Abstract: In 1974 Turnbull proposed the nonparametric MLE of the distribution function with incomplete data which may be censored either from above or below. This estimat or is also self-consistent. When the data are grouped, we showed that the distribution function of interest and the distribution functions of the two censoring variables satisfy an equation system. By studying the uniqueness of the solution of this equation system, we proved the strong consistency of the Turnbull estimator. We also proved the asymptotic normality and the law of the iterated logarithm, and have given the matrix representation of the asymptotic covariance matrix and the accurate upper bounds of the convergence rate in the law of the iterated logarithm.

Key words: doubly censoring, strong consistency, asymptotic normality, law of iterated logarithm

摘要： 在生存分析与可靠性研究中，对既有右删失，又有左删失的不完全数据，Turnbull于1974年给出了分布函数的最大似然估计。在观测数据是以分组形式得到的情形下，本文证明了此估计量的强相合性、渐近正态性和重对数律，并给出了渐近协差阵的矩阵表达式及重对数律中收敛速度的精确上限。

关键词: 强相合性, 渐近正态性, 重对数律, 左右删失