  The maximum likelihood estimation is widely used for statistical inferences.
This paper aims to reformulate Lee and Nelder's (1996) h-likelihood, so that
the maximum h-likelihood estimator resembles the maximum likelihood estimator
of the classical likelihood. We establish the statistical foundations of the
new h-likelihood. This extends classical likelihood theories to embrace broader
class of statistical models with random parameters. Maximization of the
h-likelihood yields asymptotically optimal estimators for both fixed and random
parameters achieving the generalized Cram\'er-Rao lower bound, while providing
computationally efficient fitting algorithms. Furthermore, we explore
asymptotic theory when the consistency of either fixed parameter estimation or
random parameter prediction is violated. We also study how to obtain maximum
h-likelihood estimators when the h-likelihood is not explicitly available.
