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.     

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.