Roy F. Bartlett

An approximation to the distribution of the ratio of two general quadratic forms with application

Based on mixed cumulants up to order six, this paper provides a four moment approximation to the distribution of a ratio of two general quadratic forms in normal variables. The approximation is applied to calculate the percentile points of modified F-test statistics for testing treatment effects when standard F-ratio test is misleading because of dependence among observations. For the special case, when data is generated by an AR(1) process, the approximation is evaluated by a simulation study. For the general SARMA (p,q)(P,Q)s process, a modified F-test statistic Is given, and its distribution for the (0,1)(0,l)12 process, is approximated by the moment approximation technique.

Estimating the total of a continous population

Abstract This paper discusses the estimation of the total of a continuous population defined over a finite support region. Continuous populations such as percent moisture content over a field are likely to exhibit trend and correlation along dimensions of the support region. A superpopulation model is used to incorporate both trend and correlation in the estimation of the population total. Four criterion involving the predictive mean square error of the estimator and ξ-unbiasedness used to determine estimation strategy are examined. It is shown that under constant trend and triangular correlation that the expansion estimator is optimal in conjunction with the centered systematic sample over a one dimensional support region. We introduce a set of linear interpolation estimators that are either ξ-unbiased or have minimum ξ-bias.

On estimating equations for parameters in generalized linear mixed models with application to binary data

In view of the cumbersome and often intractable numerical integrations required for a full likelihood analysis, several suggestions have been made recently for approximate inference in generalized linear mixed models and other nonlinear variance component models. For example, we refer to the penalized quasi-likelihood (PQL) approach of Breslow and Clayton (1993), the Stein-type estimating function based approach of Waclawiw and Liang (1993), and the corrected PQL approach of Breslow and Lin (1995). Recently, Sutradhar and Godambe (1998) provided a semiparametric solution to the estimation problem dealt with by Waclawiw and Liang. In the present article, we provide a semiparametric solution to the estimation problem investigated by Breslow and Lin, and Breslow and Clayton. More specifically, we propose a two-step joint estimating equations approach to estimate the model parameters. In the first step, we use an estimating function based approach to obtain the estimates of the random effects. In the second step, following Prentice and Zhao (1991), we construct the first two moment based joint estimating equations for the regression parameters and the variance component of the random effects. As the exact first and second order moments of the generalized linear mixed models are not available, these moments are obtained first by expanding the conditional moments for given random effects about their estimating function based estimates and then taking the expectation of the conditional moments over the true distribution of the random effects. Since binary mixed models arise in many biomedical application areas, for example, computations are provided in detail for this special case. We also discuss a second approach, namely, a three-step ad hoc estimating equations approach, in the context of the special binary mixed models. The performance of the proposed estimators are examined through a simulation study. The estimation procedure is also illustrated through an analysis of the child health and development study (CHDS) data from Yerushalmy (1970).

Monte carlo comparison of wald's, likelihood ratio and rao's tests

This paper, through a simulation study, examines the behaviour of Walds, the likelihood ratio and Raos tests for testing (a) simple hypotheses, (b) one-dimensional composite hypotheses, as well as (c) multi-dimensional composite hypotheses. Peers (1971) has shown that none of these three asymptotic test procedures is uniformly superior to the other two tests for testing simple hypotheses versus composite hypotheses. In a series of papers, Chandra and Joshi (1983), Chandra and Mukerjee (1985), and Chandraand Samanta (1988) have claimed that for large sample size, Raos size-adjusted test is locally more powerful than either the size-adjusted likelihood ratio test or the size-adjusted Wald test, when these tests are performed at a common and sufficiently low level of significance. In the present study, the tests were performed at the standard levels (5% and 1%) of significance, and various local alternatives were considered. For sample size as small as 20, the simulation study appears to support the resul...