Different definitions of Δ-correct selection for the indifference zone formulation

The goal of the indifference zone formulation of selection (Bechhofer, 1954) consists of selecting the t best variants out of k variants with a probability of at least 1 − β if the parameter difference between the t ‘good’ variants and the k − t ‘bad’ variants is not less than Δ. A review of generalized selection goals not using this difference condition is presented. Within some general classes of distributions, the suitable experimental designs for all these selection goals are identical. Similar results are described for the problem of selecting the best variant in comparison with a control, or standard.     

Sequential selectiok procedures based on confidence sequences for normal populations

In this paper confidence sequences are used to construct sequential procedures for selecting the population with the a common variance. These procedures are shown to provide substantial saving, particularly in the expected samplw sizes of the inferior populations,over various procedures in the literature. A new “indifference zone” formulation is given for the correct selection probability requirement, and confidence sequences are also applied to construct sequential procedures for this new selection goal.     

New procedures for selecting the k-best exponential populations

Suppose exponential populations π_i with parameters (μ_i,σ_i) (i = 1, 2, …, K) are given. The σ_i can be unknown and unequal. This article discusses how to select the k (≥1) best populations. Under the subset selection formulation, a one-stage procedure is proposed. Under the indifference zone formulation, a two-stage procedure is proposed. An appealing feature of these procedures is that no statistical tables are needed for their implementation.     

Determination of sample size for selecting the smallest of k poisson population means

A procedure for selecting a Poisson population with smallest mean is considered using an indifference zone approach. The objective is to determine the smallest sample size n required from k ≥ 2 populations in order to attain the desired probability of correct selection. Since the means procedure is not consistent with respect to the difference or ratio alone, two distance measures are used simultaneously to overcome the difficulty in obtaining the smallest probability of correct selection that is greater than some specified limit. The constants required to determine n are computed and tabulated. The asymptotic results are derived using a normal approximation. A comparison with the exact results indicates that the proposed approximation works well. Only in the extreme cases small increases in n are observed. An example of industrial accident data is used to illustrate this procedure.     

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

     

Selection from uniform populations based on sample mtdranges and an associated estimation after selection

Selection of the uniform population having the largest location parameter (point of symmetry) is considered using both the indifference zone and subset selection formulations. For the indifference zone rule selecting one of the population as the best, estimation of the parameter of the selected population is considered in the case of two given populations.     

Multiple decision rules for comparing several populations with a fixed known standard

Independent observations are available from k univariate distributions indexed by a real parameter θ. It is desired to select that distribution with the largest parameter value unless this value is smaller than some fixed standard θ₀ in which case no distribution is to be selected. Various single-stage procedures for this (k+l)-decision problem are discussed, using indifference zone, decision theoretic, Bayesian, and subset selection approaches.     

Subset selection of superior populations when the number of populations is large

The situation where k populations are partitioned into one inferior group and one superior group is considered. The statistical problem is to select a random size subset of superior populations while trying to avoid including any inferior populations. A selection procedure is assumed to satisfy the condition that the probability of selecting at least one superior population is bounded below by P¹<1. The performance of a procedure is measured by the probability of including an inferior population.The asymptotic performance, as k→∞ of Gupta's traditional maximum type procedure ψ^G is considered in the location-model. For normally distributed populations, ψ^G turns out to be asymptotically optimal, provided the size of the inferior group does not become infinitely larger than the size of the superior group.     

A Two-stage minimax procedure with screening for selecting the largest normal mean

The problem of selecting the normal population with the largest population mean when the populations have a common known variance is considered. A two-stage procedure is proposed which guarantees the same probability requirement using the indifference-zone approach as does the single-stage procedure of Bechhofer (1954). The two-stage procedure has the highly desirable property that the expected total number of observations required by the procedure is always less than the total number of observations required by the corresponding single-stage procedure, regardless of the configuration of the population means. The saving in expected total number of observations can be substantial, particularly when the configuration of the population means is favorable to the experimenter. The saving is accomplished by screening out “non-contending” populations in the first stage, and concentrating sampling only on “contending” populations in the second stage.

The two-stage procedure can be regarded as a composite one which uses a screening subset-type approach (Gupta (1956), (1965)) in the first stage, and an indifference-zone approach (Bechhofer (1954)) applied to all populations retained in the selected sub-set in the second stage. Constants to implement the procedure for various k and P? are provided, as are calculations giving the saving in expected total sample size if the two-stage procedure is used in place of the corresponding single-stage procedure.     

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.     

An alternative decision rule for quality control in sequential sampling plans

The conventional rule for acceptance sampling based on sequential samples is based on the rationale of hypothesis testing developed by Wald (1947). This type of decision rule tests the hypothesis of P=p₁ versus P=p₂ as a proxy for determining whether P>d or P<d with P₁<d<p₂. It requires a zone of indifference between the rejection and acceptance levels p₂ and p₁. In this note, we propose an alternative rule for making the decision based on the confidence level of a one-sided Bayesian interval estimate of the parameter. This method results in direct determination of whether the proportion of defects P in the population is greater or less than a prespecified level d, rather than test two points as proxy for the decision. We present a numerical illustration of the rule and an example of determining rejection and acceptance numbers. Ue also compare the results with two conventional rules.     

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     

Group selection for production yield among k manufacturing lines

Producing qualified items or products is essential to meet the requirement preset by customers. Evaluation and selection of desired manufacturing lines become challenging tasks for decision makers. Production yield is one of the important factors in measuring production performance. The goal of this paper is to screen a group of manufacturing lines and identify the best one with the highest yield. For the production lines with extremely low fraction of defectives, the yield index, S_pk, is an efficient indicator for quality level. This paper considers the production selection problem by using S_pk to compare k (k>2) manufacturing lines. A subset is constructed to contain the production lines with the highest yield. A systematic approach of test order k compares selected pairs of manufacturing lines along with the Bonferroni method is proposed to solve this problem. Each pair of production yields is compared by taking ratio. The paper provides critical values and required sample sizes of the group selection procedure. An application example on evaluating four power inductor productions is presented to illustrate the practicality of the proposed approach.     

Selecting the t-best cells in a multinomial distribution

The problem of selecting the t-best cells in a multinomial distribution with t + k cells, k > 1, 2 <= t is considered under the fixed sample-size indifference zone approach. The least favourable configuration is derived for the usual procedure of selection, for large values of N (the sample size). The result settles Conjecture I (for large N) and Conjecture IV of Chen and Hwang (Commun. Statist. - Theory Meth. 13 (10), 1289-1298, 1984) in the affirmative.     

SELECTING THE m POPULATIONS WITH LARGEST MEANS FROM k NORMAL POPULATIONS WITH UNKNOWN VARIANCES

This paper gives a two-sample procedure for selecting the m populations with the largest means from k normal populations with unknown variances. The method is a generalization of a recent work by Ofosu [1973] and hence should find wider practical applications. The experimenter takes an initial sample of preset size N₀ from each population and computes an unbiased estimate of its variance. From this estimate he determines the second sample size for the population according to a table presented for this purpose. The populations associated with the m largest overall sample means will be selected. The procedure is shown to satisfy a confidence requirement similar to that of Ofosu.     

Improved algorithms for closed step-down procedures to test heterogeneity of all subsets of means

A common problem in analysis of variance is testing for heterogeneity of different subsets of the full set of k population means. A step-down procedure tests a given subset of p means only after rejecting homogeneity for all sets that contain it. The Peritz and Gabriel closed procedure rejects homogeneity for the subset if every partition of the k means that includes the subset includes some rejected set. The Begun and Gabriel closure algorithm reduces computations, but the number of tests still increases exponentially with respect to the number of complementary means, m=k−p. We propose a new algorithm that tests only the m−1 pairs of adjacent ordered complementary sample means. Our algorithm may be used with analyses of variance test statistics in balanced and unbalanced designs, and with Studentized ranges except in extremely unbalanced designs. Seaman, Levin, and Serlin proposed a more powerful closure criterion that cannot exploit the Begun and Gabriel algorithm. We propose a new algorithm in this case as well.     

Inferences from biased samples with a memory effect

Biased sampling occurs often in observational studies. With one biased sample, the problem of nonparametrically estimating both a target density function and a selection bias function is unidentifiable. This paper studies the nonparametric estimation problem when there are two biased samples that have some overlapping observations (i.e. recaptures) from a finite population. Since an intelligent subject sampled previously may experience a memory effect if sampled again, two general 2-stage models that incorporate both a selection bias and a possible memory effect are proposed. Nonparametric estimators of the target density, selection bias, and memory functions, as well as the population size are developed. Asymptotic properties of these estimators are studied and confidence bands for the selection function and memory function are provided. Our procedures are compared with those ignoring the memory effect or the selection bias in finite sample situations. A nonparametric model selection procedure is also given for choosing a model from the two 2-stage models and a mixture of these two models. Our procedures work well with or without a memory effect, and with or without a selection bias. The paper concludes with an application to a real survey data set.     

Renewal theory and the sequential design of experiments with two states of nature

In the sequential design of experiments in experimentel is sequentially performing experiments to help him make an inference about the true state of nature. Using results from renewal twenty at derive approximations for the operations characteristics and average sample numbers for this problem when there are two states of nature.

A critical problem in the sequential design of experiments is finding a good procedure. We investigate a Bayesian formulation of this problem and use our approximations to approximate the Bayes risk. Minimization of this approximate Bayes risk over procedures is discussed as a method of finding a good procedure, but difficulties are encountered due to the discrete time character of the sequential process. To avoid these difficulties, we consider minimization of an approximation related to a diffusion process. This leads to a simple rule for the sequential selection of experiments.     

Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function

In multivariate stratified sample survey with L strata, let p-characteristics are defined on each unit of the population. To estimate the unknown p-population means of each characteristic, a random sample is taken out from the population. In multivariate stratified sample survey, the optimum allocation of any characteristic may not be optimum for others. Thus the problem arises to find out an allocation which may be optimum for all characteristics in some sense. Therefore a compromise criterion is needed to workout such allocation. In this paper, the procedure of estimation of p-population means is discussed in the presence of nonresponse when the use of linear cost function is not advisable. A solution procedure is suggested by using lexicographic goal programming problem. The numerical illustrations are given for its practical utility.     

Selecting the Best Exponential Population When the Scale Parameters are Bounded

Consider k (≥ 2) independent exponential populations with different location and scale parameters. Call a population associated with largest of unknown location parameters as the best population. For the goal of selecting the best population, it is established that if the scale parameters are completely unknown, then the indifference-zone probability requirement can not be guaranteed by any single sample decision rule which is just and translation invariant. Under the assumption that the scale parameters are bounded above by a known constant, a single sample selection procedure is proposed for which the indifference-zone probability requirement can be guaranteed. Under the same assumption, 100P*% simultaneous upper confidence intervals for all distances from the largest location parameter are also obtained.