  This paper studies Cox's regression hazard model with an unobservable random
frailty where no specific distribution is postulated for the frailty variable,
and the marginal lifetime distribution allows both parametric and
non-parametric models. Laplace's approximation method and gradient search on
smooth manifolds embedded in Euclidean space are applied, and a non-iterative
profile likelihood optimization method is proposed for estimating the
regression coefficients. The proposed method is compared with the
Expected-Maximization method developed based on a gamma frailty assumption, and
also in the case when the frailty model is misspecified.
