Glen Meeden

Measuring skewness and kurtosis

The question of how to measure the degree of skewness of a continuous random variable is addressed. In van Zwet (1964) a method for ordering two distributions with regard to skewness is given. Here, using the concept of comparative skewness, we consider properties that a measure of skewness should satisfy. Several extensions of the Bowley measure of skewness taking values on (-1, 1) are discussed. How well these measures reflect ones intuitive idea of skewness is examined. These measures of skewness are extended to measures of kurtosis for symmetric distributions.

Empirical Bayes Estimation in Finite Population Sampling

Abstract Empirical Bayes methods are becoming increasingly popular in statistics. Robbins (1955) introduced the method in the context of nonparametric estimation of a completely unspecified prior distribution. Subsequently, the method has been explored very successfully in a series of articles by Efron and Morris (1973, 1975, 1977) in a parametric framework. In the Efron—Morris setup, a family of parametric distributions is used as possible priors, but only when one or more of the parameters of the family of prior distributions is estimated from the data. Morris (1983) listed a number of areas where empirical Bayes methods are used. One of the main features of empirical Bayes analysis is to borrow strength from the ensemble—that is, use information from similar sources in constructing estimators and predictors in addition to the most directly available source of information. There are some situations in finite population sampling where such methods might be suitable. For instance, in many repetitive surve...

The Mode, Median, and Mean Inequality

Abstract An elementary method of proof of the mode, median, and mean inequality is given for skewed, unimodal distributions of continuous random variables. A proof of the inequality for the gamma, F, and beta random variables is sketched.

Fuzzy and Randomized Confidence Intervals and P-Values

The optimal hypothesis tests for the binomial distri- bution and some other discrete distributions are uniformly most powerful (UMP) one-tailed and UMP unbiased (UMPU) two- tailed randomized tests. Conventional confldence intervals are not dual to randomized tests and perform badly on discrete data at small and moderate sample sizes. We introduce a new con- fldence interval notion, called fuzzy confldence intervals, that is dual to and inherits the exactness and optimality of UMP and UMPU tests. We also introduce a new P -value notion, called fuzzy P -values or abstract randomized P -values, that also in- herits the same exactness and optimality.

Estimating the total number of distinct species using presence and absence data

SUMMARY Consider the problem of estimating the total number of distinct species in some specified region under investigation. Suppose the region is divided into N disjoint subregions or quadrats of equal area. A sample of size n quadrats is chosen, n 1. The other is to place at random n quadrats of equal area and fixed shape in the region of investigation. In both cases the n quadrats in the sample are assumed to be disjoint and are totally observed. So in fact when quadrat sampling is used, a random sample of space is taken instead of a random sample of individuals. It has been observed that species are often present with some natural clumping, which creates dependence between quadrats and within quadrats. Heltshe and Forrester (1983) and Smith and van Belle (1984) introduced some estimators for S when quadrat sampling is used; see also Burnham and Overton (1979). The estimators they proposed are nonparametric and were developed by using jackknife and bootstrap arguments. For a particular example, Palmer (1990) compared several estimators and showed that the first-order jackknife estimator is preferred within this group. In this paper we introduce some empirical Bayes estimators for S when quadrat sampling is used by imposing a grid on the region. The probabilistic model used to derive these estimators is basically a version of the Efron and Thisted (1976) model adapted to the case of quadrat

Calibration, Sufficiency, and Domination Considerations for Bayesian Probability Assessors

Abstract DeGroot and Fienberg (1982a) recently considered various aspects of the problem of evaluating the performance of probability appraisers. After briefly reviewing their notions of calibration and sufficiency we introduce related ideas of semicalibration and domination and consider their relationship to the earlier concepts. We then discuss some simple Bayesian mechanisms for making probability assessments and study their calibration, semicalibration, sufficiency, and domination properties. Finally, several results concerning the comparison of finite dichotomous experiments, relevant to the present work, are collected in an Appendix.

Selecting the Number of Bins in a Histogram: A Decision Theoretic Approach

Abstract 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.

Unbiasedness as the Dual of Being Bayes

In this note we define the notion of unbiasedness for a decision function for an arbitrary loss function. This is a generalization of Lehmanns (1951) definition. We show that this notion of unbiasedness is a dual to the notion of being Bayes; that is, if the role of the random variable and the parameter is interchanged, then unbiasedness is equivalent to being Bayes and vice versa. Some consequences of this fact are discussed.

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.

A Noninformative Bayesian Approach to Interval Estimation in Finite Population Sampling

Abstract A noninformative Bayesian approach to interval estimation in finite population sampling is discussed. Given the sample, this method introduces the Polya distribution as a pseudo posterior distribution over the unobserved members of the population. In many cases this distribution yields interval estimates similar to those of standard frequentist theory. In addition, it can be used in situations where the standard methods are difficult to apply, for example, in producing an interval estimate for the ratio of two medians. We also consider related point estimation problems and observe that estimators derived from the pseudo posterior often perform better than classical alternatives.