 2005 Royal Statistical Society 1369–7412/05/67573 J. R. Statist. Soc. B (2005) 67, Part 4, pp. 573–587 On nonparametric maximum likelihood estimation with interval censoring and left truncation Michael G. Hudgens University of North Carolina at Chapel Hill, USA [Received June 2004. Revised March 2005] Summary. A graph theoretical approach is employed to describe the support set of the non- parametric maximum likelihood estimator for the cumulative distribution function given interval- censored and left-truncated data. A necessary and sufficient condition for the existence of a nonparametric maximum likelihood estimator is then derived.Two previously analysed data sets are revisited. Keywords: Graph theory; Interval censoring; Nonparametric maximum likelihood estimation; Truncation 1. Introduction The first work on nonparametric maximum likelihood estimation of the cumulative distribu- tion function (or equivalently the survival function) in the presence of interval-censored data is usually attributed to Peto (1973), who showed that, by determining the support of the nonpara- metric maximum likelihood estimator (NPMLE) first, the problem reduces to maximizing a simpler likelihood. Subsequently, Turnbull (1976) characterized the NPMLE in the presence of interval censoring and truncation. Frydman (1994) later corrected Turnbull’s characterization and, in turn, Alioum and Commenges (1996) identified a further refinement of the set where the NPMLE could put mass. There have been several extensions of this work in recent years. Betensky and Finkelstein (1999) and Gentleman and Vandal (2001, 2002) derived the support of the NPMLE for bivariate interval-censored data. Wong and Yu (1999) presented the first work on the more general multivariate interval censoring problem. Hudgens et al. (2001b) character- ized the NPMLE for competing risks survival data subject to interval censoring and truncation. Asymptotic properties of the NPMLE for univariate interval-censored data in the absence of truncation have been derived. Generally, in this setting the NPMLE is consistent. If addition- ally the censoring mechanism is assumed discrete, the NPMLE has the usual √ n convergence rate and a normal limiting distribution (Yu et al., 1998a, b). However, if the random variables dictating the censoring are treated as continuous, the NPMLE converges slower than √ n to a non-Gaussian limiting distribution (Banerjee and Wellner, 2001; Groeneboom and Wellner, 1992; Schick and Yu, 2000; Song, 2004; van der Vaart and Wellner, 2000). Much less is known about the large sample properties of the NPMLE if the data are multivariate or if both interval censoring and truncation are present. The work by Gentleman and Vandal (2001, 2002) is noteworthy in that graph theory was employed in studying the NPMLE, providing new insights into the complex and interesting Address for correspondence: Michael G. Hudgens, Department of Biostatistics, School of Public Health, McGavran–Greenberg Hall, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA. E-mail: mhudgens@bios.unc.edu