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Constrained nonparametric maximum likelihood estimation of stochastically ordered survivor functions
Author(s) -
Park Yongseok,
Kalbfleisch John D.,
Taylor Jeremy M. G.
Publication year - 2012
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.1002/cjs.10143
Subject(s) - mathematics , estimator , nonparametric statistics , uniqueness , constraint (computer aided design) , maximum likelihood , consistency (knowledge bases) , statistics , combinatorics , discrete mathematics , mathematical analysis , geometry
This paper considers estimators of survivor functions subject to a stochastic ordering constraint based on right censored data. We present the constrained nonparametric maximum likelihood estimator (C‐NPMLE) of the survivor functions in one‐and two‐sample settings where the survivor distributions could be discrete or continuous and discuss the non‐uniqueness of the estimators. We also present a computationally efficient algorithm to obtain the C‐NPMLE. To address the possibility of non‐uniqueness of the C‐NPMLE of $S_1(t)$ when $S_1(t)\le S_2(t)$ , we consider the maximum C‐NPMLE (MC‐NPMLE) of $S_1(t)$ . In the one‐sample case with arbitrary upper bound survivor function $S_2(t)$ , we present a novel and efficient algorithm for finding the MC‐NPMLE of $S_1(t)$ . Dykstra (1982) also considered constrained nonparametric maximum likelihood estimation for such problems, however, as we show, Dykstra's method has an error and does not always give the C‐NPMLE. We corrected this error and simulation shows improvement in efficiency compared to Dykstra's estimator. Confidence intervals based on bootstrap methods are proposed and consistency of the estimators is proved. Data from a study on larynx cancer are analysed to illustrate the method. The Canadian Journal of Statistics 40: 22–39; 2012 © 2012 Statistical Society of Canada