Empirical Bayes testing for equivalence

For a fixed point θ₀ and a positive value c₀, this paper studies the problem of testing the hypotheses H₀:|θ−θ₀|≤c₀ against H₁:|θ−θ₀|>c₀ for the normal mean parameter θ using the empirical Bayes approach. With the accumulated past data, a monotone empirical Bayes test is constructed by mimicking the behavior of a monotone Bayes test. Such an empirical Bayes test is shown to be asymptotically optimal and its regret converges to zero at a rate (lnn)^2.5/n where n is the number of past data available, when the current testing problem is considered. A simulation study is also given, and the results show that the proposed empirical Bayes procedure has good performance for small to moderately large sample sizes. Our proposed method can be applied for testing close to a control problem or testing the therapeutic equivalence of one standard treatment compared to another in clinical trials.     

Empirical Bayes Testing for the Mean Lifetime of Exponential Distributions: Unequal Sample Sizes Case

This paper deals with an empirical Bayes testing problem for the mean lifetimes of exponential distributions with unequal sample sizes. We study a method to construct empirical Bayes tests {δ* _nl + 1,n }^∞ _n = 1 for the sequence of the testing problems. The asymptotic optimality of {δ* _nl + 1,n }^∞ _n = 1 is studied. It is shown that the sequence of empirical Bayes tests {δ* _nl + 1,n }^∞ _n = 1 is asymptotically optimal, and its associated sequence of regrets converges to zero at a rate (ln n)^4M?1/n, where M is an upper bound of sample sizes.     

Empirical Bayes testing for guarantee lifetime: Non identical components case

This article considers an empirical Bayes testing problem for the guarantee lifetime in the two-parameter exponential distributions with non identical components. We study a method of constructing empirical Bayes tests under a class of unknown prior distributions for the sequence of the component testing problems. The asymptotic optimality of the sequence of empirical Bayes tests is studied. Under certain regularity conditions on the prior distributions, it is shown that the sequence of the constructed empirical Bayes tests is asymptotically optimal, and the associated sequence of regrets converges to zero at a rate O(n^{? 1 + 1/[2(r + α) + 1]}) for some integer r ? 0 and 0 ? α ? 1 depending on the unknown prior distributions, where n is the number of past data available when the (n + 1)st component testing problem is considered.     

Robust empirical Bayes tests for continuous distributions

In this paper, we study the empirical Bayes two-action problem under linear loss function. Upper bounds on the regret of empirical Bayes testing rules are investigated. Previous results on this problem construct empirical Bayes tests using kernel type estimators of nonparametric functionals. Further, they have assumed specific forms, such as the continuous one-parameter exponential family for {F_θ:θΩ}, for the family of distributions of the observations. In this paper, we present a new general approach of establishing upper bounds (in terms of rate of convergence) of empirical Bayes tests for this problem. Our results are given for any family of continuous distributions and apply to empirical Bayes tests based on any type of nonparametric method of functional estimation. We show that our bounds are very sharp in the sense that they reduce to existing optimal or nearly optimal rates of convergence when applied to specific families of distributions.     

M-Estimators of Scale with Minimum Gross Errors Sensitivity

The median absolute deviation (MAD) is known to be the M-estimator of scale with minimum gross errors sensitivity (GES) when the error distribution is known to be symmetric and strongly unimodal. The problem considered here is to find the Fisher consistent M-estimator with minimum GES when the error distribution is symmetric but not necessarily unimodal. Under some general conditions, the score function χ corresponding to the minimizing M-estimator has the form χ(x) = ?1 when |x| < a; χ(x) = c when a < |x| < b; χ(x) = 1 when |x| > b. An example is given in which the M-estimator with minimum GES does not correspond to the MAD.     

A Lower Bound to the Measure of Optimality for Main Effect Plans in the Symmetric Factorial Experiments

A design d is called D-optimal if it maximizes det(M _d ), and is called MS-optimal if it maximizes tr(M _d ) and minimizes tr[(M _d )²] among those which maximize tr(M _d ), where M _d stands for the information matrix produced from d under a given model. In this article, we establish a lower bound for tr[(M _d )²] with respect to a main effects model, where d is an s-level symmetric orthogonal array of strength at least one. Non isomorphic two level MS-optimal orthogonal arrays of strength one with N = 10, 14, and 18 runs, non isomorphic three level MS-optimal orthogonal arrays of strength one with N = 6, 12, and 15 runs and non isomorphic four level MS-optimal orthogonal arrays of strength one with N = 12 runs are also presented.     

Bayes minimax ridge regression estimators

The problem of estimating of the vector β of the linear regression model y = Aβ + ? with ? ～ N_p(0, σ²I_p) 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 Maruyama, Y., and W. E. Strawderman. 2005. A new class of generalized Bayes minimax ridge regression estimators. Annals of Statistics 33:1753–70.[Crossref], [Web of Science ®] , [Google Scholar]) 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 Maruyama, Y., and W. E. Strawderman. 2005. A new class of generalized Bayes minimax ridge regression estimators. Annals of Statistics 33:1753–70.[Crossref], [Web of Science ®] , [Google Scholar]) 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 Parameters of a Clipped MA(1) Process

Let {X _t , t ∈ ?} be a sequence of iid random variables with an absolutely continuous distribution. Let a > 0 and c ∈ ? be some constants. We consider a sequence of 0-1 valued variables {ξ _t , t ∈ ?} obtained by clipping an MA(1) process X _t  ? aX _t?1 at the level c, i.e., ξ _t  = I[X _t  ? aX _t?1 < c] for all t ∈ ?. We deal with the estimation problem in this model. Properties of the estimators of the parameters a and c, the success probability p, and the 1-lag autocorrelation r ₁ are investigated. A numerical study is provided as an illustration of the theoretical results.     

Empirical Bayes vs. fully Bayes variable selection

For the problem of variable selection for the normal linear model, fixed penalty selection criteria such as AIC, _Cp${\mathrm{C}}_p$, BIC and RIC correspond to the posterior modes of a hierarchical Bayes model for various fixed hyperparameter settings. Adaptive selection criteria obtained by empirical Bayes estimation of the hyperparameters have been shown by George and Foster [2000. Calibration and Empirical Bayes variable selection. Biometrika 87(4), 731–747] to improve on these fixed selection criteria. In this paper, we study the potential of alternative fully Bayes methods, which instead margin out the hyperparameters with respect to prior distributions. Several structured prior formulations are considered for which fully Bayes selection and estimation methods are obtained. Analytical and simulation comparisons with empirical Bayes counterparts are studied.     

Quadratic subspaces and construction of Bayes invariant quadratic estimators of variance components in mixed linear models

Gnot et al. (J Statist Plann Inference 30(1):223–236, 1992) have presented the formulae for computing Bayes invariant quadratic estimators of variance components in normal mixed linear models of the form where the matrices V _i , 1 ≤ i ≤ k − 1, are symmetric and nonnegative definite and V _k is an identity matrix. These formulae involve a basis of a quadratic subspace containing MV ₁ M,...,MV _k-1 M,M, where M is an orthogonal projector on the null space of X′. In the paper we discuss methods of construction of such a basis. We survey Malley’s algorithms for finding the smallest quadratic subspace including a given set of symmetric matrices of the same order and propose some modifications of these algorithms. We also consider a class of matrices sharing some of the symmetries common to MV ₁ M,...,MV _k-1 M,M. We show that the matrices from this class constitute a quadratic subspace and describe its explicit basis, which can be directly used for computing Bayes invariant quadratic estimators of variance components. This basis can be also used for improving the efficiency of Malley’s algorithms when applied to finding a basis of the smallest quadratic subspace containing the matrices MV ₁ M,...,MV _k-1 M,M. Finally, we present the results of a numerical experiment which confirm the potential usefulness of the proposed methods. Dedicated to the memory of Professor Stanisław Gnot.     

Empirical bayes interval estimates involving uniform distributions

Bayesian and empirical Bayesian decision rules are exhibited for the interval estimation of the parameter 0 of a Uniform (0,θ) distribution. The estimate ?,δ>resulting in the interval [?,?+δ]suffers loss given by L(?,δ>,θ)=1-[?≦e≦?+δ]+c₁((?-θ)²+(?+δ?θ)²))+c₂δ. The solution is presented for prior distributions G which have bounded support, no point masses,∫θ^?mdG(θ)<∞ and for some integer m. An example is presented involving a particular parametric form for G and rates of risk convergence in the empirical Bayes problem for this example are calculated.     

Optimality of the posterior predictive p-value based on the posterior. Odds

Thompson (1997) considered a wide definition of p-value and found the Baves p-value for testing a ooint null hypothesis H₀: θ= θ₀ versus H₁: θ ≠ θ₀. In this paper, the general case of testing H₀: θ ∈ ?₀ versus H₁: θ ∈ ?^c ₀ is studied. A generalization of the concept of p-value is given, and it is proved that the posterior predictive p-value based on the posterior odds is (asymptotically) a Bayes p-value. Finally, it is suggested that this posterior predictive p-value could be used as a reference p-value     

Minimax Estimation of the Bounded Parameter of Some Discrete Distributions Under LINEX Loss Function

For a class of discrete distributions, including Poisson(θ), Generalized Poisson(θ), Borel(m, θ), etc., we consider minimax estimation of the parameter θ under the assumption it lies in a bounded interval of the form [0, m] and a LINEX loss function. Explicit conditions for the minimax estimator to be Bayes with respect to a boundary supported prior are given. Also for Bernoulli(θ)-distribution, which is not in the mentioned class of discrete distributions, we give conditions for which the Bayes estimator of θ ∈ [0, m], m < 1 with respect to a boundary supported prior is minimax under LINEX loss function. Numerical values are given for the largest values of m for which the corresponding Bayes estimators of θ are minimax.     

Nonparametric empirical Bayes for the Dirichlet process mixture model

The Dirichlet process prior allows flexible nonparametric mixture modeling. The number of mixture components is not specified in advance and can grow as new data arrive. However, analyses based on the Dirichlet process prior are sensitive to the choice of the parameters, including an infinite-dimensional distributional parameter G ₀. Most previous applications have either fixed G ₀ as a member of a parametric family or treated G ₀ in a Bayesian fashion, using parametric prior specifications. In contrast, we have developed an adaptive nonparametric method for constructing smooth estimates of G ₀. We combine this method with a technique for estimating α, the other Dirichlet process parameter, that is inspired by an existing characterization of its maximum-likelihood estimator. Together, these estimation procedures yield a flexible empirical Bayes treatment of Dirichlet process mixtures. Such a treatment is useful in situations where smooth point estimates of G ₀ are of intrinsic interest, or where the structure of G ₀ cannot be conveniently modeled with the usual parametric prior families. Analysis of simulated and real-world datasets illustrates the robustness of this approach.