Project Euclid

This paper introduces and studies the large sample properties of an estimator for the mean survival time from censored samples. Let $X_1, \cdots, X_n$ be independent identically distributed random variables with $F(x) = P\lbrack X_1 > x \rbrack.$ Let $Y_1, \cdots, Y_n$ be independent identically distributed (and independent of $X_1, \cdots, X_n$) censoring times with $G(y) = P\lbrack Y_1 > y\rbrack.$ Based on observing only $Z_i = \min(X_i, Y_i)$ and which observations are censored (i.e., $X_i > Y_i$), we give a class of estimators of the mean survival time $\mu = \int^\infty_0F(x) dx.$ The estimators are of the form $\hat{\mu} = \int^{M_n}_0 \hat{F}(x)dx,$ where $M_n \uparrow \infty$ as $n \uparrow \infty$ and $\hat{F}$ is an estimator of $F$ depending on the $Z_i$'s and the censoring pattern. Conditions of $F, G$ and $\{M_n\}$ for the asymptotic normality of $\hat{\mu}$ are stated and proved in Section 2 based on approximations detailed in Section 3. Section 4 gives conditions for strong consistency of $\hat{\mu}$ with rates, while Section 5 examines the meaning of the conditions for the case of the negative exponential distributions for $F$ and $G.$