Statistical Inference for Random Unknowns via Modifications of Extended Likelihood

Abstract

Fisher's likelihood is widely used for statistical inference for fixed unknowns. This paper aims to extend two important likelihood-based methods, namely the maximum likelihood procedure for point estimation and the confidence procedure for interval estimation, to embrace a broader class of statistical models with additional random unknowns. We propose the new h-likelihood and the h-confidence by modifying extended likelihoods. Maximization of the h-likelihood yields both maximum likelihood estimators of fixed unknowns and asymptotically optimal predictors for random unknowns, achieving the generalized Cram\'er-Rao lower bound. The h-likelihood further offers advantages in scalability for large datasets and complex models. The h-confidence could yield a valid interval estimation and prediction by maintaining the coverage probability for both fixed and random unknowns in small samples. We study approximate methods for the h-likelihood and h-confidence, which can be applied to a general class of models with additional random unknowns.