  Maximum likelihood (ML) estimation is widely used in statistics. The
h-likelihood has been proposed as an extension of Fisher's likelihood to
statistical models including unobserved latent variables of recent interest.
Its advantage is that the joint maximization gives ML estimators (MLEs) of both
fixed and random parameters with their standard error estimates. However, the
current h-likelihood approach does not allow MLEs of variance components as
Henderson's joint likelihood does not in linear mixed models. In this paper, we
show how to form the h-likelihood in order to facilitate joint maximization for
MLEs of whole parameters. We also show the role of the Jacobian term which
allows MLEs in the presence of unobserved latent variables. To obtain MLEs for
fixed parameters, intractable integration is not necessary. As an illustration,
we show one-shot ML imputation for missing data by treating them as realized
but unobserved random parameters. We show that the h-likelihood bypasses the
expectation step in the expectation-maximization (EM) algorithm and allows
single ML imputation instead of multiple imputations. We also discuss the
difference in predictions in random effects and missing data.
