Satya N. Mishra

Confidence interval estimation of overlap: equal means case

The estimators of the commonly used measures of overlap are known to be biased by an amount which depends on the unknown overlap. In general it is dicult to calculate the precision or bias of most ecological measures because there is no exact formula for the variance of the statistic and the sampling distribution is unknown (Dixon, 1993. In: Scheiner, S.M., Gurevitch, J. (Eds.), Design and Analysis of Ecological Experiments. Chapman & Hall, New York, pp. 290{318.) Two resampling techniques, namely, Jackknife and Bootstrap along with the Taylor series approximation and transformation method are considered for the construction of condence intervals. Three measures of overlap frequently used in quantitative ecology and considered in this study are Matusita’s measure , Morisita’s measure, , and Weitzman’s measure, . c 2000 Elsevier Science B.V. All rights reserved.

Overlapping coefficient: the generalized t approach

The two group t test is generalized here to produce a hypothesis testing procedure on the overlapping coefficient of two normally distributed populations with common variance, assuming that the researcher knows the direction of the population means. The confi¬dence intervals are constructed on the overlapping coefficient. An illustrative example is given using the proposed procedures

Simultaneous Testing for the Successive Differences of Exponential Location Parameters

Suppose that k (k ≥ 3) treatments under comparison are ordered in a certain way. For example, the treatments may be increasing dose levels in dose response experiments. The exponential distribution E(μ,Θ) is generally used to model the effective duration of a drug, where the location parameter μ is referred to as latency period that may decrease/increase with the increase in dose level of the drug. In such situations, the experimenter may be interested in the successive comparisons of the treatments. Let E(μ1,Θ1),…,E(μ k ,Θ k ) be k independent exponential distributions/populations with μ i (Θ i ) as the location (scale) parameter of the i th population, i = 1,…,k. In this article, we propose test procedures for simultaneously testing the family of hypotheses A recursive method for computing the critical constants is discussed. The required tables of critical constants for the implementation of the proposed test procedures are presented. The test procedure is used to derive the simultaneous confidence intervals for the successive differences between the location parameters; that is, μ2 − μ1,μ3 − μ2,…,μ k − μ k−1. We also extend these simultaneous confidence intervals for successive differences to a larger class of contrasts of the location parameters.

Simultaneous Selection of Extreme Population Means: Indifference-Zone Formulation

SYNOPTIC ABSTRACTIn his pioneering 1954 paper, R.E. Bechhofer discussed alternative selection goals for one-way classification, and the two-way classification, without interaction, when the variances are known and equal. For the one-way classification, he proposed the general problem where we want to decide which ks populations have the ks largest means, which ks-1 populations have the ks-1 largest means, and so on, with k1,k2, …, ks given positive integers with k1 + k2 + k3 + …+ ks = 1. The present paper considers selection of the “best” and “worst” populations simultaneously in the indifference-zone setting, with variances unknown (possibly unequal) as well as when they are known and equal, which is Bech-hofers s = 3, k1 = l, k2 = k-2, k3 = 1 goal. Relevant tables are included.

Inference on measures of niche overlap for heteroscedastic normal populations

SYNOPTIC ABSTRACTEcologists and economists have long been interested in measures of overlap between two populations to study disparities between resource utilizations or incomes of two groups. Also sociologists have been interested in spatial segregation of social groups, particularly racial and occupational groups. While the measures of overlap under the assumption of homogeneity of variances have received a good deal of attention, little has been written about the overlap of heteroscedastic populations. Three different measures of niche overlap between two normally distributed populations are studied in this paper. Estimation of overlap under general conditions of heterogeneity of variances is discussed. The expressions for approximate bias and variance of estimates are derived using both MLE, and unbiased estimators of population variances. The proximity of variance and bias estimates computed using the estimates of overlap coefficients to that using the actual values is studied using simulation techni...

Inference on Overlap for Two Inverse Gaussian Populations: Equal Means Case

The inverse Gaussian distribution is often suited for modeling positive and/or positively skewed data (see Chhikara and Folks, 1989) and presents an interesting alternative to the Gaussian model in such cases. We note here that overlap coefficients and their variants are widely studied in the literature for Gaussian populations (see Mulekar and Mishra, 1994, 2000, and references therein for further details). This article studies the properties and addresses point estimation for large samples of commonly used measures of overlap when the populations are described by inverse Gaussian distributions. The bias and mean square error properties of the estimators are studied through a simulation study.

A Class of Selection Procedures Based on Sample Quasi-Ranges

SYNOPTIC ABSTRACT Let Ei = E(μi, θi), i = 1,…,k be k independent exponential distributions/populations with μi(θi) as the location (scale) parameter of ith population. In literature use of quasi-range as a measure of dispersion has been advocated in censored samples because it is robust against outliers (see David (1981) section 7.4). In this paper, we propose a class of subset selection procedures based on sample quasi-ranges to select a random size subset of the populations E1…Ek, which contains a population corresponding to the least scale parameter with probability at least P*, a pre-specified value, ( < P* < 1). The constants needed to implement these procedures are tabulated. The members of the proposed class possess the monotonicity property and the maximum value of expected subset size is kP*. The procedures are very useful and easy to apply (even by persons without advanced statistical background) in reliability, engineering and quality control, where the exponential distribution is used to model the life length of components and the experimenter has censored samples (or samples containing outliers or small sample sizes) with which to work.

Selection Using Overlap Coefficients

SYNOPTIC ABSTRACT Selection of population with the largest (smallest) amount of overlap is applicable in many socio-economic, ecological as well as geological situations. From the available k(k ≥ 2) populations, the goal is to select the population corresponding to the largest (smallest) amount of overlap between two mutually exclusive subgroups of each population. A Bechhofer style selection procedure is proposed, which is carried out based on a random sample of size n taken independently from each subgroup. The relation between the Mahalanobis distance and the overlap coefficient is used to develop the selection procedure required to achieve a pre-specified probability of correct selection. The least favorable configuration and the expressions for the infimum of the probability of correct selection are given under two scenarios, namely (a) the population variances are known and (b) the population variances are unknown. A table of constants is provided for determining the optimal sample sizes needed from each population to meet the specifications. One application of this procedure to the hospitalization data for newborn babies is discussed.

ML and REML estimation of Matusita's measure for two bivariate normal distributions with missing observations

SYNOPTIC ABSTRACT The measures of niche overlap are used to assess the similarity or dissimilarity of two populations. Matusitas measure is one of the commonly used niche overlap measures. We discuss the problem of estimating Matusitas measure when the niches are bivariate normal distributions with missing observations. Under the assumption of equal variance of two variates in each population, we consider four cases depending on whether the variances and correlations for the two populations are common or different. The plug-in estimates of Matusitas measure by the Maximum Likelihood (ML) estimates and the REstricted or REsidual Maximum Likelihood (REML) estimates for dispersion parameters are considered, their asymptotic variances and bias are derived, and bias correction methods are proposed. Simulation study shows that the plug-in estimate by the REMLE tends to have smaller MSE than that by the MLE and the bias correction reduces MSE considerably.

Preface: Confluence of the Mathematical Sciences

SCMA: Confluence of the Mathematical Sciences (Statistics, Combinatorics, Mathematics, and Applications), Volume I of Proceedings of the International Conference on Statistics, Combinatorics, Mathematics, and Applications (SCMA) , sponsored by the Forum for Interdisciplinary Mathematics, Auburn University, and the University of South Alabama at Auburn University, U.S.A. (Auburn, Alabama), Volume 59 of the American Series in Mathematical and Management Sciences (ISBN 0-935950-63-X, 9780935950632, LC2009924927) & American Journal of Mathematical and Management Sciences, Volume 28 (2008), Nos. 3 & 4 (ISSN 0196-6324, CODEN AMMSDX), edited by D. Mark Carpenter (Auburn University) and Satya N. Mishra (University of South Alabama).