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.     

Selecting the better binomial population under a budget constraint

The problem of optimal non-sequential allocation of observations for the selection of the better binomial population is considered in the case of fixed sampling costs and budget. With the appropriate choice of selection rule it is shown that a 70% reduction in the probability of incorrect selection is possible by using an unequal rather than equal allocation. Simple formulae are given for the appropriate selection rule and unequal allocation in large samples.     

On selecting from k finite populations the population with the largest α -quantile

Several procedures for ranking populations according to the quantile of a given order have been discussed in the literature. These procedures deal with continuous distributions. This paper deals with the problem of selecting a population with the largest α-quantile from k ≥ 2 finite populatins, where the size of each population is known. A selection rule is given based on the sample quantiles, where he samples are drawn without replacement. A formula for the minimum probability of a correct selection for the given rule, for a certain configuration of the population α-quantiles, is given in terms of the sample numbers.     

A conservative Sample size for the comparison of several proportions

In many experimental situations we need to test the hypothesis concerning the equality of parameters of two or more binomial populations. Of special interest is the knowledge of the sample sizes needed to detect certain differences among the parameters, for a specified power, and at a given level of significance. Al-Bayyati (1971) derived a rule of thumb for a quick calculation of the sample size needed to compare two binomial parameters. The rule is defined in terms of the difference desired to be detected between the two parameters.

In this paper, we introduce a generalization of Al-Bayyatifs rule to several independent proportions. The generalized rule gives a conservative estimate of the sample size needed to achieve a specified power in detecting certain differences among the binomial parameters at a given level of significance. The method is illustrated with an example     

Error rates in classification consisting of discrete and continuous variables in the presence of covariates

We consider classifying an object based on mixed continuous and discrete variables between two populations. Mixed discrete and continuous covariates with identical means in both populations are amongst the variables. Under the location model with homogeneous location specific conditional dispersion matrices for both populations, the Bayes rule is given. Classification is implemented by a plug-in version of the Bayes rule with full covariate adjustment. An asymptotic expansion of the overall expected error of the procedure is derived. Our findings generalize several classical results.     

Model and design correspondence in finite population sampling

A correspondence rule is suggested for the choice of a sampling design when prior knowledge concerning a finite population is available. Designs satisfying the correspondence rule are discussed in the case of random permutations models. A general optimality theorem is given for strategies under such models. Approximate correspondences satisfied by systematic sampling and πps sampling are also indicated.     

Selecting the number of bins in a histogram: A decision theoretic approach

In this note we consider the problem of, given a sample, selecting the number of bins in a histogram. A loss function is introduced which reflects the idea that smooth distributions should have fewer bins than rough distributions. A stepwise Bayes rule, based on the Bayesian bootstrap, is found and is shown to be admissible. Some simulation results are presented to show how the rule works in practice.     

Selecting the best geometric distribution based on type-I censored data: a Bayesian approach

This paper considers the statistical reliability on discrete failure data and the selection of the best geometric distribution having the smallest failure probability from among several competitors. Using the Bayesian approach a Bayes selection rule based on type-I censored data is derived and its associated monotonicity is also obtained. An early selection rule which allows us to make a selection possible earlier than the censoring time of the life testing experiment is proposed. This early selection rule can be shown to be equivalent to the Bayes selection rule. An illustrative example is given to demonstrate the use and the performance of the early selection rule.     

Asymptotically pointwise optimal and asymptotically optimal stopping times in the Bayesian inference

The paper deals with the problem of determining asymptotically pointwise optimal and asymptotically optimal stopping times in the Bayesian inference. The sufficient conditions are given for a family of stopping times to be asymptotically pointwise optimal and asymptotically optimal with respect to a continuous time process. As an example a sequential estimation of the intensity of the Poisson process is considered. Under a gamma prior distribution, an asymptotically pointwise optimal and asymptotically optimal rule is given using a LINEX loss function and the cost c per unit time.     

Inference rules and inferential distributions

We introduce the concept of inferential distributions corresponding to inference rules. Fiducial and posterior distributions are special cases. Inferential distributions are essentially unique. They correspond to or represent inference rules and are defined on the parameter space. Not all inference rules can be represented by an inferential distribution. A constructive method is given to investigate its existence for any given inference rule.     

Horwitz's rule, transforming both sides and the design of experiments for mechanistic models

Summary. The paper develops methods for the design of experiments for mechanistic models when the response must be transformed to achieve symmetry and constant variance. The power transformation that is used is partially justified by a rule in analytical chemistry. Because of the nature of the relationship between the response and the mechanistic model, it is necessary to transform both sides of the model. Expressions are given for the parameter sensitivities in the transformed model and examples are given of optimum designs, not only for single-response models, but also for experiments in which multivariate responses are measured and for experiments in which the model is defined by a set of differential equations which cannot be solved analytically. The extension to designs for checking models is discussed.     

Optimal designs for the development of personalized treatment rules

We study the design of multi-armed parallel group clinical trials to estimate personalized treatment rules that identify the best treatment for a given patient with given covariates. Assuming that the outcomes in each treatment arm are given by a homoscedastic linear model, with possibly different variances between treatment arms, and that the trial subjects form a random sample from an unselected overall population, we optimize the (possibly randomized) treatment allocation allowing the allocation rates to depend on the covariates. We find that, for the case of two treatments, the approximately optimal allocation rule does not depend on the value of the covariates but only on the variances of the responses. In contrast, for the case of three treatments or more, the optimal treatment allocation does depend on the values of the covariates as well as the true regression coefficients. The methods are illustrated with a recently published dietary clinical trial.     

The move-to-partner rule for self-organizing task allocation on a linear array

The well-known Move-to-Front rule for self-organizing lists is modified to a so-called Move-to-Partner rule for dynamic task allocation on a chain of processors. We enable an analysis of the sequence of expected communication costs by reducing their computation to that of three-dimensional arrays of certain probabilities, for which a recursion formula and an asymptotic expansion can be given. The costs depend on the underlying communication profile; two types of such profiles are investigated in more detail, namely a two-dimensional version of Zipf's law, and profiles based on a class structure of tasks. In the considered cases, the Move-to-Partner rule effects fast convergence to a stationary state, but comparably high expected stationary costs.     

CLASSIFICATION AND PROPORTION ESTIMATION

Assume that a number of individuals are to be classified into one of two populations and that, at the same time, the proportion of members of each population needs to be estimated. The allocated proportions given by the Bayes classification rule are not consistent estimates of the true proportions, so a different classification rule is proposed; this rule yields consistent estimates with only a small increase in the probability of misclassification. As an illustration, the case of two normal distributions with equal covariance matrices is dealt with in detail.     

Reliability Estimation Based on System Data with an Unknown Load Share Rule

We consider a multicomponent load-sharing system in which the failure rate of a given component depends on the set of working components at any given time. Such systems can arise in software reliability models and in multivariate failure-time models in biostatistics, for example. A load-share rule dictates how stress or load is redistributed to the surviving components after a component fails within the system. In this paper, we assume the load share rule is unknown and derive methods for statistical inference on load-share parameters based on maximum likelihood. Components with (individual) constant failure rates are observed in two environments: (1) the system load is distributed evenly among the working components, and (2) we assume only the load for each working component increases when other components in the system fail. Tests for these special load-share models are investigated.     

A Bayesian procedure for the sequential estimation of the mean of a negative-binomial distribution

A Bayesian procedure for the sequential estimation of the mean of a negative-binomial distribution is considered. A reasonable sequential stopping rule, based on a one-step look-ahead procedure is derived. The procedure can be applied to estimate the infestation level of insects in a given field crop area.     

Prbposterior expected loss as a scoring rule for prior distributions

A scoring rule for evaluating the usefulness of an assessed prior distribution should reflect the purpose for which the distribution is to be used. In this paper we suppose that sample data is to become available and that the posterior distribution will be used to estimate some quantity under a quadratic loss function. The utility of a prior distribution is consequently determined by its preposterior expected quadratic loss. It is shown that this loss function has properties desirable in a scoring rule and formulae are derived for calculating the scores it gives in some common problems. Many scoring rules give a very poor score to any improper prior distribution but, in contrast, the scoring rule proposed here provides a meaningful measure for comparing the usefulness of assessed prior distributions and non-informative (improper) prior distributions. Results for making this comparison in various situations are also given.     

A simple procedure for the selection of significant effects

Summary.  Given a large number of test statistics, a small proportion of which represent departures from the relevant null hypothesis, a simple rule is given for choosing those statistics that are indicative of departure. It is based on fitting by moments a mixture model to the set of test statistics and then deriving an estimated likelihood ratio. Simulation suggests that the procedure has good properties when the departure from an overall null hypothesis is not too small.     

Exact monte carlo tests using several statistics

Concepts of ranking and boundary of multivariate statistics are discussed and applied to the simultaneous use of several test statistics calculated for data and simulated replicates. An example of residual analysis in regression is given using layer ranks and supplementary simulation with a stopping rule.     

Selection bias in working with the top genes in supervised classification of tissue samples

Currently there is much interest in using microarray gene-expression data to form prediction rules for the diagnosis of patient outcomes. A process of gene selection is usually carried out first to find those genes that are most useful according to some criterion for distinguishing between the given classes of tissue samples. However, there is a bias (selection bias) introduced in the estimate of the final version of a prediction rule that has been formed from a smaller subset of the genes that have been selected according to some optimality criterion. In this paper, we focus on the bias that arises when a full data set is not available in the first instance and the prediction rule is formed subsequently by working with the top-ranked genes from the full set. We demonstrate how large the subset of top genes must be before this selection bias is not of practical consequence.