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.     

Minimax and admissible minimax estimators of the mean of a multivariate normal distribution for unknown covariance matrix

Let X be a p-variate (p ≥ 3) vector normally distributed with mean μ and covariance Σ, and let A be a p × p random matrix distributed independent of X, according to the Wishart distribution W(n, Σ). For estimating μ, we consider estimators of the form δ = δ(X, A). We obtain families of Bayes, minimax and admissible minimax estimators with respect to the quadratic loss function (δ ? μ)′ Σ^?1(δ ? μ) where Σ is unknown. This paper extends previous results of the author [1], given for the case in which the covariance matrix of the distribution is of the form σ²I, where σ is known.     

A family of dominating minimax estimators of a multivariate normal mean

Let X have a p-variate normal distribution with mean vector θ and identity covariance matrix I. In the squared error estimation of θ, Baranchik (1970) gives a wide family G of minimax estimators. In this paper, a subfamily C of dominating estimators in G is found such that for each estimator δ₁ in G not in C, there exists an estimator δ₂ in C which which dominates δ₁.     

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.     

Contracting towards subspaces when estimating the mean of a multivariate normal distribution

The problem of estimating, under unweighted quadratic loss, the mean of a multinormal random vector X with arbitrary covariance matrix V is considered. The results of James and Stein for the case V = I have since been extended by Bock to cover arbitrary V and also to allow for contracting X towards a subspace other than the origin; minimax estimators (other than X) exist if and only if the eigenvalues of V are not “too spread out.” In this paper a slight variation of Bock's estimator is considered. A necessary and sufficient condition for the minimaxity of the present estimator is (1): the eigenvalues of (I ? P) V should not be “too spread out,” where P denotes the projection matrix associated with the subspace towards which X is contracted. The validity of (1) is then examined for a number of patterned covariance matrices (e.g., intraclass covariance, tridiagonal and first order autocovariance) and conditions are given for (1) to hold when contraction is towards the origin or towards the common mean of the components of X. (1) is also examined when X is the usual estimate of the regression vector in multiple linear regression. In several of the cases considered the eigenvalues of V are “too spread out” while those of (I ? P) V are not, so that in these instances the present method can be used to produce a minimax estimate.     

Minimax estimation of means of multivariate normal mixtures

Assume X = (X₁, …, X_p)′ is a normal mixture distribution with density w.r.t. Lebesgue measure, , where Σ is a known positive definite matrix and F is any known c.d.f. on (0, ∞). Estimation of the mean vector under an arbitrary known quadratic loss function _Q(θ, a) = (a − θ)′ Q(a − θ), Q a positive definite matrix, is considered. An unbiased estimator of risk is obatined for an arbitrary estimator, and a sufficient condition for estimators to be minimax is then achieved. The result is applied to modifying all the Stein estimators for the means of independent normal random variables to be minimax estimators for the problem considered here. In particular the results apply to the Stein class of limited translation estimators.     

On Bayes and unbiased estimators of loss

We consider estimation of loss for generalized Bayes or pseudo-Bayes estimators of a multivariate normal mean vector, θ. In 3 and higher dimensions, the MLEX is UMVUE and minimax but is inadmissible. It is dominated by the James-Stein estimator and by many others. Johnstone (1988, On inadmissibility of some unbiased estimates of loss,Statistical Decision Theory and Related Topics, IV (eds. S. S. Gupta and J. O. Berger), Vol. 1, 361–379, Springer, New York) considered the estimation of loss for the usual estimatorX and the James-Stein estimator. He found improvements over the Stein unbiased estimator of risk. In this paper, for a generalized Bayes point estimator of θ, we compare generalized Bayes estimators to unbiased estimators of loss. We find, somewhat surprisingly, that the unbiased estimator often dominates the corresponding generalized Bayes estimator of loss for priors which give minimax estimators in the original point estimation problem. In particular, we give a class of priors for which the generalized Bayes estimator of θ is admissible and minimax but for which the unbiased estimator of loss dominates the generalized Bayes estimator of loss. We also give a general inadmissibility result for a generalized Bayes estimator of loss. Research supported by NSF Grant DMS-97-04524.     

Trimmed estimates in simultaneous estimation of parameters in exponential families

Let X₁,…, X_p be p (≥ 3) independent random variables, where each X_i has a distribution belonging to the one-parameter exponential family of distributions. The problem is to estimate the unknown parameters simultaneously in the presence of extreme observations. C. Stein (Ann. Statist.9 (1981), 1135–1151) proposed a method of estimating the mean vector of a multinormal distribution, based on order statistics corresponding to the |X_i|'s, which permitted improvement over the usual maximum likelihood estimator, for long-tailed empirical distribution functions. In this paper, the ideas of Stein are extended to the general discrete and absolutely continuous exponential families of distributions. Adaptive versions of the estimators are also discussed.     

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 the estimation of a restricted normal mean

Let X ∼ N(θ,1), where θ ϵ [−m, m], for some m > 0, and consider the problem of estimating θ with quadratic loss. We show that the Bayes estimator δ_m, corresponding to the uniform prior on [−m, m], dominates δ⁰ (x) = x on [−m, m] and it also dominates the MLE over a large part of the parameter interval. We further offer numerical evidence to suggest that δ_m has quite satisfactory risk performance when compared with the minimax estimators proposed by Casella and Strawderman (1981) and the estimators proposed by Bickel (1981).     

A Grothendieck-type theorem for the space of totally measurable functions

Let Σ be a σ-algebra of subsets of a non-empty set Ω. Let X be a real Banach space and let X^* stand for the Banach dual of X. Let B(Σ, X) be the Banach space of Σ-totally measurable functions f: Ω → X, and let B(Σ, X)^* and B(Σ, X)^** denote the Banach dual and the Banach bidual of B(Σ, X) respectively. Let bvca(Σ, X^*) denote the Banach space of all countably additive vector measures ν: Σ → X^* of bounded variation. We prove a form of generalized Vitali-Hahn-Saks theorem saying that relative σ(bvca(Σ, X^*), B(Σ, X))-sequential compactness in bvca(Σ, X^*) implies uniform countable additivity. We derive that if X reflexive, then every relatively σ(B(Σ, X)^*, B(Σ, X))-sequentially compact subset of B(Σ, X)_c^~ (= the σ-order continuous dual of B(Σ, X)) is relatively σ(B(Σ, X)^*, B(Σ, X)^**)-sequentially compact. As a consequence, we obtain a Grothendieck type theorem saying that σ(B(Σ, X)^*, B(Σ, X))-convergent sequences in B(Σ, X)_c^~ are σ(B(Σ, X)^*, B(Σ, X)^**)-convergent.     

Real parabolic vector bundles over a real curve

We define real parabolic structures on real vector bundles over a real curve. Let (X, σ _X ) be a real curve, and let S???X be a non-empty finite subset of X such that σ _X (S)?=?S. Let N?≥?2 be an integer. We construct an N-fold cyclic cover p : Y→ X in the category of real curves, ramified precisely over each point of S, and with the property that for any element g of the Galois group Γ, and any y?∈?Y, one has $\sigma_Y(gy) = g^{-1}\sigma_Y(y)$ . We established an equivalence between the category of real parabolic vector bundles on (X, σ _X ) with real parabolic structure over S, all of whose weights are integral multiples of 1/N, and the category of real Γ-equivariant vector bundles on (Y, σ _Y ).     

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.     

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.     

Modulus of stability for vector fields on 3-manifolds

Let X be a vector field on M³ which exhibits a saddle connection between a singularity p₁ and a periodic orbit σ₁. We give necessary conditions and also sufficient ones in order to have the finite modulus of stability. They rely heavily upon restrictions on the behaviour of p₁ and σ₁.     

Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix

In the general Gauss-Markoff model (Y, Xβ, σ²V), when V is singular, there exist linear functions of Y which vanish with probability 1 imposing some restrictions on Y as well as on the unknown β. In all earlier work on linear estimation, representations of best-linear unbiased estimators (BLUE's) are obtained under the assumption: “L′Y is unbiased for Xβ ? L′X = X.” Such a condition is not, however, necessary. The present paper provides all possible representations of the BLUE's some of which violate the condition L′X = X. Representations of X for given classes of BLUE's are also obtained.     

Estimation of a Mean Vector in a Two-Sample Problem

We consider the problem of estimating a p-dimensional vector μ₁ based on independent variables X₁, X₂, and U, where X₁ is N_p(μ₁, σ²Σ₁), X₂ is N_p(μ₂, σ²Σ₂), and U is σ²χ²_n (Σ₁ and Σ₂ are known). A family of minimax estimators is proposed. Some of these estimators can be obtained via Bayesian arguments as well. Comparisons between our results and the one of Ghosh and Sinha (1988, J. Multivariate Anal.27 206-207) are presented.     

Adaptive estimation in circular functional linear models

We consider the problem of estimating the slope parameter in circular functional linear regression, where scalar responses Y ₁, ..., Y _n are modeled in dependence of 1-periodic, second order stationary random functions X ₁, ...,X _n . We consider an orthogonal series estimator of the slope function β, by replacing the first m theoretical coefficients of its development in the trigonometric basis by adequate estimators. We propose a model selection procedure for m in a set of admissible values, by defining a contrast function minimized by our estimator and a theoretical penalty function; this first step assumes the degree of ill-posedness to be known. Then we generalize the procedure to a random set of admissible m’s and a random penalty function. The resulting estimator is completely data driven and reaches automatically what is known to be the optimal minimax rate of convergence, in terms of a general weighted L ²-risk. This means that we provide adaptive estimators of both β and its derivatives.     

On the approximate behavior of the posterior distribution for an extreme multivariate observation

The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let X = (X₁, X₂, …, X_n) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ₁, Θ₂, …, Θ_n). Let θ_x^* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θ_x^* given X = x to converge to a multivariate normal with mean vector 0 as |x| tends to infinity. These same conditions imply that E(Θ | X = x) − θ_x^* converges to the zero vector as |x| tends to infinity.The posterior for an observation X = (X₁, X₂, …, X_n is considered for a location vector Θ = (Θ₁, Θ₂, …, Θ_n) as x gets large along a path, γ, in Rⁿ. Sufficient conditions are given for the distribution of γ(t) − Θ given X = γ(t) to converge in law as t → ∞. Slightly stronger conditions ensure that γ(t) − E(Θ | X = γ(t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ − E(Θ | X = γ(t)) given X = γ(t) converges in law to a symmetric distribution as t → ∞, it is shown that δ(γ(t)) − E(Θ | X = γ(t)) → 0 as t → ∞.