Barry C. Arnold

Parameter Estimation for a Multivariate Exponential Distribution

Simple consistent estimates are derived for the parameters of a multivariate exponential distribution. The mean squared error of the estimates is computed, and a lower bound for their efficiency is derived. A precise lower bound for the efficiency is computed in the bivariate case. It does not seem feasible to compute the efficiency precisely in the multivariate case. Some attention is given to the behavior of the estimates when observations are rounded prior to analysis.

Inadmissibility of the Usual Scale Estimate for a Shifted Exponential Distribution

Abstract Let X (1), X (2), …, X (n) be the order statistics of a sample of size n from the density f(x; μ, σ) = σ, −1e−(x-μ)1σ > μ. The statistic which is the best invariant estimate of the scale parameter σ, is shown to be inadmissible under squared error loss. It is shown to be strictly dominated by an estimate that is not translation invariant. Similarity between this estimation problem and the normal scale estimation problem treated by Stein [6] is discussed.

A Note on Multivariate Distributions with Specified Marginals

Abstract A method for generating multivariate distributions with marginals in a prescribed family is introduced. It is a straightforward generalization of a method due to Marshall and Olkin.

The Admissibility of a Preliminary Test Estimator When the Loss incorporates a Complexity Cost

Abstract Consider the problem of estimating the mean by using a random sample from a normal population. Let denote the sample mean and consider the estimator that assumes the value zero when and the value when . This is called a preliminary test estimator. For most of the usual loss functions it is inadmissible. In this article we show that for some loss functions, which include a complexity cost, the estimator is admissible. The results are related to Cohens work on hybrid estimation and hypothesis-testing problems.

Recurrence relations between expectations of functions of order statistics

Abstract General recurrence relations are derived for expectations of functions of one or more order statistics. Several examples are presented. The results also hold for order statistics from exchangeable sequences and for concomitants of order statistics.

A characterization of geometric distributions by distributional properties of order statistics

Abstract Let X 1, X 2 be independent identically distributed positive integer valued random variables. H the X i s have a geometric distribution, then the conditional distribution of R = max(X 1, X 2)-min(X 1, X 2), given R > 0, is the same as the distribution of X 1. This property is shown to characterize the geometric distribution.

On sojourn times at particular gene frequencies

The distribution of visits to a particular gene frequency in a finite population of size N with non-overlapping generations is derived. It is shown, by using well-known results from the theory of finite Markov chains, that all such distributions are geometric, with parameters dependent only on the set of b ij s, where b ij is the mean number of visits to frequency j /2 N , given initial frequency i /2 N . The variance of such a distribution does not agree with the value suggested by the diffusion method. An improved approximation is derived.

The waiting time until first duplication

Balls are drawn with replacement from an urn containing m distinguishable balls until a match is noted. The distribution of the number of drawings required is considered in the case where only the last k balls drawn are remembered. The asymptotic behavior of this distribution, as m and k become large, is investigated. Two further variants of the problem are suggested. BIRTHDAY PROBLEM; DISCRETE WAITING TIME; FINITE MEMORY; FIRST MATCH; LIMITING DENSITIES

Hypothesis Testing Incorporating a Preliminary Test of Significance

Abstract Suppose that independent samples are available from two normal populations with means μ1 and μ2 and known variances. It is desired to test H: μ1 =0. A possible preliminary test procedure in this case consists of first testing μ1 = μ2 and then testing H using either the sample mean from the first population or the pooled sample mean depending on the outcome of the test regarding equality of μ1 and μ2. This procedure is shown to be biased. A power comparison of this procedure with the generalized likelihood ratio test of H indicates that the latter test is more powerful in a fairly extensive parametric region. It is noted that biasedness is frequently encountered in preliminary test procedures. In light of the example, it is recommended that power comparisons be made between such procedures and any available unbiased tests to ascertain whether extensive power loss regions exist or not.

Bounds on deviations of estimates arising in finite population regression models

Bounds for the maximum deviation between parameters of a finite population and their corresponding sample estimates are found in the multiple regression model. The parameters considered are the vector of regression coefficients and the value ofthe regression function for given values of the independent variable (or variables). Applications are considered to several widely employed sampling methods.