Nonparametric estimation with censored data : a discrete approach
Xuecheng Liu · 2005
This dissertation principally addresses nonparametric maximal likelihood (NPML) estimation of the cumulative distribution function (CDF) given multivariate (arbitrarily) censored data (herein abbreviated MCD). The CDF nonparametric maximal likelihood estimate (NPMLE) given MCD has support on the union of all maximal intersections of the data. The CDF NPMLE can be computed numerically using the clique matrix of the intersection graph of the data; these NPMLEs can be nonunique in both a representational and a mixture sense (see Peto 1973, Turnbull 1976, Gentleman & Vandal 2001 and Gentleman & Vandal 2002). The fundamental methodology used in this dissertation consists in applying graph theory to the intersection graph of censored data and discrete mathematics to its linear algebraic representation. An optimal algorithm to determine the maximal intersections of MCD is proposed. A full discussion of measures of NPMLE mixture nonuniqueness and their computational implementations for the measures is provided. The iterative convex minorant (ICM) algorithm to obtain the NPMLE is extended to the case of MCD. The nonparametric likelihood maximization given MCD is simplified via the use of a reduction tree. The EM/X Algorithm is introduced to compute the NPMLE for large MCD set. Bounds on self-consistent estimates of the CDF (a class to which the CDF NPMLE belongs) given MCD are used to assess the degree of consistency of the CDF NPMLE. Constrained estimation and likelihood intervals computation given univariate censored data are discussed. The empirical likelihood method is also applied to construct CDF likelihood sets for MCD. An unbiased and consistent estimate is proposed for MCD with fixed censoring times.