A well-known result of Rivlin states that if p(z) is a polynomial of degree n, such that p(z) ≠ 0 in |z| < 1, then max_{|z|=r < 1} |p(z)| ≤ ((r + 1)/2)ⁿ max_{|z| = 1} |p(z)|. In this paper, we consider the polynomial p(z) = a₀ + Σⁿ_{v = μ}a_υz^υ having all its zeros in |z| ≤ k > 1 and obtain a generalization of this result. Our result improves upon a result recently proved by Bidkham and Dewan (J. Math. Anal. Appl.166 (1992), 19-324).     

A one-parameter family of unimodal, concave, polynomial maps of the interval exhibiting multiplicities

This paper shows there exists a polynomial map, p, of the interval [0, 1] onto itself that is concave, symmetric about the point and such that, when parameterized {μp}, 0 ≤ μ ≤ 1, there exist three distinct values of the parameter μ₀ < μ₁ < μ₂ such that RLR³C = K(μ₀p) ≠ K(μ₁p) ≠ K(μ₂p) = RLR³C. There is also given an explicit construction of a C¹ family with the same properties.     

On the Maximal Multiplicity of Parts in a Random Integer Partition

We study the asymptotic behavior of the maximal multiplicity μ_n = μ_n(λ) of the parts in a partition λ of the positive integer n, assuming that λ is chosen uniformly at random from the set of all such partitions. We prove that πμ_n/(6n)^1/2 converges weakly to max _jX_j/j as n→∞, where X₁, X₂, … are independent and exponentially distributed random variables with common mean equal to 1.2000 Mathematics Subject Classification: Primary—05A17; Secondary—11P82, 60C05, 60F05     

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.     

Consistency of a recursive nearest neighbor regression function estimate

Let (X, Y) be an ^d × -valued random vector and let (X₁, Y₁),…,(X_N, Y_N) be a random sample drawn from its distribution. Divide the data sequence into disjoint blocks of length l₁, …, l_n, find the nearest neighbor to X in each block and call the corresponding couple (X_i^*, Y_i^*). It is shown that the estimate m_n(X) = Σ_{i = 1}ⁿ w_niY_i^*/Σ_{i = 1}ⁿ w_ni of m(X) = E{Y|X} satisfies E{|m_n(X) − m(X)|^p} 0 (p ≥ 1) whenever E{|Y|^p} < ∞, l_n ∞, and the triangular array of positive weights {w_ni} satisfies sup_{i ≤ n}w_ni/Σ_{i = 1}ⁿ w_ni 0. No other restrictions are put on the distribution of (X, Y). Also, some distribution-free results for the strong convergence of E{|m_n(X) − m(X)|^p|X₁, Y₁,…, X_N, Y_N} to zero are included. Finally, an application to the discrimination problem is considered, and a discrimination rule is exhibited and shown to be strongly Bayes risk consistent for all distributions.     

Groups of measure space transformations and invariants of outer conjugation for automorphisms from normalizers of type III full groups

The necessary and sufficient conditions of outer conjugation for automorphisms from the normalizer of approximated III type groups are found. Let T be an automorphism of a Lebesgue space (X, μ) of the III₀ type, [T] the full group generated by T, N[T] its normalizer, {W_t(T)} the flow associated with T and α → mod α the homomorphism from N[T] to C{W} the centralizer of the associated flow. The following results are obtained: such that mod ga = α; automorphisms α₁, and α₂ from N[T] are outer conjugate if and only if p(α₁) = p(α₂), mod α₁ = γ mod α₂γ⁻¹, where γ C{W} and p(·) is the outer period; the canonical form of the elements from N[T] is found. The case is also considered where T is types III_λ (0 < λ < 1) and III₁.     

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.