A quantile variant of the expectation–maximization algorithm and its application to parameter estimation with interval data

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 Op(1/ √ 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.

A Note on Weibull Parameter Estimation with Interval Censoring Using the EM Algorithm

“…However, it is often the case that the implementation of the EM algorithm is quite difficult because the expectation of the log-likelihood in the E-step is generally complex or unavailable in closed form. This problem has been studied by several authors, including Panahi and Asadi [14], Guure et al [15], Pradhan and Kundu [16], Ferreira and Silva [17], Park [18], Saeed and Elfaki [19], Kurniawan et al [20], Ameen and Akkash [21], and Almetwally et al [22].…”

Parameter Estimation for Grouped Data Using EM and MCEM Algorithms

“…In a similar manner to [11,12,14], to find θ that maximizes log L(θ) in (3) within the EM framework, the vector of x i = (x i1 , x i2 , . .…”

“…Calculating the exact form of the expectations in ( 6) and ( 7) can be seen as a tedious and difficult task. In this case, an alternative way is to use the Monte-Carlo EM (MCEM) algorithm, in which the required expectations are replaced with an average over simulations [12,14]. The unobserved data x = (x 1 , .…”

“…In another study, Heitjan [9] has applied Newton-Raphson's method and the EM algorithm to find parameter estimates of bivariate regression analysis for grouped data. Chanseok Park [14] has proposed a quantile variant of the EM (QEM) algorithm for parameter estimation of interval-valued data (grouped data with one interval) and demonstrated that their proposed algorithm is more efficient than Monte-Carlo EM (MCEM) [12]. Wengrzik and Timm [19] have studied the fitting of two-component Gaussian mixtures to grouped data.…”

Load-Sharing Model under Lindley Distribution and Its Parameter Estimation Using the Expectation-Maximization Algorithm

“…Amari et al [21] used the EM method to evaluate incomplete warranty data. Recently, Park [22] analyzed interval-data using the quantile implementation of the EM algorithm which improves the Monte Carlo EM (MCEM) algorithm's convergence and stability properties and Ouyang [23] considered an interval-based model for analyzing incomplete data in a different viewpoint. These observations motivate us to develop an EM-type MLE for the parameters in the considered system to overcome the possible difficulties mentioned above when using the standard numerical methods.…”