Andrew L. Rukhin

Estimation of a Common Mean and Weighted Means Statistics

Abstract Measurements made by several laboratories may exhibit nonnegligible between-laboratory variability, as well as different within-laboratory variances. Also, the number of measurements made at each laboratory often differ. Questions of fundamental importance in the analysis of such data are how to form a best consensus mean, and what uncertainty to attach to this estimate. An estimation equation approach due to Mandel and Paule is often used at the National Institute of Standards and Technology (NIST), particularly when certifying standard reference materials. Primary goals of this article are to study the theoretical properties of this method, and to compare it with some alternative methods, in particular to the maximum likelihood estimator (MLE). Toward this end, we show that the Mandel-Paule solution can be interpreted as a simplified version of the maximum likelihood method. A class of weighted means statistics is investigated for situations where the number of laboratories is large. This class...

Restricted maximum likelihood estimation of a common mean and the Mandel–Paule algorithm

Abstract The estimation of a common normal mean on the basis of interlaboratory evaluations is studied when there is an interlaboratory effect. An estimation equation approach due to Mandel and Paule is examined and its theoretical properties are studied. In particular, we show that the Mandel–Paule solution can be interpreted as a simplified version of the restricted maximum likelihood method. It is also demonstrated that the Mandel–Paule algorithm is a generalized Bayes procedure. The results of numerical comparison of these estimators for a special distribution of within-laboratory variances are also reported.

Risk behavior of variance estimators in multivariate normal distribution

In this paper we consider the estimation problem of unknown variance of a multivariate normal vector under quadratic loss and entropy loss. The behavior of risk functions of the Brewster--Zidek estimator and the original Stein estimator is examined. Numerical studies show that an asymptotically inadmissible Stein estimator provides a larger degree of improvement than an admissible Brewster--Zidek estimator.

Distribution of the number of words with a prescribed frequency and tests of randomness

The goal of this paper is to investigate properties of statistical procedures based on numbers of different patterns by using generating functions for the probabilities of a prescribed number of occurrences of given patterns in a random text. The asymptotic formulae are derived for the expected value of the number of words occurring a given number of times and for the covariance matrix. The form of the optimal linear test based on these statistics is established. These problems appear in testing for the randomness of a string of binary bits, DNA sequencing, source coding, synchronization, quality control protocols, etc. Indeed, the probabilities of repeated (overlapping) patterns are important in information theory (the second-order properties of relative frequencies) and molecular biology problems (finding patterns with unexpectedly low or high frequencies).

Distribution of the number of visits of a random walk

The distribution of the number of visits to a given state within an excursion of a simple random walk is derived. This distribution is shown to be a zero-modified geometric law, which extends a result of Revesz

Bahadur Efficiency of Tests of Separate Hypotheses and Adaptive Test Statistics

The notion of separability of hypotheses was first introduced by Cox (1961); two families of hypotheses are separate if no distribution in one family can be obtained as a limit of distributions from the other family. For testing separate hypotheses, Cox (1961, 1962) suggested the test based on the comparison of the logarithm of the likelihood ratio with its expected value (or estimate thereof) under each hypothesis. The tests of such hypotheses are required to have high power against the specified alternatives; when nuisance parameters are present, this concept leads to the notion of an adaptive test, which by definition must be asymptotically efficient for any value of the unknown nuisance parameter. A testing problem of separate hypotheses for an exponential family is studied from the standpoint of Bahadur asymptotic optimality. A formula for the Bahadur-exact slope of any smooth test statistic is obtained, and in the example of testing lognormality versus exponentiality, it is shown that Coxs test can...

Generalized Bayes estimators of a normal discriminant function

It is shown that the traditional estimator of a discriminant coefficient vector is the generalized Bayes estimator with respect to a prior which can be approximated by proper priors. As a corollary the admissibility of this procedure in the class of all scale equivariant estimators is derived.

Limiting Distributions in Sequential Occupancy Problem

Abstract In a general sequential allocation scheme the limiting distribution of the instant at which a treatment receives the given number of subjects is derived. Classical occupancy problems, in particular, the Banach match-box problem and the birthday problem are shown to be closely related and are discussed.

Perpetuities and asymptotic change-point analysis

The distribution of stochastically discounted sums (perpetuities) is studied. For Bernoulli-type variables a canonical representation of this distribution is obtained, and it is proven to be singular continuous. In the asymptotic setting of the change-point estimation problem the limiting behavior of the posterior distribution is shown to be given by two independent perpetuities.

ESTIMATING EXPONENTIAL RELIABILITY FUNCTION AND EXPONENTIAL DENSITY

The estimation problem of the reliability function and the density at a point of two-parameter exponential population is considered. The Bayes nature of the best unbiased estimator of the reliability is demonstrated. The nonexistence of the unbiased density estimates is noticed and the admissibility of related generalized Bayes estimators in the class of all scale-equivariant procedures is established. Asymptotical formulae for quadratic risk of these estimators and the maximum likelihood estimator are obtained.