Generalized Bayes Minimax Estimators Research Articles

Bayes minimax ridge regression estimators

ABSTRACTThe problem of estimating of the vector β of the linear regression model y = Aβ + ϵ with ϵ ∼ Np(0, σ2Ip) under quadratic loss function is considered when common variance σ2 is unknown. We first find a class of minimax estimators for this problem which extends a class given by Maruyama and Strawderman (2005) and using these estimators, we obtain a large class of (proper and generalized) Bayes minimax estimators and show that the result of Maruyama and Strawderman (2005) is a special case of our result. We also show that under certain conditions, these generalized Bayes minimax estimators have greater numerical stability (i.e., smaller condition number) than the least-squares estimator.

Estimation of the mean vector in a singular multivariate normal distribution

This paper addresses the problem of estimating the mean vector of a singular multivariate normal distribution with an unknown singular covariance matrix. The maximum likelihood estimator is shown to be minimax relative to a quadratic loss weighted by the Moore–Penrose inverse of the covariance matrix. An unbiased risk estimator relative to the weighted quadratic loss is provided for a Baranchik type class of shrinkage estimators. Based on the unbiased risk estimator, a sufficient condition for the minimaxity is expressed not only as a differential inequality, but also as an integral inequality. Also, generalized Bayes minimax estimators are established by using an interesting structure of singular multivariate normal distribution.

Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions with residual vector

We consider estimation of the mean vector, \(\theta \), of a spherically symmetric distribution with known scale parameter under quadratic loss and when a residual vector is available. We show minimaxity of generalized Bayes estimators corresponding to superharmonic priors with a non decreasing Laplacian of the form \(\pi (\Vert \theta \Vert ^{2})\), under certain conditions on the generating function \(f(\cdot )\) of the sampling distributions. The class of sampling distributions includes certain variance mixtures of normals.

Minimax estimation of normal precisions via expansion estimators

In this paper, the simultaneous estimation of the precision parameters of k normal distributions is considered under the squared loss function in a decision-theoretic framework. Several classes of minimax estimators are derived by using the chi-square identity, and the generalized Bayes minimax estimators are developed out of the classes. It is also shown that the improvement on the unbiased estimators is characterized by the superharmonic function. This corresponds to Stein's [1981. Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9, 1135–1151] result in simultaneous estimation of normal means.

Generalized Bayes minimax estimators of the mean of multivariate normal distribution with unknown variance

We construct a broad class of generalized Bayes minimax estimators of the mean of a multivariate normal distribution with covariance equal to σ 2 I p , with σ 2 unknown, and under the invariant loss ‖ δ ( X ) − θ ‖ 2 / σ 2 . Examples that illustrate the theory are given. Most notably it is shown that a hierarchical version of the multivariate Student- t prior yields a Bayes minimax estimate.

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 δ 0 ( X ) = X under certain conditions on f ( ) . The class of priors includes priors of the form 1 A + ∥ θ ∥ 2 k for k ⩽ p 2 - 1 and hence includes the fundamental harmonic prior 1 ∥ θ ∥ p - 2 . The class of sampling distributions includes certain variance mixtures of normals and other functions f ( t ) of the form e - α t β and e - α t + β φ ( 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 δ π ( X ) = 1 - ar ( t ) t X .

Bayes minimax estimators of the mean of a scale mixture of multivariate normal distributions

Bayes estimation of the mean of a variance mixture of multivariate normal distributions is considered under sum of squared errors loss. We find broad class of priors (also in the variance mixture of normal class) which result in proper and generalized Bayes minimax estimators. This paper extends the results of Strawderman [Minimax estimation of location parameters for certain spherically symmetric distribution, J. Multivariate Anal. 4 (1974) 255–264] in a manner similar to that of Maruyama [Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distribution, J. Multivariate Anal. 21 (2003) 69–78] but somewhat more in the spirit of Fourdrinier et al. [On the construction of bayes minimax estimators, Ann. Statist. 26 (1998) 660–671] for the normal case, in the sense that we construct classes of priors giving rise to minimaxity. A feature of this paper is that in certain cases we are able to construct proper Bayes minimax estimators satisfying the properties and bounds in Strawderman [Minimax estimation of location parameters for certain spherically symmetric distribution, J. Multivariate Anal. 4 (1974) 255–264]. We also give some insight into why Strawderman's results do or do not seem to apply in certain cases. In cases where it does not apply, we give minimax estimators based on Berger's [Minimax estimation of location vectors for a wide class of densities, Ann. Statist. 3 (1975) 1318–1328] results. A main condition for minimaxity is that the mixing distributions of the sampling distribution and the prior distribution satisfy a monotone likelihood ratio property with respect to a scale parameter.

A new class of generalized Bayes minimax ridge regression estimators

Let y=Aβ+ɛ, where y is an N×1 vector of observations, β is a p×1 vector of unknown regression coefficients, A is an N×p design matrix and ɛ is a spherically symmetric error term with unknown scale parameter σ. We consider estimation of β under general quadratic loss functions, and, in particular, extend the work of Strawderman [J. Amer. Statist. Assoc. 73 (1978) 623–627] and Casella [Ann. Statist. 8 (1980) 1036–1056, J. Amer. Statist. Assoc. 80 (1985) 753–758] by finding adaptive minimax estimators (which are, under the normality assumption, also generalized Bayes) of β, which have greater numerical stability (i.e., smaller condition number) than the usual least squares estimator. In particular, we give a subclass of such estimators which, surprisingly, has a very simple form. We also show that under certain conditions the generalized Bayes minimax estimators in the normal case are also generalized Bayes and minimax in the general case of spherically symmetric errors.

Stein's idea and minimax admissible estimation of a multivariate normal mean

We consider estimation of a multivariate normal mean vector under sum of squared error loss.We propose a new class of minimax admissible estimator which are generalized Bayes with respect to a prior distribution which is a mixture of a point prior at the origin and a continuous hierarchical type prior. We also study conditions under which these generalized Bayes minimax estimators improve on the James–Stein estimator and on the positive-part James–Stein estimator.

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 ∫ 1 (2πσ 2) e −( 1 2σ 2 )‖X−θ‖ 2 dG(σ) , 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 (1 − ar(X′X) E 0( 1 X′X )X′X)X where 0 ≤ a ≤ 2, r( X′ X) is nondecreasing, and r(X′X) X′X is nonincreasing. Generalized Bayes minimax estimators are found for certain G(·)'s.

Generalized Bayes Minimax Estimators of the Multivariate Normal Mean with Unknown Covariance Matrix

Let $\mathbf{X}$ be a $p$-variate $(p \geqq 3)$ vector normally distributed with mean $\mathbf{\theta}$ and covariance matrix $\Sigma$, positive definite but unknown. Let $A$ be a $p \times p$ Wishart matrix with parameters $(n, \Sigma)$, independent of $\mathbf{X}$. To estimate $\mathbf{\theta}$ relative to quadratic loss function $(\hat{\mathbf{\theta}} - \mathbf{\theta})'\Sigma^{-1}(\hat{\mathbf{\theta}} - \mathbf{\theta})$, we obtain a family of minimax estimators $\mathbf{\delta}(\mathbf{X}, \mathbf{A})$ based on $\mathbf{X}$ and $\mathbf{A}$ through $\mathbf{X}$ and $\mathbf{X}'\mathbf{A}^{-1}\mathbf{X}$. It is shown that there are minimax estimators of the form $\mathbf{\delta}(\mathbf{X}, \mathbf{A})$ which are also generalized Bayes. A special case where $\Sigma = \sigma^2\mathbf{I}$ is also considered.