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.     

Classification with discrete and continuous variables via general mixed-data models

We study the problem of classifying an individual into one of several populations based on mixed nominal, continuous, and ordinal data. Specifically, we obtain a classification procedure as an extension to the so-called location linear discriminant function, by specifying a general mixed-data model for the joint distribution of the mixed discrete and continuous variables. We outline methods for estimating misclassification error rates. Results of simulations of the performance of proposed classification rules in various settings vis-à-vis a robust mixed-data discrimination method are reported as well. We give an example utilizing data on croup in children.     

Certain nonparametric classification rules and their asymptotic efficiencies

Two nonparametric classification rules for e-univariace populations are proposed, one in which the probability of correct classification is a specified number and the other in which one has to evaluate the probability of correct classification. In each case the classification is with respect to the Chernoff and Savage (1958) class of statistics, with possible specialization to populations having different location shifts and different changes of scale. An optimum property, namely the consistency of the classification procedure, is established for the second rule, when the distributions are either fixed or “near” in the Pitman sense and are tending to a common distribution at a specified rate. A measure of asymptotic efficiency is defined for the second rule and its asymptotic efficiency based on the Chernoff-Savage class of statistics relative to the parametric competitors ie the case of location shifts and scale changes is shown to be equal to the analogous Pitman efficiency.     

A Classification Method for Directional Data with Application to the Human Skull

There are many well-known methods applied in classification problem for linear data with both known and unknown distribution. Here, we deal with classification involving data on torus and cylinder. A new method involving a generalized likelihood ratio test is developed for classifying in two populations using directional data. The approach assumes that one of the probabilities of misclassification is known. The procedure is constructed by applying Gibbs sampler on the conditionally specified distribution. A parametric bootstrap approach is also presented. An application to data involving linear and circular measurements on human skull from two tribal populations is given.     

A two-stage procedure for estimating a linear function of K multinormal mean vectors when covariance matrices are unknown

We consider the problem of constructing a fixed-size confidence region for a linear function of mean vectors of k multinormal populations, where all covariance matrices are completely unknown. A two-stage procedure is proposed to construct such a confidence region. It is shown that the proposed two-stage procedure is consistent and its asymptotic property for the expected sample size is also given. A Monte Carlo simulation study is given for an illustration.     

A two-stage minimax procedure with screening for selecting the largest normal mean (ii): an improved pcs lower bound and associated tables

This paper is a follow-up to an earlier article by the authors in which they proposed a two-stage procedure with screening to select the normal population with the largest population mean when the populations have a common known variance. 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 of Bechhofer (1954), regardless of the configuration of the population means. The present paper contains new results which make possible the more efficient implementation of the two-stage procedure. Tables for this purpose are given, and the improvements achieved (which are substantial) are assessed.     

Segmented classification analysis with a class of rectangle-screened elliptical populations

In many practical situations, a statistical practitioner often faces a problem of classifying an object from one of the segmented (or screened) populations where the segmentation was conducted by a set of screening variables. This paper addresses this problem, proposing and studying yet another optimal rule for classification with segmented populations. A class of q-dimensional rectangle-screened elliptically contoured (RSEC) distributions is considered for flexibly modeling the segmented populations. Based on the properties of the RSEC distributions, a parametric procedure for the segmented classification analysis (SCA) is proposed. This includes motivation for the SCA as well as some theoretical propositions regarding its optimal rule and properties. These properties allow us to establish other important results which include an efficient estimation of the rule by the Monte Carlo expectation–conditional maximization algorithm and an optimal variable selection procedure. Two numerical examples making use of utilizing a simulation study and a real dataset application and advocating the SCA procedure are also provided.     

Classification of an observation into linear manifolds,their closed subsets and closed convex subsets

In this paper, we generalize the notion of classification of an observation (sample), into one of the given n populations to the case where some or all of the populations into which the new observation is to be classified may be new but related in a simple way to the given n populations. The discussion is in the frame-work of the given set of observations obeying the usual multivariate general linear hypothesis model. The set ofpopulations into which the new observation may be classified could be linear manifolds of the parameter space or their closed subsets or closed convex subsets or a combination of them or simply t subsets of the parameter space each of which has a finite number of elements. In the last case alikelihood ratio procedure can be obtained easily. Classification procedures given here are based on Mahalanobis distance. Bonferroni lower bound estimate of the probability of correctly classifying an observation is given for the case when the covariance matrix is known or is estimated from a large sample. A numerical example relating to the classification procedures suggested her is given.     

A simple procedure for testing linear hypotheses about the parameters of a nonlinear model using weighted least squares

Suppose the same nonlinear function involving k parameters is fit to each of t populations. Suppose further it is of interest to compare a specific parameter of the models across the populations. Such comparisons can be expressed as linear hypotheses about the parameters of the nonlinear models. A weighted linear least squares (WLLS) procedure is proposed to test these linear hypotheses. The advantages and disadvantages of the WLLS procedure are discussed. This procedure is also compared to a nonlinear least squares procedure for testing these hypotheses in nonlinear models.     

ON SELECTING VARIABLES AND ASSESSING THEIR PERFORMANCE IN LINEAR DISCRIMINANT ANALYSIS

Linear discriminant analysis between two populations is considered in this paper. Error rate is reviewed as a criterion for selection of variables, and a stepwise procedure is outlined that selects variables on the basis of empirical estimates of error. Problems with assessment of the selected variables are highlighted. A leave-one-out method is proposed for estimating the true error rate of the selected variables, or alternatively of the selection procedure itself. Monte Carlo simulations, of multivariate binary as well as multivariate normal data, demonstrate the feasibility of the proposed method and indicate its much greater accuracy relative to that of other available methods.     

A Permutation Based Procedure for Classification Assessment

This article proposes a permutation procedure for evaluating the performance of different classification methods. In particular, we focus on two of the most widespread and used classification methodologies: latent class analysis and k-means clustering. The classification performance is assessed by means of a permutation procedure which allows for a direct comparison of the methods, the development of a statistical test, and points out better potential solutions. Our proposal provides an innovative framework for the validation of the data partitioning and offers a guide in the choice of which classification procedure should be used     

On the performance of some subset selection procedures

In this paper, we restrict attention to the problem of subset selection of normal populations. The approaches and results of some previous comparison studies of subset selection procedures are discussed briefly. And then the result of a new Monte Carlo study comparing the performance of two classical procedures and the Bayes procedure is presented.     

A three-population constrained discrimination procedure

ABSTRACT

Classification rules with a reserve judgment option provide a way to satisfy constraints on the misclassification probabilities when there is a high degree of overlap among the populations. Constructing rules which maximize the probability of correct classification while satisfying such constraints is a difficult optimization problem. This paper uses the form of the optimal solution to develop a relatively simple and computationally fast method for three populations which has a non parametric quality in controlling the misclassification probabilities. Simulations demonstrate that this procedure performs well.     

A comparison of the classical and the linear programming approaches to the classification problem in discriminant analysis

Several mathematical programming approaches to the classification problem in discriminant analysis have recently been introduced. This paper empirically compares these newly introduced classification techniques with Fisher's linear discriminant analysis (FLDA), quadratic discriminant analysis (QDA), logit analysis, and several rank-based procedures for a variety of symmetric and skewed distributions. The percent of correctly classified observations by each procedure in a holdout sample indicate that while under some experimental conditions the linear programming approaches compete well with the classical procedures, overall, however, their performance lags behind that of the classical procedures.     

A Bayesian Approach for Selecting the Best Exponential Population with Interval Censored Samples

In this article, we study the problem of selecting the best population from among several exponential populations based on interval censored samples using a Bayesian approach. A Bayes selection procedure and a curtailed Bayes selection procedure are derived. We show that these two Bayes selection procedures are equivalent. A numerical example is provided to illustrate the application of the two selection procedure. We also use Monte Carlo simulation to study performance of the two selection procedures. The numerical results of the simulation study demonstrate that the curtailed Bayes selection procedure has good performance because it can substantially reduce the duration time of life test experiment.     

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.     

Selecting logistic populations using the sample medians

This paper is concerned primarily with subset selection procedures based on the sample mediansof logistic populations. A procedure is given which chooses a nonempty subset from among kindependent logistic populations, having a common known variance, so that the populations with thelargest location parameter is contained in the subset with a pre‐specified probability. Theconstants required to apply the median procedure with small sample sizes (≤= 19) are tabulated and can also be used to construct simultaneous confidence intervals. Asymptotic formulae are provided for application with larger sample sizes. It is shown that, under certain situations, rules based on the median are substantially more efficient than analogous procedures based either on sample means or on the sum of joint ranks.     

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.     

A comparison of the post selection error rate behaviour of the normal linear and quadratic discriminant rules

The normal linear discriminant rule (NLDR) and the normal quadratic discriminant rule (NQDR) are popular classifiers when working with normal populations. Several papers in the literature have been devoted to a comparison of these rules with respect to classification performance. An aspect which has, however, not received any attention is the effect of an initial variable selection step on the relative performance of these classification rules. Cross model validation variabie selection has been found to perform well in the linear case, and can be extended to the quadratic case. We report the results of a simulation study comparing the NLDR and the NQDR with respect to the post variable selection classification performance. It is of interest that the NQDR generally benefits from an initial variable selection step. We also comment briefly on the problem of estimating the post selection error rates of the two rules.     

Two-stage procedures for the difference of two multinormal means with covariance matrices different only by unknown scalar multipliers

We consider the problem of constructing a fixed-size confidence region for the difference of means of two multivariate normal populations It is assumed that the variance-covariance matrices of two populations are different only by unknown scalar multipliers Two-stage procedures are presented to derive such a confidence region We also discuss the asymptotic efficiency of the procedure.