SELECTING THE t BEST POPULATIONS — SCALE PARAMETER CASE

Selection from k independent populations of the t (< k) populations with the smallest scale parameters has been considered under the Indifference Zone approach by Bechhofer k Sobel (1954). The same problem has been considered under the Subset Selection approach by Gupta & Sobel (1962a) for the normal variances case and by Carroll, Gupta & Huang (1975) for the more general case of stochastically increasing distributions. This paper uses the Subset Selection approach to place confidence bounds on the probability of selecting all “good” populations, or only “good” populations, for the Case of scale parameters, where a “good” population is defined to have one of the t smallest scale parameters. This is an extension of the location parameter results obtained by Bofinger & Mengersen (1986). Special results are obtained for the case of selecting normal populations based on variances and the necessary tables are presented.     

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.     

COMPARISONS AND SELECTION OF TWO-PARAMETER EXPONENTIAL POPULATIONS

For two-parameter exponential populations with the same scale parameter (known or unknown) comparisons are made between the location parameters. This is done by constructing confidence intervals, which can then be used for selection procedures. Comparisons are made with a control, and with the (unknown) “best” or “worst” population. Emphasis is laid on finding approximations to the confidence so that calculations are simple and tables are not necessary. (Since we consider unequal sample sizes, tables for exact values would need to be extensive.)     

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.     

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.     

On selecting the best population

This paper deals with the problem of selecting the “best” population from a given number of populations in a decision theoretic framework. The class of selection rules considered is based on a suitable partition of the sample space. A selection rule is given which is shown to have certain optimum properties among the selection rules in the given class for a mal rules are known.     

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.     

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.     

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.     

Post model-fitting exploration via a “Next-Door” analysis

We propose a simple method for evaluating the model that has been chosen by an adaptive regression procedure, our main focus being the lasso. This procedure deletes each chosen predictor and refits the lasso to get a set of models that are “close” to the chosen “base model,” and compares the error rates of the base model with that of nearby models. If the deletion of a predictor leads to significant deterioration in the model's predictive power, the predictor is called indispensable; otherwise, the nearby model is called acceptable and can serve as a good alternative to the base model. This provides both an assessment of the predictive contribution of each variable and a set of alternative models that may be used in place of the chosen model. We call this procedure “Next-Door analysis” since it examines models “next” to the base model. It can be applied to supervised learning problems with ℓ₁ penalization and stepwise procedures. We have implemented it in the R language as a library to accompany the well-known glmnet library. The Canadian Journal of Statistics 48: 447–470; 2020 © 2020 Statistical Society of Canada     

Step Down Procedures for Comparison With a Control

“Step down” or “sequentially rejective” procedures for comparisons with a control are considered for both one sided and two sided comparisons. Confidence bounds (in terms of the control) are derived for those (location) parameters not in a selected set. Special results are derived for the normal distribution with unknown variance where the sample numbers are (possibly) unequal.     

A weighted least-squares cross-validation bandwidth selector for kernel density estimation

Since the late 1980s, several methods have been considered in the literature to reduce the sample variability of the least-squares cross-validation bandwidth selector for kernel density estimation. In this article, a weighted version of this classical method is proposed and its asymptotic and finite-sample behavior is studied. The simulation results attest that the weighted cross-validation bandwidth performs quite well, presenting a better finite-sample performance than the standard cross-validation method for “easy-to-estimate” densities, and retaining the good finite-sample performance of the standard cross-validation method for “hard-to-estimate” ones.     

Estimation of the population mean when the coefficient of variation is known

Theory has been developed to provide an optimum estimator of the population mean based on a “mean per unit” estimator and the estimated standard deviation, assuming that the form of the distribution as well as its coefficient of variation (c.v.) are known. Theory has been extended to the case when an estimate of c.v. is available from an independent sample drawn in the past; the case when the form of the distribution is not known is also discussed. It is shown that the relative efficiency of the estimator with respect to “mean per unit estimator” is generally high for normal or near normal populations. For log-normal populations, an increase in efficiency of about 17 percent can be achieved. The results have been illustrated with data from biological populations.     

BIAS REDUCTION AND ELIMINATION WITH KERNEL ESTIMATORS

A great deal of research has focused on improving the bias properties of kernel estimators. One proposal involves removing the restriction of non-negativity on the kernel to construct “higher-order” kernels that eliminate additional terms in the Taylor's series expansion of the bias. This paper considers an alternative that uses a local approach to bandwidth selection to not only reduce the bias, but to eliminate it entirely. These so-called “zero-bias bandwidths” are shown to exist for univariate and multivariate kernel density estimation as well as kernel regression. Implications of the existence of such bandwidths are discussed. An estimation strategy is presented, and the extent of the reduction or elimination of bias in practice is studied through simulation and example.     

The efficiency of Efron's “Bootstrap” Approach Applied to Error Rate Estimation in Discriminant Analysis

The “bootstrap” approach of Efron is considered in its application to the estimation of error rates in discriminant analysis. Its efficiency relative to parametric estimation is investigated by simulation for Fisher's linear discriminant function in the context of two multivariate normal populations with a common covariance matrix.     

Learning from a lot: Empirical Bayes for high‐dimensional model‐based prediction

Empirical Bayes is a versatile approach to “learn from a lot” in two ways: first, from a large number of variables and, second, from a potentially large amount of prior information, for example, stored in public repositories. We review applications of a variety of empirical Bayes methods to several well‐known model‐based prediction methods, including penalized regression, linear discriminant analysis, and Bayesian models with sparse or dense priors. We discuss “formal” empirical Bayes methods that maximize the marginal likelihood but also more informal approaches based on other data summaries. We contrast empirical Bayes to cross‐validation and full Bayes and discuss hybrid approaches. To study the relation between the quality of an empirical Bayes estimator and p, the number of variables, we consider a simple empirical Bayes estimator in a linear model setting. We argue that empirical Bayes is particularly useful when the prior contains multiple parameters, which model a priori information on variables termed “co‐data”. In particular, we present two novel examples that allow for co‐data: first, a Bayesian spike‐and‐slab setting that facilitates inclusion of multiple co‐data sources and types and, second, a hybrid empirical Bayes–full Bayes ridge regression approach for estimation of the posterior predictive interval.     

A domain outlier robust design and smooth estimation approach

Outliers that commonly occur in business sample surveys can have large impacts on domain estimates. The authors consider an outlier‐robust design and smooth estimation approach, which can be related to the so‐called “Surprise stratum” technique [Kish, “Survey Sampling,” Wiley, New York (1965)]. The sampling design utilizes a threshold sample consisting of previously observed outliers that are selected with probability one, together with stratified simple random sampling from the rest of the population. The domain predictor is an extension of the Winsorization‐based estimator proposed by Rivest and Hidiroglou [Rivest and Hidiroglou, “Outlier Treatment for Disaggregated Estimates,” in “Proceedings of the Section on Survey Research Methods,” American Statistical Association (2004), pp. 4248–4256], and is similar to the estimator for skewed populations suggested by Fuller [Fuller, Statistica Sinica 1991;1:137–158]. It makes use of a domain Winsorized sample mean plus a domain‐specific adjustment of the estimated overall mean of the excess values on top of that. The methods are studied in theory from a design‐based perspective and by simulations based on the Norwegian Research and Development Survey data. Guidelines for choosing the threshold values are provided. The Canadian Journal of Statistics 39: 147–164; 2011 © 2010 Statistical Society of Canada     

On Mixture Failure Rates Ordering

Mixtures of increasing failure rate distributions (IFR) can decrease at least in some intervals of time. Usually, this property can be observed asymptotically as t → ∞. This is due to the fact that the mixture failure rate is “bent down” compared with the corresponding unconditional expectation of the baseline failure rate, which was proved previously for some specific cases. We generalize this result and discuss the “weakest populations are dying first” property, which leads to the change in the failure rate shape. We also consider the problem of mixture failure rate ordering for the ordered mixing distributions. Two types of stochastic ordering are analyzed: ordering in the likelihood ratio sense and ordering in variances when the means are equal.     

On some properties of variable size simple random sampling and a limit theorem

This paper introduces a sampling plan for finite populations herein called “variable size simple random sampling” and compares properties of estimators based on it with results from the usual fixed size simple random sampling without replacement. Necessary and sufficient conditions (in the spirit of Hajek (1960)) for the limiting distribution of the sample total (or sample mean) to be normal are given.     

Variable selection for semiparametric proportional hazards model under progressive Type-II censoring

Variable selection is an effective methodology for dealing with models with numerous covariates. We consider the methods of variable selection for semiparametric Cox proportional hazards model under the progressive Type-II censoring scheme. The Cox proportional hazards model is used to model the influence coefficients of the environmental covariates. By applying Breslow’s “least information” idea, we obtain a profile likelihood function to estimate the coefficients. Lasso-type penalized profile likelihood estimation as well as stepwise variable selection method are explored as means to find the important covariates. Numerical simulations are conducted and Veteran’s Administration Lung Cancer data are exploited to evaluate the performance of the proposed method.