Bayesian Reliability Estimation of a Two-parameter Cauchy Distribution

This paper gives an approximate Bayes procedure for the estimation of the reliability function of a two-parameter Cauchy distribution using Jeffreys' non-informative prior with a squared-error loss function, and with a log-odds ratio squared-error loss function. Based on a Monte Carlo simulation study, two such Bayes estimators of the reliability are compared with the maximum likelihood estimator.     

On the maximum likelihood method for estimating molecular trees: Uniqueness of the likelihood point

Summary Studies are carried out on the uniqueness of the stationary point on the likelihood function for estimating molecular phylogenetic trees, yielding proof that there exists at most one stationary point, i.e., the maximum point, in the parameter range for the one parameter model of nucleotide substitution. The proof is simple yet applicable to any type of tree topology with an arbitrary number of operational taxonomic units (OTUs). The proof ensures that any valid approximation algorithm be able to reach the unique maximum point under the conditions mentioned above. An algorithm developed incorporating Newton's approximation method is then compared with the conventional one by means of computers simulation. The results show that the newly developed algorithm always requires less CPU time than the conventional one, whereas both algorithms lead to identical molecular phylogenetic trees in accordance with the proof. Contribution No. 1780 from the National Institute of Genetics, Mishima 411, Japan     

On the exact null distribution of the generalised likelihood ratio test for analysing unreplicated factorial designs

Al-Shiha and Yang (1999) proposed a multistage procedure for analysing unreplicated factorial experiments, which is based on the statistic that is derived from the generalised likelihood ratio test statistic under the assumption of normality. It was shown by their simulation study that the method is quite competitive with Lenth's (1989) method. In their paper, because of the difficulty of determining the null distribution analytically, the quantiles of the null distribution were empirically simulated. In this paper, we give the exact null distribution of their test statistic, which makes it possible to calculate the critical values of the test.     

A note on profile likelihood for exponential tilt mixture models

Suppose that independent observations are drawn from multipledistributions, each of which is a mixture of two component distributionssuch that their log density ratio satisfies a linear model witha slope parameter and an intercept parameter. Inference forsuch models has been studied using empirical likelihood, andmixed results have been obtained. The profile empirical likelihoodof the slope and intercept has an irregularity at the null hypothesisso that the two component distributions are equal. We derivea profile empirical likelihood and maximum likelihood estimatorof the slope alone, and obtain the usual asymptotic propertiesfor the estimator and the likelihood ratio statistic regardlessof the null. Furthermore, we show the maximum likelihood estimatorof the slope and intercept jointly is consistent and asymptoticallynormal regardless of the null. At the null, the joint maximumlikelihood estimator falls along a straight line through theorigin with perfect correlation asymptotically to the firstorder.     

On the finite sample distribution of the likelihood ratio statistic for testing heterogeneity in meta-analysis

In meta-analysis, hypothesis testing is one of the commonly used approaches for assessing whether heterogeneity exists in effects between studies. The literature concluded that the Q-statistic is clearly the best choice and criticized the performance of the likelihood ratio test in terms of the type I error control and power. However, all the criticism for the likelihood ratio test is based on the use of a mixture of two chi-square distributions with 0 and 1 degrees of freedom, which is justified only asymptotically. In this study, we develop a novel method to derive the finite sample distribution of the likelihood ratio test and restricted likelihood ratio test statistics for testing the zero variance component in the random effects model for meta-analysis. We also extend this result to the heterogeneity test when metaregression is applied. A numerical study shows that the proposed statistics have superior performance to the Q-statistic, especially when the number of studies collected for meta-analysis is small to moderate.     

Stable and invariant adjusted profile likelihood and directed likelihood for curved exponential models

For curved exponential models, the modified profile likelihoodand the modified directed likelihood are approximated, witherror 0(n^-1) under ordinary repeated sampling, by explicitlydefined and invariant quantities which do not require specificationof an ancillary and which are stable in the sense of havingthe appropriate conditionality property with respect to anyreasonable ancillary.     

Applications of likelihood asymptotics for nonlinear regression in herbicide bioassays

Dose-response models are intensively used in herbicide bioassays. Despite recent advancements in the development of new herbicides, statistical analyses are commonly based on asymptotic approximations that are sometimes poor. This paper presents the use of recent results in higher order asymptotics for likelihood-based inference in nonlinear regression. The methods presented provide accurate approximation for the distribution of test statistics and for prediction limits. Analyses of the fit and measures of detection limits of the bioassays are considered, and the potential of the methods is illustrated by examples with real data.