Asymptotic near admissimlity and asymptotic near optimality by the “two armed bandit” problem

The classical “Two Armed Bandit” problem with Bernoulli-distributed outcomes is being considered. First the terms “asymptotic nearly admissibility” and “asymptotic nearly optimality” are defined. A nontrivial asymptotic nearly admissible and (with respect to a certain Bayes risk) asymptotic nearly optimal strategy is presented, then these properties are shown. Finally, it is discussed how these results generalize to the non-Bernoulli cases and the “k-Armed Bandit” problem (;k≧2).     

m个方差分量的非负二次同时估计的可容许性

给出了ｍ个方差分量的模型Ｅ（Ｙｎ×１）＝Ｘｎ×ｐβｐ×１,ｃｏｖ（Ｙ）＝∑ｍｉ＝１σ２ｉＶｉ的方差分量（σ２１,σ２２,…,σ２ｍ）的非负二次同时估计可容许的一个必要条件,推广了相关论文中的结果     

A test of the mean square error criterion for shrinkage estimators

A test for choosing between a shrinkage estimator and the least squares estimator is described and a central-F approximation to the test statistic is considered. An example from the literature was analysed using the test procedure proposed here. The power of the test was studied by means of simulation.     

Admissible two-stage designs for phase II cancer clinical trials that incorporate the expected sample size under the alternative hypothesis

two‐stage studies may be chosen optimally by minimising a single characteristic like the maximum sample size. However, given that an investigator will initially select a null treatment e?ect and the clinically relevant di?erence, it is better to choose a design that also considers the expected sample size for each of these values. The maximum sample size and the two expected sample sizes are here combined to produce an expected loss function to ?nd designs that are admissible. Given the prior odds of success and the importance of the total sample size, minimising the expected loss gives the optimal design for this situation. A novel triangular graph to represent the admissible designs helps guide the decision‐making process. The H ₀‐optimal, H ₁‐optimal, H ₀‐minimax and H ₁‐minimax designs are all particular cases of admissible designs. The commonly used H ₀‐optimal design is rarely good when allowing stopping for e?cacy. Additionally, the δ‐minimax design, which minimises the maximum expected sample size, is sometimes admissible under the loss function. However, the results can be varied and each situation will require the evaluation of all the admissible designs. Software to do this is provided. Copyright © 2012 John Wiley & Sons, Ltd.     

Admissibility in Games1

Suppose that each player in a game is rational, each player thinks the other players are rational, and so on. Also, suppose that rationality is taken to incorporate an admissibility requirement—that is, the avoidance of weakly dominated strategies. Which strategies can be played? We provide an epistemic framework in which to address this question. Specifically, we formulate conditions of rationality and mth‐order assumption of rationality (RmAR) and rationality and common assumption of rationality (RCAR). We show that (i) RCAR is characterized by a solution concept we call a “self‐admissible set”; (ii) in a “complete” type structure, RmAR is characterized by the set of strategies that survive m+1 rounds of elimination of inadmissible strategies; (iii) under certain conditions, RCAR is impossible in a complete structure.     

Linearly admissible estimators on linear functions of regression coefficient under balanced loss function

Linearly admissible estimators on linear functions of regression coefficient are studied in a singular linear model and balanced loss when the design matrix has not full column rank. The sufficient and necessary conditions for linear estimators to be admissible are obtained respectively in homogeneous and inhomogeneous classes.     

Limiting admissible estimators for variance components

Simultaneous estimation of the vector of the variance components for mixed and random models under the quadratic loss function is considered. For a large class of such models there are identified classes of admissible biased invariant quadratic estimators that are better than some admissible unbiased estimators. Numerous numerical results presented in the paper show that for many of the commonly used balanced models the improvements in the quadratic risk may be considerable over a large set of the parameter space.     

Monotonic minimax estimators of a 2×2 covariance matrix

Let S : 2 × 2 have a nonsingular Wishart distribution with unknown matrix σ and n degrees of freedom. For estimating σ two families of mimmax estimators, with respect to the entropy loss, are presented. These estimators are of the form σ(S) = Rø(L)R^t where R is orthogonal, L and Φ are diagonal, and RLR^T = S. Conditions under which the components of Φ and L follow the same order relation [i.e., writing Φ = diag(Φ₁,Φ₂) and L = diag(l₁,/₂) with l₁ ≥ l₂, we have Φ₁ ≥ Φ₂] are established. Comparisons with Stein's estimators and other orthogonally invariant estimators are discussed.     

Locally minimax tests for multiple correlations

Let X be a normally distributed p-dimensional column vector with mean μ and positive definite covariance matrix σ. and let X α, α = 1,…, N, be a random sample of size N from this distribution. Partition X as ( X ₁, X ₍₂₎', X '₍₃₎)', where X₁ is one-dimension, X₍₂₎ is p₂- dimensional, and so 1 + p₁ + p₂ = p. Let ρ₁ and ρ be the multiple correlation coefficients of X₁ with X₍₂₎ and with ( X '₍₂₎, X '₍₃₎)', respectively. Write ρ2/2 = ρ² - ρ2/1. We shall cosider the following two problems     

A note on the admissibility of hammersley's estimator of an integer mean

Let X₁, X₂, …, X_n be identically, independently distributed N(i,1) random variables, where i = 0, ±1, ±2, … Hammersley (1950) showed that d = [X?_n], the nearest integer to the sample mean, is the maximum likelihood estimator of i. Khan (1973) showed that d is minimax and admissible with respect to zero-one loss. This note now proves a conjecture of Stein to the effect that in the class of integer-valued estimators d is minimax and admissible under squared-error loss.