Jewell Nicholas P, Kalbfleisch John D
Division of Biostatistics, School of Public Health, University of California, Berkeley, USA.
Biostatistics. 2004 Apr;5(2):291-306. doi: 10.1093/biostatistics/5.2.291.
The pool adjacent violator algorithm Ayer et al. (1955, The Annals of Mathematical Statistics, 26, 641-647) has long been known to give the maximum likelihood estimator of a series of ordered binomial parameters, based on an independent observation from each distribution (see Barlow et al., 1972, Statistical Inference under Order Restrictions, Wiley, New York). This result has immediate application to estimation of a survival distribution based on current survival status at a set of monitoring times. This paper considers an extended problem of maximum likelihood estimation of a series of 'ordered' multinomial parameters p(i)= (p(1i),p(2i),.,p(mi)) for 1 <or=i <ro=k, where ordered means that p(j1) <or=p(j2) <or=<or=p(jk) for each j with 1 <or=j <or=m-1. The data consist of k independent observations X(1),., X(k) where X(i) has a multinomial distribution with probability parameter p(i) and known index n(i)\geq 1. By making use of variants of the pool adjacent violator algorithm, we obtain a simple algorithm to compute the maximum likelihood estimator of p(1),., p(k), and demonstrate its convergence. The results are applied to nonparametric maximum likelihood estimation of the sub-distribution functions associated with a survival time random variable with competing risks when only current status data are available (Jewell et al. 2003, Biometrika, 90, 183-197).
相邻合并违反者算法（Ayer等人，1955年，《数理统计年鉴》，第26卷，641 - 647页）长期以来已知，基于来自每个分布的独立观测值，可给出一系列有序二项式参数的最大似然估计量（见Barlow等人，1972年，《序约束下的统计推断》，Wiley出版社，纽约）。该结果可直接应用于基于一组监测时间点的当前生存状态来估计生存分布。本文考虑一个扩展问题，即对于(1\leq i\leq k)，对一系列“有序”多项参数(p(i)=(p(1i),p(2i),\cdots,p(mi)))进行最大似然估计，其中“有序”意味着对于每个(1\leq j\leq m - 1)，有(p(j1)\leq p(j2)\leq\cdots\leq p(jk))。数据由(k)个独立观测值(X(1),\cdots,X(k))组成，其中(X(i))具有参数为(p(i))且已知指标(n(i)\geq1)的多项分布。通过利用相邻合并违反者算法的变体，我们得到一种简单算法来计算(p(1),\cdots,p(k))的最大似然估计量，并证明其收敛性。这些结果应用于在仅可获得当前状态数据时，对与具有竞争风险的生存时间随机变量相关的子分布函数进行非参数最大似然估计（Jewell等人，2003年，《生物统计学》，第90卷，183 - 197页）。
