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.     

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.     

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.     

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.     

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.     

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.     

On selecting a subset which contains all populations better than a control

Let π₁…, π_k denote k(≥ 2) populations with unknown means μ₁ , …, μ_k and variances σ₁ ² , …, σ_k ² , respectively and let π_o denote the control population having mean μ_o and variance σ_o ² . It is assumed that these populations are normally distributed with correlation matrix {ρ_ij}. The goal is to select a subset, of populations of π₁ , …, π_k which contains all the populations with means larger than or equal to the mean of the control one. Procedures are given for selecting such a subset so that the probability that all the populations with means larger than or equal to the mean of the control one are included in the selected subset is at least equal to a predetermined value P?(l/k < P? < 1). The goal treated here is a first step screening procedure that allows the experimenter to choose a subset and withhold judgement about which one has the largest mean. Then, if the one with the largest mean is desired it can be chosen from the selected subset on the basis of cost and other considerations. Percentage points are also included.     

Complete ranking procedures with quadratic loss

Suppose that one wishes to rank k normal populations, each with common variance σ² and unknown means θ_i (i=1,2,…,k). Independent samples of size n are taken from each population, and the sample averages are used to rank the populations. In this paper, we investigate what sample sizes, n, are necessary to attain “good” rankings under various loss functions. Section discusses various loss functions and their interpretation. Section 2 gives the solution for a reasonable non-parametric loss function. Section 3 gives the solution for a reasonable parameteric loss function.     

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.     

Bayesian design of experiment for quantal responses: What is promised versus what is delivered

This article considers a design problem in quantal response analysis, where an experimenter must choose a set of dose levels and number of independent observations to take at these levels, subject to some total sample size, in order to minimize the expected or predicted posterior variance of some characteristics ø of the tolerance distribution F_θ, with unknown parameters θ. An exact solution to this problem is demonstrated when ø is the unknown LD50 of the one parameter logistic tolerance distribution, under the restriction that an equal number of observations are taken at each of a set of equally spaced levels. The solution is based on a combination of simulated outcomes and Monte Carlo integration to evaluate the predicted variance. The numerical results are compared to those obtained previously by asymptotic approximations in Tsutakawa (1972), (J. Amer. Statist. Assoc. 67 584–590). The wide variability in the simulated posterior variance suggests that the expected posterior variance alone is not a good criterion for design selection.     

Estimation of the location parameter and the average worth of the selected subset of two parameter exponential populations under LINEX loss function

Suppose a subset of populations is selected from k exponential populations with unknown location parameters θ₁, θ₂, …, θ_k and common known scale parameter σ. We consider the estimation of the location parameter of the selected population and the average worth of the selected subset under an asymmetric LINEX loss function. We show that the natural estimator of these parameters is biased and find the uniformly minimum risk-unbiased (UMRU) estimator of these parameters. In the case of k = 2, we find the minimax estimator of the location parameter of the smallest selected population. Furthermore, we compare numerically the risk of UMRU, minimax, and the natural estimators.     

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.     

Some Pitman closeness properties pertinent to symmetric populations

In this paper, we focus on Pitman closeness probabilities when the estimators are symmetrically distributed about the unknown parameter θ. We first consider two symmetric estimators θ?₁ and θ?₂ and obtain necessary and sufficient conditions for θ?₁ to be Pitman closer to the common median θ than θ?₂. We then establish some properties in the context of estimation under the Pitman closeness criterion. We define Pitman closeness probability which measures the frequency with which an individual order statistic is Pitman closer to θ than some symmetric estimator. We show that, for symmetric populations, the sample median is Pitman closer to the population median than any other independent and symmetrically distributed estimator of θ. Finally, we discuss the use of Pitman closeness probabilities in the determination of an optimal ranked set sampling scheme (denoted by RSS) for the estimation of the population median when the underlying distribution is symmetric. We show that the best RSS scheme from symmetric populations in the sense of Pitman closeness is the median and randomized median RSS for the cases of odd and even sample sizes, respectively.     

Selecting the normal population with the smallest variance: A restricted subset selection rule

Consider k( ? 2) normal populations whose means are all known or unknown and whose variances are unknown. Let σ²_[1] ? ??? ? σ_[k]² denote the ordered variances. Our goal is to select a non empty subset of the k populations whose size is at most m(1 ? m ? k ? 1) so that the population associated with the smallest variance (called the best population) is included in the selected subset with a guaranteed minimum probability P* whenever σ²_[2]/σ_[1]² ? δ* > 1, where P* and δ* are specified in advance of the experiment. Based on samples of size n from each of the populations, we propose and investigate a procedure called R_BCP. We also derive some asymptotic results for our procedure. Some comparisons with an earlier available procedure are presented in terms of the average subset sizes for selected slippage configurations based on simulations. The results are illustrated by an example.     

Bayesian estimation of the difference between two proportions

This paper is concerned with the problem of allocating a fixed number of trials between two independent binomial populations with unknown success probabilities θ₁ and θ₂, in order to estimate θ₁ - θ₂ with squared error loss. Introducing independent beta priors on θ₁ and θ₂, a heuristic allocation procedure is introduced and compared both with the optimal and with the best fixed allocation procedure. Numerical and asymptotic results of these comparisons are given and seem to indicate that there are situations when the best fixed allocation procedure performs almost as well as the optimal procedure.     

SELECTION PROCEDURES FOR SCALE PARAMETERS USING TWO-SAMPLE U-STATISTICS

Let be k independent populations having the same known quantile of order p (0 p 1) and let F(x)=F(x/_i) be the absolutely continuous cumulative distribution function of the ith population indexed by the scale parameter ₁, i = 1,…, k. We propose subset selection procedures based on two-sample U-statistics for selecting a subset of k populations containing the one associated with the smallest scale parameter. These procedures are compared with the subset selection procedures based on two-sample linear rank statistics given by Gill & Mehta (1989) in the sense of Pitman asymptotic relative efficiency, with interesting results.     

Simultaneous selection of extreme populations:a bayesian approach

The problem of simultaneously selecting two non-empty subsets, S_Land S_U, of k populations which contain the lower extreme population (LEP) and the upper extreme population (UEP), respectively, is considered. Unknown parameters θ₁,…,θ_kcharacterize the populations π₁,…,π_kand the populations associated with θ_[1]=min θ_i. and θ_[k]= max θ_i. are called the LEP and the UEP, respectively. It is assumed that the underlying distributions possess the monotone likelihood ratio property and that the prior distribution of θ= (θ₁,…,θ_k) is exchangeable. The Bayes rule with respect to a general loss function is obtained. Bayes rule with respect to a semi-additive and non-negative loss function is also determined and it is shown that it is minimax and admissible. When the selected subsets are required to be disjoint, it shown that the Bayes rule with respect to a specific loss function can be obtained by comparing certain computable integrals, Application to normal distributions with unknown means θ₁,…,θ_kand a common known variance is also considered.     

Subset selection based on likelihood from uniform and related popupulations

Let π_i (i=1,2,…, k) be charceterized by the uniform distribution on (a_i;b_i), where exactly one of a_i and b_i is unknown. With unequal sample sizes, suppose that from the k (>=2) given populations, we wish to select a random-size subset containing the one with the smllest value of θ_i= b_i - a_i. RuleR_i selects π if a likelihood-based k-dimensional confidence region for the unknown (θ₁,… θ_k) contains at least one point having θ_i as its smallest component. A second rule, R , is derived through a likelihood ratio and turns out to be that of Barr and prabhu whenthe sample sizes are equal. Numerical comparisons are made. The results apply to the larger class of densities g ( z ; θ_i) =M(z)Q(θ_i) if a(θ_i) < z <b(θ_i). Extensions to the cases when both a_i and b_i are unknown and when θ_j isof interest are indicated. 1<=j<=k     

SELECTING "DEMONSTRABLY BEST" OR "DEMONSTRABLY WORST" EXPONENTIAL POPULATIONS

Consider k independent observations Y_i (i= 1,., k) from two-parameter exponential populations _i with location parameters μ and the same scale parameter If the μ_i are ranked as consider population as the “worst” population and II_p(k) as the “best” population (with some tagging so that p{) and p(k) are well defined in the case of equalities). If the Y_i are ranked as we consider the procedure, “Select provided Y_R(k) Y_r(k) is sufficiently large so that is demonstrably better than the other populations.” A similar procedure is studied for selecting the “demonstrably worst” population.