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.     

Testing the Equality of the Intra‐Subject Treatment Covariance Matrices in a Crossover Study

In a 2 × 2 crossover bioavailability study, the sets of estimates of the pharmacokinetic parameters quite often have a symmetric covariance structure between the two treatments. For testing the equality of the intra‐subject covariance matrices for the two treatments in such studies, we suggest in this paper some statistical tests. When the response vectors are bivariate, we propose an exact test. Since the statistical procedures depend on the assumption of a symmetric covariance structure between the two treatments, we put forth some statistical tests for this assumption. We then apply the discussed tests to real data from a crossover bioavailability trial.     

The Wald Statistic in Proportional Hazards Hypothesis Testing

In survivorship modelling using the proportional hazards model of Cox (1972, Journal of the Royal Statistical Society, Series B, 34, 187–220), it is often desired to test a subset of the vector of unknown regression parameters β in the expression for the hazard rate at time t. The likelihood ratio test statistic is well behaved in most situations but may be expensive to calculate. The Wald (1943, Transactions of the American Mathematical Society 54, 426–482) test statistic is easier to calculate, but has some drawbacks. In testing a single parameter in a binomial logit model, Hauck and Donner (1977, Journal of the American Statistical Association 72, 851–853) show that the Wald statistic decreases to zero the further the parameter estimate is from the null and that the asymptotic power of the test decreases to the significance level. The Wald statistic is extensively used in statistical software packages for survivorship modelling and it is therefore important to understand its behavior. The present work examines empirically the behavior of the Wald statistic under various departures from the null hypothesis and under the presence of Type I censoring and covariates in the model. It is shown via examples that the Wald statistic's behavior is not as aberrant as found for the logistic model. For the single parameter case, the asymptotic non-null distribution of the Wald statistic is examined.     

Evaluation of the Subsample from the Non-respondents by a Random Response Design

The Random Response Design is used for obtaining information from the non respondents. The existence of a refusal to respond may be the cause of non responses. The model obtained seems to be a good alternative for reducing non sampling errors due to dishonest answers.     

A Note on Maximum Likelihood Ratio Test of No Outliers in Regression Models

A derivation of the maximum likelihood ratio test for testing no outliers in regression models is given using the method of WETHERILL (1981, pp. 106–107) for estimating the regression parameters. This method is essentially similar to the one outlined in BARNETT and LEWIS (1978, p. 263), although by our detailed derivation it is easier to see that the maximum likelihood estimate of θ of model (3) under the hypothesis that the i^th observation in an outlier is the same as that obtained from model (1) when the i^th observation is removed.