Minimax estimation of a multivariate normal mean under arbitrary quadratic loss

Let X be a p-variate (p ≥ 3) vector normally distributed with mean θ and known covariance matrix

. It is desired to estimate θ under the quadratic loss (δ ? θ)^tQ(δ ? θ), where Q is a known positive definite matrix. A broad class of minimax estimators for θ is developed.     

Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions

Let X∼f(∥x-θ∥²) and let δ_π(X) be the generalized Bayes estimator of θ with respect to a spherically symmetric prior, π(∥θ∥²), for loss ∥δ-θ∥². We show that if π(t) is superharmonic, non-increasing, and has a non-decreasing Laplacian, then the generalized Bayes estimator is minimax and dominates the usual minimax estimator δ₀(X)=X under certain conditions on . The class of priors includes priors of the form for and hence includes the fundamental harmonic prior . The class of sampling distributions includes certain variance mixtures of normals and other functions f(t) of the form e^-αtβ and e^-αt+βφ(t) which are not mixtures of normals. The proofs do not rely on boundness or monotonicity of the function r(t) in the representation of the Bayes estimator as .     

Minimax estimation of location parameters for certain spherically symmetric distributions

Families of minimax estimators are found for the location parameters of a p-variate distribution of the form

, where G(·) is a known c.d.f. on (0, ∞), p ≥ 3 and the loss is sum of squared errors. The estimators are of the form $\text{(1 ? ar(X′X)}\text{E}{}_0\text{(}\text{1}\text{X′X}\text{)X′X)X}$ where 0 ≤ a ≤ 2, r(X′X) is nondecreasing, and $\text{r(X′X)}\text{X′X}$ is nonincreasing. Generalized Bayes minimax estimators are found for certain G(·)'s.     

On Bayes estimators with uniform priors on spheres and their comparative performance with maximum likelihood estimators for estimating bounded multivariate normal means

For independently distributed observables: X_i∼N(θ_i,σ²),i=1,…,p, we consider estimating the vector θ=(θ₁,…,θ_p)^′ with loss ‖d−θ‖² under the constraint , with known τ₁,…,τ_p,σ²,m. In comparing the risk performance of Bayesian estimators δ_α associated with uniform priors on spheres of radius α centered at (τ₁,…,τ_p) with that of the maximum likelihood estimator , we make use of Stein’s unbiased estimate of risk technique, Karlin’s sign change arguments, and a conditional risk analysis to obtain for a fixed (m,p) necessary and sufficient conditions on α for δ_α to dominate . Large sample determinations of these conditions are provided. Both cases where all such δ_α’s and cases where no such δ_α’s dominate are elicited. We establish, as a particular case, that the boundary uniform Bayes estimator δ_m dominates if and only if m≤k(p) with , improving on the previously known sufficient condition of Marchand and Perron (2001) [3] for which . Finally, we improve upon a universal dominance condition due to Marchand and Perron, by establishing that all Bayesian estimators δ_π with π spherically symmetric and supported on the parameter space dominate whenever m≤c₁(p) with .     

Minimax estimators that shift towards a hypersphere for location vectors of spherically symmetric distributions

Let X be a p-dimensional random vector with density f(6X?θ6) where θ is an unknown location vector. For p ≥ 3, conditions on f are given for which there exist minimax estimators θ?(X) satisfying 6Xt6 · 6θ?(X) ? X6 ≤ C, where C is a known constant depending on f. (The positive part estimator is among them.) The loss function is a nondecreasing concave function of 6θ?? θ6². If θ is assumed likely to lie in a ball in $\text{R}$^p, then minimax estimators are given which shrink from the observation X outside the ball in the direction of P(X) the closest point on the surface of the ball. The amount of shrinkage depends on the distance of X from the ball.     

The completion of a B-convex normed Riesz space is reflexive

If

is a B-convex normed Riesz space, then the topological completion of

is a closed subspace of

⁷, the second Banach dual of

. If $\text{N=}\textcolor{red}{}{}^7$ or $\text{N=}\textcolor{red}{}{}^{\mathrm{7x}}$, then $\text{N}$ is a B-convex σ-Dedekind complete normed Riesz space which is the Banach dual of a normed Riesz space. In such a $\text{N}$, if u₁ ? u₂ ? … ? 0 and inf_n{u_n} = 0, then lim_n∥u_n∥ = 0. This is the key step in verifying that Ogasawara's criteria that a normed Riesz space be reflexive are satisfied by

⁷. Thus the topological completion of

as a closed subspace of

⁷ is also reflexive.     

Finite embedding theorems for partial Steiner triple systems

Let P be a finite set and (P,

₁), (P,

₂),…, (P,

_k) any collection of mutually disjoint partial Steiner triple systems. Then these partial triple systems can be embedded in finite mutually disjoint triple systems (S,

₁), (S,

₂),…, (S,

_k). This result is then used to prove the following more general result. If (P,

₁), (P,

₂),…, (P,

_k) are any collection of finite partial Steiner triple systems, then these partial triple systems can be embedded in finite triple systems (S,

₁), (S,

_j =

_i ∩

_j for all i ≠ j = 1, 2,…, k.     

Endomorphism semigroups of finite subset algebras

For an arbitrary set X, if $\textcolor{red}{}\text{(X)}$ denotes the collection of finite subsets of X, then $\text{(}\textcolor{red}{}\text{(X),∩,∪,-)}$ is called the finite subset algebra of X (which we also denote by $\textcolor{red}{}\text{(X))}$. Because of the many natural occurrences of finite subset algebras, the intent of this paper is to initiate an investigation of the semigroup, End $\textcolor{red}{}\text{(X)}$, of endomorphisms of $\textcolor{red}{}\text{(X)}$.Several representations of End $\textcolor{red}{}\text{(X)}$ are obtained and these representations are in turn used to obtain information about the structure of End $\textcolor{red}{}\text{(X)}$. In particular, the Green relations are characterized; for finite X, the maximal subgroups of End $\textcolor{red}{}\text{(X)}$ are found and the number of idempotents is given. As a side result, an abstract characterization of a certain semigroup of (0,1)-matrices is obtained.     

Approximation numbers of Sobolev imbeddings over unbounded domains

Let Ω ? $\text{R}$^N be an open set with $\text{dist}\text{(x, ?Ω) = O(¦ x ¦}{}^{\mathrm{?l}}\text{)}\text{for}\text{x ? Ω}$ and some l > 0 satisfying an additional regularity condition. We give asymptotic estimates for the approximation numbers α_n of Sobolev imbeddings

over these quasibounded domains Ω. Here

denotes the Sobolev space obtained by completing $\text{C}{}_0{}^{\mathrm{staggered\infty }}\text{(Ω)}$ under the usual Sobolev norm. We prove $\text{α}{}_{\mathrm{n}}\text{(I}{}_{\mathrm{p,q}}{}^{\mathrm{m}}\text{)}\text{\$}\text{?}\text{n}{}^{\mathrm{?\gamma }}$, where

. There are quasibounded domains of this type where γ is the exact order of decay, in the case p ? q under the additional assumption that either 1 ? p ? q ? 2 or 2 ? p ? q ? ∞. This generalizes the known results for bounded domains which correspond to l = ∞. Similar results are indicated for the Kolmogorov and Gelfand numbers of I_p,q^m. As an application we give the rate of growth of the eigenvalues of certain elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded domain of the above type.     

Majors of geometric strong maps

Most of the constructions in the theory of combinatorial geometries take place in the category of pregeometries and strong maps. In this present paper, we study these constructions and the structure of pregeometries by factoring strong maps into elementary maps, after the work of Dowling and Kelly. Using the modular cuts of the factorization to determine certain single-element extensions, we associate each factorization Φ to a unique labeled pregeometry, called the major of Φ. Every major of a factorization of a strong map f:H→G has H and G as distinguished minors: H as a subgeometry; G as a contraction.A partial order is defined on the set of all factorizations of f,

, and a compatible partial order is defined on the set of all majors of f,

. The map Φ :

→

is shown to be an increasing surjection, while the map Y :

→

is shown to be an increasing injection. The composition Φ ° Y is shown to be the identity on

, while Y ° Φ is a decreasing map on

, defining a co-closure on majors. The partial order on

, although seemingly more restrictive than necessary, is shown to be necessary to reflect the order on factorizations.Special consideration is given the zero element of

, called the Higgs factorization of f, which is constructed using the lift construction of Higgs. We explore the connection between the majors of this theory and a particular major constructed by Higgs, and show that this Higgs major is the major of the Higgs factorization. As the unique smallest major of any strong map, the Higgs major thus defines a unary operator on pregeometries: Y(G) is the Higgs major of f :

→G.     

Generic periodic solutions of functional differential equations

Consider the class of retarded functional differential equations

$\text{x(t) = f(x}{}_{\mathrm{t}}\text{)}$

, (1) where x_t(θ) = x(t + θ), ?1 ? θ ? 0, so x_t?C = C([?1, 0], Rⁿ), and $\text{f∈}\textcolor{red}{}\text{=C}{}^{\mathrm{r}}\text{(C,R}{}^{\mathrm{n}}\text{)}$. Let 2 ? r ? ∞ and give $\text{X}$ the appropriate (Whitney) topology. Then the set of $\text{f∈}\textcolor{red}{}$ such that all fixed points and all periodic solutions of (¹) are hyperbolic is residual in

.     

Estimation of location parameters for spherically symmetric distributions

Estimation of the location parameters of a p×1 random vector with a spherically symmetric distribution is considered under quadratic loss. The conditions of Brandwein and Strawderman [Ann. Statist. 19(1991) 1639-1650] under which estimators of the form dominate are (i) where -h is superharmonic, (ii) is nonincreasing in R, where has a uniform distribution in the sphere centered at with a radius R, and (iii) . In this paper, we not only drop their condition (ii) to show the dominance of over but also obtain a new bound for a which is sometimes better than that obtained by Brandwein and Strawderman. Specifically, the new bound of a is 0<a<[μ₁/(p²μ_-1)][1-(p-1)μ₁/(pμ_-1μ₂)]^-1 with for i=-1,1,2. The generalization to concave loss functions is also considered. Additionally, we investigate estimators of the location parameters when the scale is unknown and the observation contains a residual vector.     

Asymptotic properties of Bayes estimators for Gaussian Itô-processes with noisy observations

The estimation of a real parameter θ in a linear stochastic differential equation of the simple type is investigated, based on noisy, time continuous observations of X_t. Sufficient conditions on the continuous functions β and σ are given such that the (conditionally normal) Bayes estimators of θ satisfy certain error bounds and are strongly consistent.     

Generalized inverse of linear transformations: a geometric approach

A generalized inverse of a linear transformation A:

→

, where

and

are arbitrary finite dimensional vector spaces, is defined using only geometrical concepts of linear transformations. The inverse is uniquely defined in terms of specified subspaces $\text{L}$ ?

, $\text{M}$ ?

and a linear transformation N satisfying some conditions. Such an inverse is called the $\text{LM}$N-inverse. A Moore-Penrose type inverse is obtained by choosing N = 0. Some optimization problems are considered by choosing

and

as inner product spaces. Our results extend without any major modification of proofs to bounded linear operators with closed range on Hilbert spaces.     

Limit sets of dynamical systems on the space of probability measures

Let m be a dynamical system on the space of probability measures $\text{M}$₁(R^d), and let Λ + (?) be the positive limit set for ? ∈ $\text{M}$₁(R^d), where ? has compact support K ?R^d. The main result of this paper states that support of Λ⁺(?) ?

,support of Λ + (δ_x), where δ_x is the Dirac measure at point x.     

Confidence intervals for marginal parameters under fractional linear regression imputation for missing data

Item nonresponse occurs frequently in sample surveys and other applications. Imputation is commonly used to fill in the missing item values in a random sample {Y_i;i=1,…,n}. Fractional linear regression imputation, based on the model with independent zero mean errors ?_i, is used to create one or more imputed values in the data file for each missing item Y_i, where {X_i,i=1,…,n}, is observed completely. Asymptotic normality of the imputed estimators of the mean μ=E(Y), distribution function θ=F(y) for a given y, and qth quantile θ_q=F^-1(q),0<q<1 is established, assuming that Y is missing at random (MAR) given X. This result is used to obtain normal approximation (NA)-based confidence intervals on μ,θ and θ_q. In the case of θ_q, a Bahadur-type representation and Woodruff-type confidence intervals are also obtained. Empirical likelihood (EL) ratios are also obtained and shown to be asymptotically scaled variables. This result is used to obtain asymptotically correct EL-based confidence intervals on μ,θ and θ_q. Results of a simulation study on the finite sample performance of NA-based and EL-based confidence intervals are reported.     

The Dirichlet Markov Ensemble

We equip the polytope of n×n Markov matrices with the normalized trace of the Lebesgue measure of Rⁿ². This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean (1/n,…,1/n). We show that if is such a random matrix, then the empirical distribution built from the singular values of tends as n→∞ to a Wigner quarter-circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of tends as n→∞ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of is of order when n is large.     

Successive approximation method for quasilinear impulsive differential equations with control

We introduce a technique to define successive approximations to solutions of the control problem with implulse actions on surfaces

where μ is a small positive parameter, ζ_i + θ_i + μτ_i(x(ζ_i), μ), x ϵ Rⁿ and Δx(θ) := x(θ+) − x(θ). A sequence of piecewise continuous functions with discontinuities of the first kind that converges to a solution of the above problem is constructed.