Abstract: The optimal number of regressors is determined to minimize mean squared prediction error and is shown to be a small fraction of the number of data points. As the number of regressors grows large, the Sp criterion provides an asymptotically optimal rule for the number of variables to enter.

Abstract: A simple class of quantizers is introduced which are asymptotically optimal, as the number of quantization levels increases to infinity, with respect to a mean r th power absolute error distortion measure. These asymptotically optimal quantizers are very easy to compute. Their performance is evaluated for several distributions and compares favorably with the performance of the optimal quantizers in all cases for which the latter have been computed. In addition their asymptotic robustness is studied under location, scale, and shape mismatch for several families of distributions.

Abstract: Sequential procedures are proposed and studied for point and interval estimation of the difference between the means of two normal populations with unknown variances. The costs of sampling are allowed to be different for the two populations and to depend on the difference between the means. The procedures are shown to be asymptotically optimal, and small sample behaviour is studied by computer simulation.

Abstract: Suppose random variables X 1, X 2, …, with distribution depending on parameter θ1, are observable from population 1, independent of random variables Y 1, Y 2, …, with distribution depending on parameter θ2, and observed from population 2. The simultaneous estimation of parameters θ1 and θ2 is of interest. An experiment is conducted where at each stage either an X or a Y is observed, and the experiment may be stopped at any stage (allocation and decision depending on information at that stage). In such an experiment, a sequential allocation procedure or policy determines whether to observe X or Y at each stage, and a stopping rule determines when to stop, and hence the pair (policy, stopping time) constitutes the design of the experiment. The purpose of this work is the study of pairs of policies and stopping times for simultaneous estimation of θ1 and θ2 using a Bayesian approach with additive estimation loss and unit sampling costs. In this context, optimal and asymptotically optimal pairs are d...

Abstract: Cubature formulas are obtained which are optimal or asymptotically optimal on given sets of functions. These formulas consist of line integrals which may be evaluated by optimal or asymptotically optimal quadrature formulas. The advantage of these formulas over the optimal and asymptotically optimal cubature formulas with rectangular-lattices of knots is shown.

Abstract: : A sequential design for estimating the percentiles of a quantal response curve is proposed. Its updating rule is based on an efficient summary of all the data available via a parametric model. Its efficiency in terms of saving the number of runs and its robustness against the distributional assumption are demonstrated heuristically and in a simulation study. A linear approximation to the logit-MLE version of the proposed sequential design is shown to be equivalent to an asymptotically optimal stochastic approximation method, thereby providing a large sample justification. For sample size between 12 and 35, the simulation study shows that the logit-MLE version of the general sequential procedure substantially outperforms an adaptive (and asymptotically optimal) version of the Robbins-Monro method, which in turn outperforms the nonadaptive Robbins-Munro and Up-and-Down methods. A nonparametric sequential design, via the Spearman-Karber estimator, for estimating the median is also proposed. (Author)

Abstract: A method of generating asymptotically optimal test statistics using consistent estimators is presented and is useful when full maximum likelihood estimation is difficult under null and alternative hypotheses. Generalizing Neyman's (1959) work, statistics are derived for hypotheses of the constraint equation and parameter type and my be computed from simple GLS or OLS regressions. Testing strategies are given for an ordered sequence of hypotheses and for evaluating a null model against several alternatives. These methods are applied to dynamic equation systems and common factor restrictions in a single dynamic equation model.

Abstract: In the mathematical learning literature, reward-penalty rules have been studied in various decision-theoretic and game-theoretic contexts, the multi-armed bandit problem included. Here we propose an elaboration of Bather's randomised allocation indices which yields rules for the multi-armed bandit which are both reward-penalty and asymptotically optimal. GITTINS INDEX: MATHEMATICAL LEARNING This note concerns rules for sampling one at a time from k(-2) Bernoulli populations, population i having unknown probability of success pi, 15 i i k. Our concern is with rules of the reward-penalty type. The central idea of such rules may be stated as follows: Do not decrease (do not increase) the probability of sampling the ith population at time t + 1 if it was sampled at time t and the outcome was a success (failure). Plainly the 'play the winner' rule introduced by Robbins for the case k = 2 is rewardpenalty. In this rule if a success is observed with pi then the same pi is used in the next trial; otherwise we switch to the other one. Reward-penalty rules have been studied in various contexts (this one included) in the mathematical learning literature--see, for example, Meybodi and Lackshmivarahan (1982). In the class of all reward-penalty rules we seek those which are asymptotically optimal, i.e. which will guarantee that the observed proportion of successes converges to max pi when the total number of trials becomes infinite. From Bather (1981) we know that for a special version of the problem with k = 2 no deterministic stationary rule can be asymptotically optimal and that (excepting the case pi = P2) the play-the-winner rule is not asymptotically optimal. Bather (1981) proposed a class of asymptotically optimal rules based on randomised allocation indices. Although these rules are not in general reward-penalty they can be elaborated in such a way as to make them so while preserving their asymptotic optimality. The new class of rules thus obtained samples one at a time from k(?2) Bernoulli populations as follows: on the (t + 1)th occasion sample from population j if and only if j is the smallest integer such that Qi(t) = maxi Q,(t), where (1) Q,(t) = rl {si (t), fi (t)} + A,{s, (t), f (t)}Xj (t) and where for each i

Abstract: In the mathematical learning literature, reward–penalty rules have been studied in various decision-theoretic and game-theoretic contexts, the multi-armed bandit problem included. Here we propose an elaboration of Bather's randomised allocation indices which yields rules for the multi-armed bandit which are both reward-penalty and asymptotically optimal.

Abstract: We present a family of tests for detecting initialization bias in the mean of a simulation output series using a hypothesis testing framework. The null hypothesis is that the output mean does not change throughout the simulation run. The alternative hypothesis specifies a general transient mean function. The tests are asymptotically optimal based on cumulative sums of deviations about the sample mean. A particular test in this family is applied to a variety of simulation models. The test requires very modest computation and appears to be both robust and powerful.

Abstract: Multi-stage hypothesis tests are studied as competitors of sequential tests. A class of three-stage tests for the one-dimensional exponential family is shown to be asymptotically efficient, whereas two-stage tests are not. Moreover, in order to be asymptotically optimal, three-stage tests must mimic the behavior of sequential tests. Similar results are obtained for the problem of testing two simple hypotheses.