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.     

A comparison of selection procedures for selecting the best normal population under heterogeneity of variance

This study examines the comparative probabilities of making a correct selection when using the means procedure (M), the medians procedure (D) and the rank-sum procedure (S) to correctly select the normal population with the largest mean under heterogeneity of variance. The comparison is conducted by using Monte-Carlo simulation techniques for 3, 4, and 5 normal populations under the condition that equal sample sizes are taken from each population. The population means and standard deviations are assumed to be equally-spaced. Two types of heterogeneity of variance are considered: (1) associating larger means with larger variances, and (2) associating larger means with smaller variances.     

A restricted subset selection procedure for Weibull populations

Consider k(k ≥ 2) two-parameter Weibull populations. We want to select a subset of the populations not exceeding m in size such that the subset contains at least ? of the t best populations. We have proposed a procedure which uses either the maximum likelihood estimators or ‘simplified’ linear estimators of the parameters. The estimators are based on type II censored data. The ranking of the populations is done by comparing their reliabilities at a certain fixed time. In selected cases the constants for the procedure are tabulated using Monte Carlo methods.     

On two-stage selection procedures and related probability-inequalities

In this paper we discuss a modification of the Dudewicz-Dalal procedure for the problem of selecting the population with the largest mean from k normal populations with unknown variances. We derive some inequalities and use them to lower-bound the probability of correct selection. These bounds are applied to the determination of the second-stage sample size which is required in order to achieve a prescribed probability of correct selection. We discuss the resulting procedure and compare it to that of Dudewicz and Dalai (1975).     

Some subset selection problems

Consider that we have a collection of k populations π₁, π₂…,π_k. The quality of the ith population is characterized by a real parameter θ_i and the population is to be designated as superior or inferior depending on how much the θ_i differs from θ_max = max{θ₁, θ₂,…,θ_k}. From the set {π₁, π₂,…,π_k}, we wish to select the subset of superior populations. In this paper we devise rules of selection which have the property that their selected set excludes all the inferior populations with probability at least 1?α, where a is a specified number.     

On lower confidence bounds for pcs in truncated location parameter models

We are concerned with deriving lower confidence bounds for the probability of a correct selection in truncated location-parameter models. Two cases are considered according to whether the scale parameter is known or unknown. For each case, a lower confidence bound for the difference between the best and the second best is obtained. These lower confidence bounds are used to construct lower confidence bounds for the probability of a correct selection. The results are then applied to the problem of seleting the best exponential populationhaving the largest truncated location-parameter. Useful tables are provided for implementing the proposed methods.     

Equal probability of correct selection for bernoulli selection procedures

The problem of selecting the Bernoulli population which has the highest "success" probability is considered. It has been noted in several articles that the probability of a correct selection is the same, uniformly in the Bernoulli p-vector (P1,P2,….,Pk), for two or more different selection procedures. We give a general theorem which explains this phenomenon.

An application of particular interest arises when "strong" curtailment of a single-stage procedure (as introduced by Bechhofer and Kulkarni (1982a) )is employed; the corresponding result for "weak" curtailment of a single-stage procedure needs no proof. The use of strong curtailment in place of weak curtailment requires no more (and usually many less) observations to achieve the same.     

Sampling strategy for optimal classification into one of two correlated normal populations

A unit ω is to be classified into one of two correlated homoskedastic normal populations by linear discriminant function known as W classification statistic [T.W. Anderson, An asymptotic expansion of the distribution of studentized classification statistic, Ann. Statist. 1 (1973), pp. 964–972; T.W. Anderson, An Introduction to Multivariate Statistical Analysis, 2nd edn, Wiley, New York, 1984; G.J. Mclachlan, Discriminant Analysis and Statistical Pattern Recognition, John Wiley and Sons, New York, 1992]. The two populations studied here are two different states of the same population, like two different states of a disease where the population is the population of diseased patient. When a sample unit is observed in both the states (populations), the observations made on it (which form a pair) become correlated. A training sample is unbalanced when not all sample units are observed in both the states. Paired and also unbalanced samples are natural in studies related to correlated populations. S. Bandyopadhyay and S. Bandyopadhyay [Choosing better training sample for classifying an individual into one of two correlated normal populations, Calcutta Statist. Assoc. Bull. 54(215–216) (2003), pp. 167–180] studied the effect of unbalanced training sample structure on the performance of W statistics in the univariate correlated normal set-up for finding optimal sampling strategy for a better classification rate. In this study, the results are extended to the multivariate case with discussion on application in real scenario.     

SHOULD WE ESTIMATE THE PROBABILITY OF CORRECT SELECTION?

Various authors, given k location parameters, have considered lower confidence bounds on (standardized) dserences between the largest and each of the other k - 1 parameters. They have then used these bounds to put lower confidence bounds on the probability of correct selection (PCS) in the same experiment (as was used for finding the lower bounds on differences). It is pointed out that this is an inappropriate inference procedure. Moreover, if the PCS refers to some later experiment it is shown that if a non-trivial confidence bound is possible then it is already possible to conclude, with greater confidence, that correct selection has occurred in the first experiment. The short answer to the question in the title is therefore ‘No’, but this should be qualified in the case of a Bayesian analysis.     

Selection and ranking procedures: a brief introduction

This paper presents a brief introduction to selection and ranking methodology. Both indifference zone and subset selection approaches are discussed along with some modifications and generalizations. Two examples are provided to illustate the use of subset selection and the indifference zone approaches. The paper concludes with the remark that selection and ranking methodology is a realistic approach in statistical analyses involving comparisons among two or more treatments.     

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.     

Indifference-zone ranking and selection: confidence intervals for true achieved P(CD)

In this paper we present procedures which attempt to allow experimenters to take at least partial advantage of more-favorable configurations of the population parameters (without, in the process, sacrificing precise probability statements about the inferences made) by obtaining confidence intervals for the true probability of a correct decision, regarded as a function of the true unknown underlying parameter values.     

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

     

On rules based on sample medians for selection of the largest location parameter

The problem of selection of a subset containing the largest of several location parameters is considered, and a Gupta-type selection rule based on sample medians is investigated for normal and double exponential populations. Numerical comparisons between rules based on medians and means of small samples are made for normal and contaminated normal populations, assuming the popula-tion means to be equally spaced. It appears that the rule based on sample means loses its superiority over the rule based on sample medians in case the samples are heavily contaminated. The asymptotic relative efficiency (ARE) of the medians procedure relative to the means procedure is also computed, assuming the normal means to be in a slippage configuration. The means proce-dure is found to be superior to the median procedure in the sense of ARE. As in the small sample case, the situation is reversed if the normal populations are highly contaminate.     

Subset selection of the t best populations

Selection of the “best” t out of k populations has been considered in the indifferece zone formulation by Bachhofer (1954) and in the subset selection formulation by Carroll, Gupta and Huang (1975). The latter approach is used here to obtain conservative solutions for the goals of selecting (i) all the “good” or (ii) only “good” populations, where “good” means having a location parameter among the largest t. For the case of normal distributions, with common unknown variance, tables are produced for implementing these procedures. Also, for this case, simulation results suggest that the procedure may not be too conservative.     

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.     

On a strengthening of the indifference zone approach to a generalized selection goal

The problem of selecting s out of k given compounts which contains at least c of the t best ones is considered. In the case of underlying distribution families with location or scale parameter it is shown that the indiffence zone approach can be strengthened to confidence statements for the parameters of the selected components. These confidence statements are valid over the entire parameter space without decreasing the infimum of the probability of a correct selection.     

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     

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.