Murphy E A, Trojak J E, Berger K R, Foster E C
Division of Medical Genetics, Johns Hopkins University School of Medicine, Baltimore, Maryland 21205.
Am J Med Genet. 1987 Nov;28(3):691-701. doi: 10.1002/ajmg.1320280317.
The survivorship (time to death or failure) of a bingo-gamma (BG) model is defined as the minimum among the waiting times for completion among k independent gamma processes. The ith process is of order ni, with a mean rate for the occurrence of hits of ai. In this paper we address the case where, for all competing processes, the order and the rate at which hits occur are the same but both they and k are unknown. We denote by k the multiplicity, by n the order or the number of hits to failure, and by a the transition parameter. The joint maximum likelihood estimator (MLE) of the three parameters of this BG process is developed. An algorithm for calculating it has been devised and a computer program in BASIC has been written. The properties of the MLE have been explored systematically, mainly by Monte Carlo simulation. The distributions, means, variances, covariances, and correlation coefficients of the three parameters are explored for samples of size 25 and samples of size 100. Also, the simple average of the observed survival times (which gives a method of moments estimator of the mean survival) is compared with the MLE of the mean survival; the two estimators seem to be unbiased and about equally efficient.
宾果 - 伽马（BG）模型的生存时间（直至死亡或失效的时间）定义为k个独立伽马过程中完成等待时间的最小值。第i个过程的阶数为ni，命中发生的平均速率为ai。在本文中，我们处理的情况是，对于所有竞争过程，命中发生的阶数和速率相同，但它们以及k均未知。我们用k表示多重性，用n表示阶数或失效的命中次数，用a表示转移参数。推导了该BG过程三个参数的联合最大似然估计量（MLE）。设计了一种计算它的算法，并编写了一个BASIC语言的计算机程序。主要通过蒙特卡罗模拟系统地探索了MLE的性质。针对大小为25的样本和大小为100的样本，研究了三个参数的分布、均值、方差、协方差和相关系数。此外，将观察到的生存时间的简单平均值（它给出了平均生存的矩估计方法）与平均生存的MLE进行比较；这两个估计量似乎都是无偏的，并且效率大致相同。
