The maximum likelihood estimation is widely used for statistical inferences. In this study, we reformulate the h-likelihood proposed by Lee and Nelder in 1996, whose maximization yields maximum likelihood estimators for fixed parameters and asymptotically best unbiased predictors for random parameters. We establish the statistical foundations for h-likelihood theories, which extend classical likelihood theories to embrace broad classes of statistical models with random parameters. The maximum h-likelihood estimators asymptotically achieve the generalized Cramer-Rao lower bound. Furthermore, we explore asymptotic theory when the consistency of either fixed parameter estimation or random parameter prediction is violated. The introduction of this new h-likelihood framework enables likelihood theories to cover inferences for a much broader class of models, while also providing computationally efficient fitting algorithms to give asymptotically optimal estimators for fixed parameters and predictors for random parameters.

翻译：最大似然估计广泛用于统计推断。本研究重新阐述了Lee和Nelder于1996年提出的h似然方法，其最大化方法可得到固定参数的最大似然估计量以及随机参数的渐近最优无偏预测量。我们建立了h似然理论的统计学基础，将经典似然理论扩展至涵盖更广泛的含随机参数统计模型类别。最大h似然估计量渐近地达到广义Cramer-Rao下界。此外，我们探讨了当固定参数估计或随机参数预测的相合性被违反时的渐近理论。这一新h似然框架的引入使得似然理论能够覆盖更广泛模型类别的推断，同时提供计算高效的拟合算法，为固定参数和随机参数分别给出渐近最优估计量与预测量。