## Adaptive Statistical Procedures and Related Topics: Proceedings of a Symposium in Honor of Herbert Robbins, June 7-11, 1985, Brookhaven National Laboratory, Upton, New York |

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### Common terms and phrases

1980 subject classifications adaptive RM algorithm AMS 1980 subject applications assume assumption asymptotically optimal Bayes estimator Bayes risk Bayes rules bounded censoring central limit theorem component compound decision problem computing consider consistent estimators constant convergence Corollary decision problem defined denote density distribution function empirical Bayes approach empirical Bayes estimation empirical Bayes rules equation equivariant error example exists finite fixed follows given Hence holds implies independent inequality integer iterated Lemma linear logit logit-MLE Math maximum likelihood estimate mean method minimax ML recursion nonadaptive nonparametric observations obtain optimal stopping order statistics parameter population posterior Primal prior distribution probability procedures projection pursuit quantile random variables Robbins-Monro Ryzin sample satisfies score function Section sequence sequential solution stochastic approximation stopping rule Suppose symmetric tests Theorem 3.1 theory uniformly unknown variance vector Wiener process zero

### Popular passages

Page 116 - A01 CSCL 12/1 This paper deals with the problem of selecting good binomial populations compared with a standard or a control through the empirical Bayes approach. Two cases have been studied: one with the prior distribution completely unknown and the other with the prior distribution symmetrical about p = 1/2, but otherwise unknown. In each case, empirical Bayes rules are derived and their rates of convergence are shown to be of order O(exp(-cn)) for some c > O, where n is the number of accumulated...

Page 458 - The main result of this section is contained in the following theorem.

Page 286 - Blum, JR (1954). Approximation methods which converge with probability one. Ann. Math. Statist. 25 382-386.

Page 117 - Bayes rules, prior distribution, asymptotically optimal, rate of convergence. to be asymptotically optimal In the sense that the risk for the n-th decision problem converges to the optimal Bayes risk which would have been obtained if the prior distribution was known and the Bayes rule with respect to this prior distribution was used. Empirical Bayes rules have been derived for multiple decision problems by Deely (1965) for selecting a subset containing the best population.

Page 337 - Some Limit Theorems in Statistics. SIAM, Philadelphia. Bahadur, RR , and Raghavachari , M. (1972). "Some Asymptotic Properties of Likelihood Ratios on General Sample Spaces,

Page 321 - This research was supported in part by National Science Foundation Grant No.

Page 282 - white noise" is a stochastic process which satisfies the following three assumptions. i) w(t) is a martingale difference sequence with respect to an increasing sequence of cr-algebras {Ft}', ie, E[w(t) \ /-,_,] = Oa.s.,V<.

Page 116 - Bayes approach in statistical decision theory is appropriate when one is confronted repeatedly and independently with the same decision problem. In such instances, it is reasonable to formulate the component problem in the sequence as a Bayes decision problem with respect to an unknown prior distribution on the parameter space and then use the accumulated observations to improve the decision rule at each stage. This approach is due to Robbins (1956, 1964, 1983).

Page 304 - ... discrete) . The proposed approach is based on design updating with the maximum likelihood estimate via a parametric model. It is dubbed the maximum likelihood (ML) recursion approach. In several important situations it is shown to be closely related to the stochastic approximation approach of Robbins and Monro (1951). The problem can be described as follows . The response y is related to an underlying "design

Page 390 - O. (1976). Maximum likelihood estimation in the three-parameter lognormal distribution, J. Roy. Statist. Soc., Ser. B, 38, 257-264.