An EM algorithm for smoothing the self-consistent estimator of survival functions with interval-censored data

Interval-censored data arise in a wide variety of application and research areas such as, for example, AIDS studies (Kim et al., 1993) and cancer research (Finkelstein, 1986; Becker & Melbye, 1991). Peto (1973) proposed a Newton-Raphson algorithm for obtaining a generalized maximum likelihood estimate (GMLE) of the survival function with interval-censored observations. Turnbull (1976) proposed a self-consistent algorithm for interval-censored data and obtained the same GMLE. Groeneboom & Wellner (1992) used the convex minorant algorithm for constructing an estimator of the survival function with "case 2" interval-censored data. However, as is known, the GMLE is not uniquely defined on the interval [0, infinity). In addition, Turnbull's algorithm leads to a self-consistent equation which is not in the form of an integral equation. Large sample properties of the GMLE have not been previously examined because of, we believe, among other things, the lack of such an integral equation. In this paper, we present an EM algorithm for constructing a GMLE on [0, infinity). The GMLE is expressed as a solution of an integral equation. More recently, with the help of this integral equation, Yu et al. (1997a, b) have shown that the GMLE is consistent and asymptotically normally distributed. An application of the proposed GMLE is presented.