  Claeskens and Hjort (2003) constructed the focused information criterion
(FIC) and developed frequentist model averaging methods using maximum
likelihood estimators assuming the observations to be independent and
identically distributed. Towards the immediate extensions and generalizations
of these results, the present article is aimed at providing the focused model
selection and model averaging methods using general maximum likelihood type
estimators, popularly known as $M$-estimators. The necessary asymptotic theory
is derived in a setup of stationary and strong mixing stochastic processes
employing von Mises functional calculus of empirical processes and Le Cam's
contiguity lemmas. We illustrate the proposed focused stochastic modeling
methods using three well-known spacial cases of $M$-estimators, namely,
conditional maximum likelihood estimators, conditional least square estimators
and estimators based on method of moments. For the sake of simulation
exercises, we consider two simple applications of FIC. The first application
discusses the simultaneous selection of order of autoregression and symmetry of
innovations in asymmetric Laplace autoregressive models. The second application
demonstrates the FIC based choice between general scale-shape Gamma density and
exponential density with shape being unity. We observe that in terms of the
correct selections, FIC outperforms classical Akaike's information criterion
AIC and performs at par with Bayesian information criterion BIC.
