A Quantile Variant of the EM Algorithm and Its Applications to Parameter Estimation with Interval Data

TL;DR

This paper introduces a quantile-based variant of the EM algorithm that offers faster and more stable convergence than the Monte Carlo EM, especially useful for parameter estimation with interval-censored data.

Contribution

The paper proposes the QEM algorithm with proven higher accuracy and stability compared to MCEM, enhancing parameter estimation in incomplete data scenarios.

Findings

QEM has an $O(1/K^2)$ accuracy, outperforming MCEM's $O_p(1/ ext{sqrt}(K))$.

QEM demonstrates faster convergence in numerical simulations.

Practical applications include improved estimation in interval-censored data problems.

Abstract

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.