The power functions of the likelihood ratio tests for a simply ordered trend in normal means

Likelihood ratio tests for the homogeneity of k normal means with the alternative restricted by an increasing trend are considered as well as the likelihood ratio tests of the null hypothesis that the means satisfy the trend. While the work is primarily a survey of results concerning the power functions of these tests, the extensions of some results to the case of not necessarily equal sample sizes are presented. For the case of known or unknown population variances, exact expressions are given for the power functions for k=3,4, and approximations are discussed for larger k. The topics of consistency, bias and monotonicity of the power functions are included. Also, Bartholomew's conjectures concerning minimal and maximal powers are investigated, with results of a new numerical study given.     

Recurrence relations for moments of different orders of extremes from gamma distribution

Certain recurrence relations for the moments of different orders of the largest order statistic from a gamma distribution with shape parameter p are obtained. By using this it is shown that for obtaining the moment of any order of each order statistic of a sample of size n from the gamma distribution, one has to evaluate at most n-2 single integrals.     

Bayesian credibility limits for small finite population proportions:stratified and cluster sampling

Bayes credibility limits for small proportions from stratified and fixed size cluster samples are discussed. Ericson’s (JRSS B (1969)) Beta Binomial and Dirichlet-Multinomial priors are used. Approximate limits that are appropriate for large samples and small proportions are derived in both cases. These allow asymptotic comparisons of the efficacy of stratified and cluster sampling relative to simple random sampling for estimating small proportions. Procedures for the selection of hyper parameters are also presented.     

Estimation of mean using double sampling for stratification and multivariate auxiliary information

Several estimators for estimating the mean of a principal variable are proposed based on double sampling for stratification (DSS) and multivariate auxiliary information. The general properties of the proposed estimators are studied, search for optimum estimators is made and the proposed estimators are compared with the corresponding estimators based on unstratified double sampling (USDS).     

Improved estimation of a finite population mean based on paired observations

This paper considers an improvement of the customary estimator of a finite population mean under a single stage sampling design when paired data, are available on each unit of the sample. Guided by the well known problem of “corninon mean”, a mixture i.e. a weighted combination of the mean of the principal characteristic and that of the auxiliary (possibly transformed) characteristic is proposed. It is shown that, under some conditions, improveinent (with respect to MSE) over the traditional estimator is possible for a broad range of the values of the mixing constant. An estimator of the MSE of the proposed estimator is also provided.     

Semiparametric estimation for count data through weighted distributions

This paper is concerned with semiparametric discrete kernel estimators when the unknown count distribution can be considered to have a general weighted Poisson form. The estimator is constructed by multiplying the Poisson estimate with a nonparametric discrete kernel-type estimate of the Poisson weight function. Comparisons are then carried out with the ordinary discrete kernel probability mass function estimators. The Poisson weight function is thus a local multiplicative correction factor, and is considered as the uniform measure to detect departures from the equidispersed Poisson distribution. In this way, the effects of dispersion and zero-proportion with respect to the standard Poisson distribution are also minimized. This method of estimation is also applied to the weighted binomial form for the count distribution having a finite support. The proposed estimators, in addition to being simple, easy-to-implement and effective, also outperform the competing nonparametric and parametric estimators in finite-sample situations. Two examples illustrate this new semiparametric estimation.     

Analysis of all-zero binomial outcomes with borderline and equilibrium priors

This article is concerned with the analysis of a random sample from a binomial distribution when all the outcomes are zero (or unity). We discuss how elicitation of the prior can be reduced to asking the expert whether (and which of) the so-called borderline or equilibrium priors are plausible.     

Estimating the propagation rate of a viral infection of potato plants via mixtures of regressions

A problem arising from the study of the spread of a viral infection among potato plants by aphids appears to involve a mixture of two linear regressions on a single predictor variable. The plant scientists studying the problem were particularly interested in obtaining a 95% confidence upper bound for the infection rate. We discuss briefly the procedure for fitting mixtures of regression models by means of maximum likelihood, effected via the EM algorithm. We give general expressions for the implementation of the M-step and then address the issue of conducting statistical inference in this context. A technique due to T. A. Louis may be used to estimate the covariance matrix of the parameter estimates by calculating the observed Fisher information matrix. We develop general expressions for the entries of this information matrix. Having the complete covariance matrix permits the calculation of confidence and prediction bands for the fitted model. We also investigate the testing of hypotheses concerning the number of components in the mixture via parametric and 'semiparametric' bootstrapping. Finally, we develop a method of producing diagnostic plots of the residuals from a mixture of linear regressions.