Selecting the best normal population better than a standard under unequal variances

In this article we consider a problem of selecting the best normal population that is better than a standard when the variances are unequal. Single-stage selection procedures are proposed when the variances are known. Wilcox (1984) and Taneja and Dudewicz (1992) proposed two-stage selection procedures when the variances are unknown. In addition to these procedures, we propose a two-stage selection procedure based on the method of Lam (1988). Comparisons are made between these selection procedures in terms of the sample sizes.     

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.     

Screening for Multiple Target Detection

We apply statistical selection theory to multiple target detection problems by analyzing the Mahalanobis distances between multivariate normal populations and a desired standard (a known characteristic of a target). We want to achieve the goal of selecting a subset that contains no non target (negative) sites, which entails screening out all non targets. Correct selection (CS) is defined according to this goal. We consider two cases: (1) that all covariance matrices are known; and (2) that all covariance matrices are unknown, including both heteroscedastic and homoscedastic cases. Optimal selection procedures are proposed in order to reach the selection goal. The least favorable configurations (LFC) are found. Tables and figures are presented to illustrate the properties of our proposed procedures. Simulation examples are given to show that our procedures work well. The log-concavity results of the operating characteristic functions are also given.     

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.     

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.     

Lower bounds for expected sample size of sequential procedures for the multinomial selection problems

In this article, lower bounds for expected sample size of sequential selection procedures are constructed for the problem of selecting the most probable event of k-variate multinomial distribution. The study is based on Volodin’s universal lower bounds for expected sample size of statistical inference procedures. The obtained lower bounds are used to estimate the efficiency of some selection procedures in terms of their expected sample sizes.     

Selecting a Good Stochastic System for the Large Number of Alternatives

In this article, we present the problem of selecting a good stochastic system with high probability and minimum total simulation cost when the number of alternatives is very large. We propose a sequential approach that starts with the Ordinal Optimization procedure to select a subset that overlaps with the set of the actual best m% systems with high probability. Then we use Optimal Computing Budget Allocation to allocate the available computing budget in a way that maximizes the Probability of Correct Selection. This is followed by a Subset Selection procedure to get a smaller subset that contains the best system among the subset that is selected before. Finally, the Indifference-Zone procedure is used to select the best system among the survivors in the previous stage. The numerical test involved with all these procedures shows the results for selecting a good stochastic system with high probability and a minimum number of simulation samples, when the number of alternatives is large. The results also show that the proposed approach is able to identify a good system in a very short simulation time.     

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     

A Class of Two-Stage Selection Procedures Using L-Statistics

Summary: A class of selection procedures for selecting the least dispersive distribution from k available distributions has been proposed. This problem finds applications in reliability and engineering. In engineering, for example, the goal of the experimenter is to select a firm whose components have least dispersive distribution from the available set of competing firms manufacturing the components of the desired specifications meant for the same purpose. The proposed procedures can be used even when the underlying distributions belong to different families. Applications of the proposed selection procedures are discussed by taking exponential, gamma and Lehmann type distributions. Performance of the proposed selection procedures is assessed through simulation study. Implementation of the proposed selection procedure is illustrated through an example. * The authors are very grateful to the editor and referees for their valuable comments.     

On selecting the treatment with the largest probability of survival

Suppose there are k(>= 2) treatments and each treatment is a Bernoulli process with binomial sampling. The problem of selecting a random-sized subset which contains the treatment with the largest survival probability (reliability or probability of success) is considered. Based on the ideas from both classical approaches and general Bayesian statistical decision approach, a new subset selection procedure is proposed to solve this kind of problem in both balanced and unbalanced designs. Comparing with the classical procedures, the proposed procedure has a significantly smaller selected subset. The optimal properties and performance of it were examined. The methods of selecting and fitting the priors and the results of Monte Carlo simulations on selected important cases are also studied.     

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.     

Multinomial Selection Problem: A Study of BEM and AVC Algorithms

The two well-known and widely used multinomial selection procedures Bechhofor, Elmaghraby, and Morse (BEM) and all vector comparison (AVC) are critically compared in applications related to simulation optimization problems.

Two configurations of population probability distributions in which the best system has the greatest probability p _i of yielding the largest value of the performance measure and has or does not have the largest expected performance measure were studied.

The numbers achieved by our simulations clearly show that none of the studied procedures outperform the other in all situations. The user must take into consideration the complexity of the simulations and the performance measure probability distribution properties when deciding which procedure to employ.

An important discovery was that the AVC does not work in populations in which the best system has the greatest probability p _i of yielding the largest value of the performance measure but does not have the largest expected performance measure.     

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

     

Some multiple decision problems in analysis of variance

In most practical situations to which the analysis of variance tests are applied, they do not supply the information that the experimenter aims at. If, for example, in one-way ANOVA the hypothesis is rejected in actual application of the F-test, the resulting conclusion that the true means θ₁,…,θ_k are not all equal, would by itself usually be insufficient to satisfy the experimenter. In fact his problems would begin at this stage. The experimenter may desire to select the “best” population or a subset of the “good” populations; he may like to rank the populations in order of “goodness” or he may like to draw some other inferences about the parameters of interest.

The extensive literature on selection and ranking procedures depends heavily on the use of independence between populations (block, treatments, etc.) in the analysis of variance. In practical applications, it is desirable to drop this assumption or independence and consider cases more general than the normal.

In the present paper, we derive a method to construct optimal (in some sense) selection procedures to select a nonempty subset of the k populations containing the best population as ranked in terms of θ_i’s which control the size of the selected subset and which maximizes the minimum average probability of selecting the best. We also consider the usual selection procedures in one-way ANOVA based on the generalized least squares estimates and apply the method to two-way layout case. Some examples are discussed and some results on comparisons with other procedures are also obtained.     

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.     

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.     

An empirical bayes approach to estimating the probability of correct selection

In the problem of selecting the best of k populations, Olkin, Sobel, and Tong (1976) have introduced the idea of estimating the probability of correct selection. In an attempt to improve on their estimator we consider anempirical Bayes approach. We compare the two estimators via analytic results and a simulation study.     

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.     

△-correct decision for location and scale parameters

In selecting t out of k populations, a △-correct decision is said to be made if the smallest location parameter for the selected populations is not more than △ below the largest location parameter for the non-selected populations. (For seal? parameters there is a similar definition in terms of ratio3.) The minimum probability of △-correct decision over the entire parameter pace is shown to be equal to the minimum probability of correct selection over a preference zone determined by △.