Applications of a probability inequality to ranking and selection and other related problems

A probability inequality of conditionally independent and identically distributed (i.i.d.) random variables obtained recently by the author is applied to ranking and selection problems. It is shown that under both the indifference-zone and the subset formulations, the probability of a correct selection (PCS) is a cumulative probability of conditionally i.i.d, random variables. Therefore bounds on both the PCS and the sample size required can be obtained from that probability inequality. Applications of that inequality to other multiple decision problems are also considered. It is illustrated that general results concerning conditionally i.i.d. random variables are applicable to many problems in multiple decision theory.     

Extended tables for moments of gamma distribution order statistics

Apart from having intrinsic mathematical interest, order statistics are also useful in the solution of many applied sampling and analysis problems. For a general review of the properties and uses of order statistics, see David (1981). This paper provides tabulations of means and variances of certain order statistics from the gamma distribution, for parameter values not previously available. The work was motivated by a particular quota sampling problem, for which existing tables are not adequate. The solution to this sampling problem actually requires the moments of the highest order statistic within a given set; however the calculation algorithm used involves a recurrence relation, which causes all the lower order statistics to be calculated first. Therefore we took the opportunity to develop more extensive tables for the gamma order statistic moments in general. Our tables provide values for the order statistic moments which were not available in previous tables, notably those for higher values of m, the gamma distribution shape parameter. However we have also retained the corresponding statistics for lower values of m, first to allow for checking accuracy of the computtions agtainst previous tables, and second to provide an integrated presentation of our new results with the previously known values in a consistent format     

Which Wald statistic? Choosing a parameterization of the Wald statistic to maximize power in k-sample generalized estimating equations

The Wald statistic is known to vary under reparameterization. This raises the question: which parameterization should be chosen, in order to optimize power of the Wald statistic? We specifically consider k-sample tests of generalized linear models (GLMs) and generalized estimating equations (GEEs) in which the alternative hypothesis contains only two parameters. An example is presented in which such an alternative hypothesis is of interest. Amongst a general class of parameterizations, we find the parameterization that maximizes power via analysis of the non-centrality parameter, and show how the effect on power of reparameterization depends on sampling design and the differences in variance across samples. There is no single parameterization with optimal power across all alternatives. The Wald statistic commonly used under the canonical parameterization is optimal in some instances but it performs very poorly in others. We demonstrate results by example and by simulation, and describe their implications for likelihood ratio statistics and score statistics. We conclude that due to poor power properties, the routine use of score statistics and Wald statistics under the canonical parameterization for GEEs is a questionable practice.     

On simultaneous closeness probabilities of order statistics from odd sample sizes to the population median

In this note, we present alternative derivations for the probability that an individual order statistic is closest to the target parameter among all order statistics from a complete random sample. This approach is simpler than the geometric arguments used earlier. We also provide a simple direct proof for the symmetry property of the simultaneous closeness probabilities among order statistics for the estimation of percentiles from a symmetric family. Finally, we offer an alternative simpler proof for the result that sample medians from larger odd sample sizes are Pitman closer to the population median than sample medians from smaller odd sample sizes.     

Directional monotonicity properties of the power functions of likelihood ratio tests for cone-restricted hypotheses of normal means

In some cone-restricted testing problems, it is known that the power function of the likelihood ratio test has a certain monotonicity property with respect to the direction of the alternatives. In this paper, we establish a unified result concerning such directional monotonicity by using an argument based on a cone ordering and a related probability inequality. From this result, we show that the power function is unimodal in direction when the cone is suitably symmetric.     

Asymptotic properties of numbers of observations in random regions determined by central order statistics

In this paper, we extend the concept of near order statistic observation by considering observations that fall into a random region determined by a given order statistic and a Borel set. We study asymptotic properties of numbers of such observations as the sample size tends to infinity and the order statistic is a central one. We show that then proportions of these numbers converge in probability to some population probabilities. We also prove that these numbers can be centered and normalized to yield normal limit law. First, we derive results for one order statistic; next we give extensions to the multivariate case of two or more order statistics.     

Transform orders and stochastic monotonicity of statistical functionals

In some inferential statistical methods, such as tests and confidence intervals, it is important to describe the stochastic behavior of statistical functionals, aside from their large sample properties. We study such a behavior in terms of the usual stochastic order. For this purpose, we introduce a generalized family of stochastic orders, which is referred to as transform orders, showing that it provides a flexible framework for deriving stochastic monotonicity results. Given that our general definition makes it possible to obtain some well known ordering relations as particular cases, we can easily apply our method to different families of functionals. These include some prominent inequality measures, such as the generalized entropy, the Gini index, and its generalizations. We also illustrate the applicability of our approach by determining the least favorable distribution, and the behavior of some bootstrap statistics, in some goodness-of-fit testing procedures.     

p-values for the Discrete Scan Statistic through Slack Variables

The discrete scan statistic is used in many areas of applied probability and statistics to study local clumping of patterns. Testing based on the statistic requires tail probabilities. Whereas the distribution has been studied extensively, most of the results are approximations, due to the difficulties associated with the computation. Results for exact p-values for the statistic have been given for a binary sequence that is independent or first-order Markovian. We give an algorithm to obtain probabilities for the statistic over multi-state trials that are Markovian of a general order of dependence, and explore the algorithm's usefulness.     

On Estimating the Residual Rényi Entropy under Progressive Censoring

In this article, the residual Rényi entropy (RRE) as a measure of uncertainty is considered in progressively Type II censored samples and some properties of it are investigated. The RRE of sth order statistic from a continuous distribution function is represented in terms of the RRE of the sth order statistic from uniform distribution. In general, we do not have a closed form for RRE of order statistics in most of distributions. This gives us a motivation for obtaining some bounds for RRE in progressively censored samples. In addition, two estimators are proposed for RRE. The performance of these estimators is compared using simulation studies.     

Diagrams for Set Theory & Probability Problems of Four or More Variables

It is pointed out that in many one-sided testing situations for a real-valued parameter θ, the monotonicity of the power function hinges on the stochastic order of the underlying family of distributions [F_θ] rather than on the stronger property of monotone likelihood ratio of the family. An elementary proof, accessible to students of introductory probability and statistics, is presented.     

Interval estimation for the breakpoint in segmented regression: a smoothed score‐based approach

This paper is concerned with interval estimation for the breakpoint parameter in segmented regression. We present score‐type confidence intervals derived from the score statistic itself and from the recently proposed gradient statistic. Due to lack of regularity conditions of the score, non‐smoothness and non‐monotonicity, naive application of the score‐based statistics is unfeasible and we propose to exploit the smoothed score obtained via induced smoothing. We compare our proposals with the traditional methods based on the Wald and the likelihood ratio statistics via simulations and an analysis of a real dataset: results show that the smoothed score‐like statistics perform in practice somewhat better than competitors, even when the model is not correctly specified.     

A Unified Approach to Likelihood Inference on Linear Inequality Constraints in Contingency Tables

In contingency table analysis, a likelihood ratio test for linear inequality constraints is discussed. The restriction condition considered in this article is much more general than usual stochastic order restriction conditions. Asymptotic property of test statistic is shown. Simulation study is conducted to compare the empirical power and size performed via the proposed method and others. Several real data sets are used to illustrate our theoretical result. The idea used in this article can be applied to test the problems with nonlinear inequality constraints.     

Order statistics from overlapping samples: bivariate densities and regression properties

In this paper, we are interested in the joint distribution of two order statistics from overlapping samples. We give an explicit formula for the distribution of such a pair of random variables under the assumption that the parent distribution is absolutely continuous. We are also interested in the question to what extent conditional expectation of one of such order statistic given another determines the parent distribution. In particular, we provide a new characterization by linearity of regression of an order statistic from the extended sample given the one from the original sample, special case of which solves a problem explicitly stated in the literature. It appears that to describe the correct parent distribution it is convenient to use quantile density functions. In several other cases of regressions of order statistics we provide new results regarding uniqueness of the distribution in the sample.     

Quasi-nonparametric upper tolerance regions based on the bootstrap

In this paper the upper tolerance problem for random samples will be investigated. Here we will be concerned with populations that are skewed to the right and possess a finite minimum (e.g. the Exponential distribution). The method developed here takes the form of a multiplicative factor times a quantile estimate. The multiplicative factor is simple in form and is based on bootstrapped quantiles of order statistics, though no resampling is required. The quantile estimate itself could be of any desired form; for example, it could be nonparametric, and, therefore based on order statistics as well. The proposed tolerance limit has the desirable property of allowing for the possibility of exceeding the sample maximum, and therefore capturing more probability content, while avoiding, in general, use of the extreme order statistics.     

Simultaneous closeness among order statistics to population quantiles

We derive expressions for the probability that an individual order statistic is closest to the target parameter among the order statistics from a complete random sample. Results are given for random variables with bounded and complete support. We then apply these general results to location-scale parameter families of distributions with specific applications to estimation of percentiles. In this case, simultaneous-closeness probabilities depend upon the parameters through the value of p in the percentile and the sample size, n. Results are finally illustrated with the estimation of percentiles for normal and exponential distributions.     

Small-sample comparisons for the Rukhin goodness-of-fit-statistics

Rukhin's statistic family for goodness-of-fit, under the null hypothesis, has asymptotic chi-squared distribution; however, for small samples the chi-squared approximation in some cases does not well agree with the exact distribution. In this paper we consider this approximation and other three to get appropriate test levels in comparison with the exact level. Moreover, exact power comparisons for several values of the parameter under specified alternatives provide that the classical Pearson's statistic, obtained as a particular case of Rukhin statistic, can be improved by choosing other statistics from the family. An explanation is proposed in terms of the effects of individual cell frequencies on the Rukhin statistic. This work was supported in part by the DGCYT grants No. PR156/97-7159 and PB96-0635     

Exact Non-parametric Confidence Intervals for Quantiles with Progressive Type-II Censoring

It is shown how various exact non-parametric inferences based on order statistics in one or two random samples can be generalized to situations with progressive type-II censoring, which is a kind of evolutionary right censoring. Ordinary type-II right censoring is a special case of such progressive censoring. These inferences include confidence intervals for a given parent quantile, prediction intervals for a given order statistic of a future sample, and related two-sample inferences based on exceedance probabilities. The proposed inferences are valid for any parent distribution with continuous distribution function. The key result is that each observable uncensored order statistic that becomes available with progressive type-II censoring can be represented as a mixture with known weights of underlying ordinary order statistics. The importance of this mixture representation lies in that various properties of such observable order statistics can be deduced immediately from well-known properties of ordinary order statistics.     

Inference from PseudoLikelihoods with Plug‐In Estimates

Effective implementation of likelihood inference in models for high‐dimensional data often requires a simplified treatment of nuisance parameters, with these having to be replaced by handy estimates. In addition, the likelihood function may have been simplified by means of a partial specification of the model, as is the case when composite likelihood is used. In such circumstances tests and confidence regions for the parameter of interest may be constructed using Wald type and score type statistics, defined so as to account for nuisance parameter estimation or partial specification of the likelihood. In this paper a general analytical expression for the required asymptotic covariance matrices is derived, and suggestions for obtaining Monte Carlo approximations are presented. The same matrices are involved in a rescaling adjustment of the log likelihood ratio type statistic that we propose. This adjustment restores the usual chi‐squared asymptotic distribution, which is generally invalid after the simplifications considered. The practical implication is that, for a wide variety of likelihoods and nuisance parameter estimates, confidence regions for the parameters of interest are readily computable from the rescaled log likelihood ratio type statistic as well as from the Wald type and score type statistics. Two examples, a measurement error model with full likelihood and a spatial correlation model with pairwise likelihood, illustrate and compare the procedures. Wald type and score type statistics may give rise to confidence regions with unsatisfactory shape in small and moderate samples. In addition to having satisfactory shape, regions based on the rescaled log likelihood ratio type statistic show empirical coverage in reasonable agreement with nominal confidence levels.     

Tests for a family of random-effects covariance structures in a multivariate growth curve model

In this paper we propose test statistics for a general hypothesis concerning the adequacy of multivariate random-effects covariance structures in a multivariate growth curve model with differing numbers of random effects (Lange, N., N.M. Laird, J. Amer. Statist. Assoc. 84 (1989) 241–247). Since the exact likelihood ratio (LR) statistic for the hypothesis is complicated, it is suggested to use a modified LR statistic. An asymptotic expansion of the null distribution of the statistic is obtained. The exact LR statistic is also discussed.     

Two-sample Pitman closeness comparison under progressive Type-II censoring

In this paper, we consider two problems concerning two independent progressively Type-II censored samples. We first consider the Pitman closeness (PC) of order statistics from two independent progressively censored samples to a specific population quantile. We then consider the point prediction of a future progressively censored order statistic and discuss the determination of the closest progressively censored order statistic from the current sample according to the simultaneous closeness probabilities. For both these problems, explicit expressions are derived for the pertinent PC probabilities, and then special cases are given as examples. For various censoring schemes, we also present numerical results for the standard uniform, standard exponential, and standard normal distributions. Finally, a distribution-free result for the median is obtained.