  The expectation-maximization (EM) algorithm is a powerful computational
technique for finding the maximum likelihood estimates for parametric models
when the data are not fully observed. The EM is best suited for situations
where the expectation in each E-step and the maximization in each M-step are
straightforward. A difficulty with the implementation of the EM algorithm is
that each E-step requires the integration of the log-likelihood function in
closed form. The explicit integration can be avoided by using what is known as
the Monte Carlo EM (MCEM) algorithm. The MCEM uses a random sample to estimate
the integral at each E-step. However, the problem with the MCEM is that it
often converges to the integral quite slowly and the convergence behavior can
also be unstable, which causes a computational burden. In this paper, we
propose what we refer to as the quantile variant of the EM (QEM) algorithm. We
prove that the proposed QEM method has an accuracy of $O(1/K^2)$ while the MCEM
method has an accuracy of $O_p(1/\sqrt{K})$. Thus, the proposed QEM method
possesses faster and more stable convergence properties when compared with the
MCEM algorithm. The improved performance is illustrated through the numerical
studies. Several practical examples illustrating its use in interval-censored
data problems are also provided.
