Minimax estimation of a multivariate normal mean under polynomial loss

Let X be an observation from a p-variate (p ≥ 3) normal random vector with unknown mean vector θ and known covariance matrix

. The problem of improving upon the usual estimator of θ, δ⁰(X) = X, is considered. An approach is developed which can lead to improved estimators, δ, for loss functions which are polynomials in the coordinates of (δ ? θ). As an example of this approach, the loss L(δ, θ) = |δ ? θ|⁴ is considered, and estimators are developed which are significantly better than δ⁰. When

is the identity matrix, these estimators are of the form $\text{δ(X) = (1 ? (}\text{b}\text{(d + |X|}{}^2\text{)}\text{))X}$.     

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.     

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 .     

Asymptotic behavior of the maximal zero of a polynomial orthogonal on a segment with a nonclassical weight

Let {p _n (t)} _n=0 ^t8 be a system of algebraic polynomials orthonormal on the segment [?1, 1] with a weight p(t); let {x _n,ν ^(p) } _ν=1 ⁿ be zeros of a polynomial p _n (t) (x _x,ν ^(p) = cosθ _n,ν ^(p) ; 0 < θ _n,1 ^(p) < θ _n,2 ^(p) < ... < θ _n,n ^(p) < π). It is known that, for a wide class of weights p(t) containing the Jacobi weight, the quantities θ _n,1 ^(p) and 1 ? x _n,1 ^(p) coincide in order with n ^?1 and n ^?2, respectively. In the present paper, we prove that, if the weight p(t) has the form p(t) = 4(1 ? t ²)^?1{ln²[(1 + t)/(1 ? t)] + π ²}^?1, then the following asymptotic formulas are valid as n → ∞:

$$\theta _{n,1}^{(p)} = \frac{{\sqrt 2 }}{{n\sqrt {\ln (n + 1)} }}\left[ {1 + {\rm O}\left( {\frac{1}{{\ln (n + 1)}}} \right)} \right],x_{n,1}^{(p)} = 1 - \left( {\frac{1}{{n^2 \ln (n + 1)}}} \right) + O\left( {\frac{1}{{n^2 \ln ^2 (n + 1)}}} \right).$$

     

A dependent theory with few indiscernibles

We give a full solution to the question of existence of indiscernibles in dependent theories by proving the following theorem: For every θ there is a dependent theory T of size θ such that for all κ and δ, κ → (δ) _T,1 iff κ → (δ) _θ ^<ω . This means that unless there are good set theoretical reasons, there are large sets with no indiscernible sequences.     

A landau-type construction concerning (L)′ ⊂ L

Let 1 ? p < ∞ and 1/p + 1/q = 1. For a locally finite measure space (X, S, μ) and a measurable complex-valued function f ∉ L^q functions g ∈ L^p may be constructed explicitly which satisfy

     

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.     

The behavior of \sum\nolimits_{n = 1}^\infty {{{\zeta ^{ \llcorner n\theta \lrcorner } } \mathord{\left/ {\vphantom {{\zeta ^{ \llcorner n\theta \lrcorner } } n}} \right. \kern-\nulldelimiterspace} n}} for particular values of θ

Let ζ be a primitive q′-root of unity. We prove that the series $ \sum\nolimits_{n = 1}^\infty {{{\zeta ^{ \llcorner n\theta \lrcorner } } \mathord{\left/ {\vphantom {{\zeta ^{ \llcorner n\theta \lrcorner } } n}} \right. \kern-0em} n}} $ for θ ∈ Q converges if and only if θ = p/q with (p,q) = 1 and q′ ? p, and that there exists an uncountable set S of Liouville’s numbers such that the series does not converge when θ ∈ S.     

On the optimal robust solution of IVPs with noisy information

We investigate the optimal solution of systems of initial-value problems with smooth right-hand side functions f from a Hölder class \(F^{r,\varrho }_{\text {reg}}\), where r ≥ 0 is the number of continuous derivatives of f, and ? ∈ (0, 1] is the Hölder exponent of rth partial derivatives. We consider algorithms that use n evaluations of f, the ith evaluation being corrupted by a noise δ_i of deterministic or random nature. For δ ≥ 0, in the deterministic case the noise δ_i is a bounded vector, ∥δ_i∥≤δ. In the random case, it is a vector-valued random variable bounded in average, (E(∥δ_i∥^q))^1/q ≤ δ, q ∈ [1, + ∞). We point out an algorithm whose L_p error (p ∈ [0, + ∞]) is O(n ^{? (r + ?)} + δ), independently of the noise distribution. We observe that the level n ^{? (r + ?)} + δ cannot be improved in a class of information evaluations and algorithms. For ε > 0, and a certain model of δ-dependent cost, we establish optimal values of n(ε) and δ(ε) that should be used in order to get the error at most ε with minimal cost.     

Description of the Space of Riesz Potentials of Functions in a Grand Lebesgue Space on ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

The Riesz potentials L^af, 0 < α < ∞, are considered in the framework of a grand Lebesgue space L_a^p),θ, 1 < p < ∞, θ > 0, on Rn with grandizers a ∈ L¹(?ⁿ), which are understood in the case α ≥ n/p in terms of distributions on test functions in the Lizorkin space. The images under I^α of functions in a subspace of the grand space which satisfy the so-called vanishing condition is studied. Under certain assumptions on the grandizer, this image is described in terms of the convergence of truncated hypersingular integrals of order α in this subspace.     

Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces

Let θ ∈ (0, 1), λ ∈ [0, 1) and p, p ₀, p ₁ ∈ (1,∞] be such that (1 ? θ)/p ₀ + θ/p ₁ = 1/p, and let φ, φ₀, φ₁ be some admissible functions such that φ, φ₀ ^p/p0 and φ₁ ^p/p1 are equivalent. We first prove that, via the ± interpolation method, the interpolation L _φ0 ^p0),λ (X), L _φ1 ^{p1), λ} (X), θ> of two generalized grand Morrey spaces on a quasi-metric measure space X is the generalized grand Morrey space L _φ ^p),λ (X). Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces.     

Densities for random balanced sampling

A random balanced sample (RBS) is a multivariate distribution with n components X_k, each uniformly distributed on [-1,1], such that the sum of these components is precisely 0. The corresponding vectors lie in an (n-1)-dimensional polytope M(n). We present new methods for the construction of such RBS via densities over M(n) and these apply for arbitrary n. While simple densities had been known previously for small values of n (namely 2,3, and 4), for larger n the known distributions with large support were fractal distributions (with fractal dimension asymptotic to n as n→∞). Applications of RBS distributions include sampling with antithetic coupling to reduce variance, and the isolation of nonlinearities. We also show that the previously known densities (for n?4) are in fact the only solutions in a natural and very large class of potential RBS densities. This finding clarifies the need for new methods, such as those presented here.     

Estimating the mean function of a Gaussian process and the Stein effect

The problem of global estimation of the mean function θ(·) of a quite arbitrary Gaussian process is considered. The loss function in estimating θ by a function a(·) is assumed to be of the form L(θ, a) = ∫ [θ(t) ? a(t)]²μ(dt), and estimators are evaluated in terms of their risk function (expected loss). The usual minimax estimator of θ is shown to be inadmissible via the Stein phenomenon; in estimating the function θ we are trying to simultaneously estimate a larger number of normal means. Estimators improving upon the usual minimax estimator are constructed, including an estimator which allows the incorporation of prior information about θ. The analysis is carried out by using a version of the Karhunen-Loéve expansion to represent the original problem as the problem of estimating a countably infinite sequence of means from independent normal distributions.     

Spectral multipliers on 2-step groups: topological versus homogeneous dimension

Let G be a 2-step stratified group of topological dimension d and homogeneous dimension Q. Let \({\mathcal{L}}\) be a homogeneous sub-Laplacian on G. By a theorem due to Christ and to Mauceri and Meda, an operator of the form \({F(\mathcal{L})}\) is of weak type (1, 1) and bounded on L ^p (G) for all p ∈ (1, ∞) whenever the multiplier F satisfies a scale-invariant smoothness condition of order s > Q/2. It is known that, for several 2-step groups and sub-Laplacians, the threshold Q/2 in the smoothness condition is not sharp and in many cases it is possible to push it down to d/2. Here we show that, for all 2-step groups and sub-Laplacians, the sharp threshold is strictly less than Q/2, but not less than d/2.