Whittemore A S, Keller J B
Biometrics. 1986 Sep;42(3):495-506.
A nonparametric maximum likelihood procedure is given for estimating the survivor function from right-censored data. It approximates the hazard rate by a simple function such as a spline, with different approximations yielding different estimators. A special case is that proposed by Nelson (1969, Journal of Quality Technology 1, 27-52) and Altshuler (1970, Mathematical Biosciences 6, 1-11). The estimators are uniformly consistent and have the same asymptotic weak convergence properties as the Kaplan-Meier (1958, Journal of the American Statistical Association 53, 457-481) estimator. However, in small and in heavily censored samples, the simplest spline estimators have uniformly smaller mean squared error than do the Kaplan-Meier and Nelson-Altshuler estimators. The procedure is extended to estimate the baseline hazard rate and regression coefficients in the Cox (1972, Journal of the Royal Statistical Society, Series B 34, 187-220) proportional hazards model and is illustrated using experimental carcinogenesis data.
给出了一种用于从右删失数据估计生存函数的非参数最大似然方法。它通过样条等简单函数来近似风险率，不同的近似方式会产生不同的估计量。一个特殊情况是纳尔逊（1969年，《质量技术杂志》第1卷，第27 - 52页）和阿尔特舒勒（1970年，《数学生物科学》第6卷，第1 - 11页）所提出的情况。这些估计量是一致收敛的，并且具有与卡普兰 - 迈耶（1958年，《美国统计协会杂志》第53卷，第457 - 481页）估计量相同的渐近弱收敛性质。然而，在小样本和删失严重的样本中，最简单的样条估计量的均方误差比卡普兰 - 迈耶和纳尔逊 - 阿尔特舒勒估计量的均方误差要小。该方法被扩展用于估计考克斯（1972年，《皇家统计学会杂志，B辑》第34卷，第187 - 220页）比例风险模型中的基线风险率和回归系数，并通过实验致癌数据进行了说明。
