On selecting a subset containing the most probable multinomial event

Panchapakesan's procedure is considered for the problem of selectinga subset containing the most probable multinomial event. We use the type-2 Dirichlet integral to express the probability of a correct selection and propose a much simpler proof for the worst configuration. We also show that the supremum of the expected subset size occurs at the equal configuration.     

On a sequential subset selection procedure for the least probable multinomial cell

This paper studies a sequential procedure R for selecting a random size subset that contains the multinomial cell which has the smallest cell probability. The stopping rule of the proposed procedure R is the composite of the stopping rules of curtailed sampling, inverse sampling, and the Ramey-Alam sampling. A reslut on the worst configuration is shown and it is employed in computing the procedure parameters that guarantee certain probability requirements. Tables of these procedure parameters, the corresponding probability of correct selection, the expected sample size, and the expected subset size are given for comparison purpose.     

A comparison of two procedures to select the best binomial population with sequential elimination of inferior populations

We compare the selection procedure of Levin and Robbins [1981. Selecting the highest probability in binomial or multinomial trials. Proc. Nat. Acad. Sci. USA 78, 4663–4666.] with the procedure of Paulson [1994. Sequential procedures for selecting the best one of k Koopman–Darmois populations. Sequential Analysis 13, 207–220.] to identify the best of several binomial populations with sequential elimination of unlikely candidates. We point out situations in which the Levin–Robbins procedure dominates the Paulson procedure in terms of the duration of the experiment, the expected total number of observations, and the expected number of failures. Because the Levin–Robbins procedure is also easier to implement than Paulson's procedure and gives a tighter guarantee for the probability of correct selection, we conclude that it holds a competitive edge over Paulson's procedure.     

Conditional probability of correct selection under the continuum partition with applications

The problem of selecting the best population from among a finite number of populations in the presence of uncertainty is a problem one faces in many scientific investigations, and has been studied extensively, Many selection procedures have been derived for different selection goals. However, most of these selection procedures, being frequentist in nature, don't tell how to incorporate the information in a particular sample to give a data-dependent measure of correct selection achieved for this particular sample. They often assign the same decision and probability of correct selection for two different sample values, one of which actually seems intuitively much more conclusive than the other. The methodology of conditional inference offers an approach which achieves both frequentist interpret ability and a data-dependent measure of conclusiveness. By partitioning the sample space into a family of subsets, the achieved probability of correct selection is computed by conditioning on which subset the sample falls in. In this paper, the partition considered is the so called continuum partition, while the selection rules are both the fixed-size and random-size subset selection rules. Under the distributional assumption of being monotone likelihood ratio, results on least favourable configuration and alpha-correct selection are established. These re-sults are not only useful in themselves, but also are used to design a new sequential procedure with elimination for selecting the best of k Binomial populations. Comparisons between this new procedure and some other se-quential selection procedures with regard to total expected sample size and some risk functions are carried out by simulations.     

An essentially complete class of multiple decision procedures

Let π₁,…, π_k represent k(?2) independent populations. The quality of the ith population π_i is characterized by a real-valued parameter θ_i, usually unknown. We define the best population in terms of a measure of separation between θ_i's. A selection of a subset containing the best population is called a correct selection (CS). We restrict attention to rules for which the size of the selected subset is controlled at a given point and the infimum of the probability of correct selection over the parameter space is maximized. The main theorem deals with construction of an essentially complete class of selection rules of the above type. Some classical subset selection rules are shown to belong to this class.     

On some optimal selection procedures for weibull populations

Consider k (k >(>)2) Weibull populations. We shall derive a method of constructing optimal selection procedures to select a subset of the k populations containing the best population which control the size of the selected subset and which maximises the minimum probability of making a correct selection. Procedures and results are derived for the case when sample sizes are unequal. Some tables and figures are given at the end of this paper.     

On Some Multiple Decision Procedures for Normal Variances

In this article, we propose a multiple decision procedure to test the homogeneity of normal variances. If the null-hypothesis is rejected, our goal is to select a subset containing the population associated with the largest variance. An approximation for the critical value is obtained by deriving an approximate distribution for a linear combination of independent log-gamma distributed random variables. A lower bound for the probability of correct decision is obtained. We also study the determination of the common sample size in order to satisfy a given probability of correct decision when the largest variance is “sufficiently” larger than the rest.     

Combining two classical approaches for statistical selection

This paper presents a selection procedure that combines Bechhofer's indifference zone selection and Gupta's subset selection approaches, by using a preference threshold. For normal populations with common known variance, a subset is selected of all populations that have sample sums within the distance of this threshold from the largest sample sum. We derive the minimal necessary sample size and the value for the preference threshold, in order to satisfy two probability requirements for correct selection, one related to indifference zone selection, the other to subset selection. Simulation studies are used to illustrate the method.     

Bayesian sequential methods for choosing the best multinomial cell: some simulation results

Suboptimal Bayesian sequential methods for choosing the best (i.e. largest probability) multinomial cell are considered and their performance is studied using Monte Carlo simulation. Performance characteristics, such as the probability of correct selection and some other associated with the sample size distribution, are evaluated assuming a maximum sample size. Single observation sequential rules as well as rules, where groups of observations are taken, and fixed sample size rules are discussed.     

Subset selection of the best treatment via many-one tests

The method of Gupta (1956, 1965) was developed to select a subset from k normal populations that contains the best populations with given probability. This paper shows a duality between the general goal of selecting a subset for the best population and many-one tests. A population should be regarded as ‘candidate’ for the best population and thus retained in the subset if the samples from the other populations are not significantly better. Based on this ‘idea’ a general selection procedure is proposed using many-one tests for the comparison of each population against the remaining ones.     

Multinomial Selection for Comparison with a Standard

The multinomial selection problem is considered under the formulation of comparison with a standard, where each system is required to be compared to a single system, referred to as a “standard,” as well as to other alternative systems. The goal is to identify systems that are better than the standard, or to retain the standard when it is equal to or better than the other alternatives in terms of the probability to generate the largest or smallest performance measure. We derive new multinomial selection procedures for comparison with a standard to be applied in different scenarios, including exact small-sample procedure and approximate large-sample procedure. Empirical results and the proof are presented to demonstrate the statistical validity of our procedures. The tables of the procedure parameters and the corresponding exact probability of correct selection are also provided.     

Closed inverse sampling procedure for selecting the largest multinomial cell probability

A class of closed inverse sampling procedures R(n,m) for selecting the multinomial cell with the largest probability is considered; here n is the maximum sample size that an experimenter can take and m is the maximum frequency that a multinomial cell can have. The proposed procedures R(n,m) achieve the same probability of a correct selection as do the corresponding fixed sample size procedures and the curtailed sequential procedures when m is at least n/2. A monotonicity property on the probability of a correct selection is proved and it is used to find the least favorable configurations and to tabulate the necessary probabilities of a correct selection and corresponding expected sample sizes     

On eliminating inferior regression models

Consider a linear regression model with [p-1] predictor variables which is taken as the "true" model.The goal Is to select a subset of all possible reduced models such that all inferior models ‘to be defined’ are excluded with a guaranteed minimum probability.A procedure is proposed for which the exact evaluation of the probability of a correct decision 1s difficult; however, 1t is shown that the probability requirement can be met for sufficiently large sample size.Monte Carlo evaluation of the constant associated with the procedure and some ways to reduce the amount of computations Involved in the Implementation of the procedure are discussed.     

Selection of the best with a preliminary test for two-parameter exponential distributions

This paper deals with the problem of selecting the best population from among k(≥ 2) two-parameter exponential populations. New selection procedures are proposed for selecting the unique best. The procedures include preliminary tests which allow the xperimenter to have an option to not select if the statistical evidence is not significant. Two probabilities, the probability to make a selection and the probability of a correct selection, are controlled by these selection procedures. Comparisons between the proposed selection procedures and certain earlier existing procedures are also made. The results show the superiority of the proposed selection procedures in terms of the required sample size.     

A New Formulation For The Multinomial Selection Problem

The two approaches to a multinomial ranking and selection problem (for selecting the t best cells out of k are combined to form a new apprach. In this new approach there is a preference zone (PZ) and an indifference zone (IZ), and the concept of a correct selection (CS) is defined differentlv in eac of these zones. Lower bounds for the probability of correct selection p(CS) are then guaranteed in each of these zones using a single experiment. The procedure on the ordered frequencies in the cells. The principle tool used to derive expressins for the p(CS). for the expected sample size EN, for the expected subsct size ES and for other probabilities. is the Dirichlet integral (Type 2) which was recent tabulated. These Dirichlet integrals are used to prove that the multiplicative slippage configuratin is leas favorable in the PZ and, for t = l, that the IZ. Numerical calculations are carried out for an illustrative example but extensive tables are not yet avalable

     

Designing selection experiments with bernoulli populations

This paper is concerned with a fixed size subset selection problem for Bernoulli populations in the framework of the indifference zone approach. The goal is to select s populationswhich contain at least c of those with the t largest success probabilities. In order to control the probability of correct selection over the preference zone extensive tables of exact minimum sample sizes have been prepared to implement the single-stage procedure generalized from the well-known Sobel-Huyett procedure. It is shown how the tables can also be employed to design certain closedsequential procedures. These procedures curtail the sampling process of the single-stage procedureand may differ in their sampling rules. Two procedures working with play-the-winner rules are described in detail     

Robust and nonparametric subset selection procedures

Subset selection procedures based on ranks have been investigated by a number of authors previously. Their methods are based on ranking the samples from all the populations jointly. However, as was pointed out by Rizvi and Woodworth (1970), the procedures they proposed cannot control the probability of a correct selection over the entire parameter space. In this paper, we propose a subset selection procedure based on pairwise rather than joint ranking of the samples. It is shown that this procedure controls the probability of a correct selection over the entire parameter space. It is also shown that the Pitman efficiency of this nonparametric procedure relative to the multivariate t procedure of Gupta (1956, 1965) is the same as the Pitman efficiency of the Mann-Whitney-Wilcoxon test relative to the t-test.     

Subset selection procedures for ΔP-superior populations

In some ranking and selection problems it is reasonable to consider any population which is inferior but sufficiently close to the best (t-th best) as acceptable. Under this assumption, this paper studies classes of procedures to meet two possible goals. A and B. Goal A is to select a subset which contains only good populations, while Goal B is of a screening nature and requires selection of a subset of size not exceeding m (1 ≤ m ≤ k) and containing at least one good population. In each case results loading to the determination of the sample size required to attain the goals above with prespecified probability are obtained. Properties of the procedures are discussed.     

Some computational problems related to multinomial trials

Efficient numerical algorithms are developed to evaluate several probabilities related to multinomial trials.In the first part of the paper, the probability distribution of the number of trials until the alternatives j, j = 1,… m, have occurred at least i_j times is computed. The multinomial trials involve the m alternatives l,…, m, with positive probabilities P_l-P_m of occurrence. In the second part, several aspects of a multinomial subset selection problem, discussed by S. S. Gupta and K. Nagel, are investigated.     

Average worth estimation of the selected subset of Poisson populations

Let Π₁, …, Π _p be p(p≥2) independent Poisson populations with unknown parameters θ₁, …, θ _p , respectively. Let X _i denote an observation from the population Π _i , 1≤i≤p. Suppose a subset of random size, which includes the best population corresponding to the largest (smallest) θ _i , is selected using Gupta and Huang [On subset selection procedures for Poisson populations and some applications to the multinomial selection problems, in Applied Statistics, R.P. Gupta, ed., North-Holland, Amsterdam, 1975, pp. 97–109] and (Gupta et al. [On subset selection procedures for Poisson populations, Bull. Malaysian Math. Soc. 2 (1979), pp. 89–110]) selection rule. In this paper, the problem of estimating the average worth of the selected subset is considered under the squared error loss function. The natural estimator is shown to be biased and the UMVUE is obtained using Robbins [The UV method of estimation, in Statistical Decision Theory and Related Topics-IV, S.S. Gupta and J.O. Berger, eds., Springer, New York, vol. 1, 1988, pp. 265–270] UV method of estimation. The natural estimator is shown to be inadmissible, by constructing a class of dominating estimators. Using Monte Carlo simulations, the bias and risk of the natural, dominated and UMVU estimators are computed and compared.