Large Sample Theory for an Estimator of the Mean Survival Time from Censored Samples

This paper introduces and studies the large sample properties of an estimator for the mean survival time from censored samples. Let X₁, ⋯, Xₙ be independent identically distributed random variables with F(x) = P X₁ > x . Let Y₁, ⋯, Yₙ be independent identically distributed (and independent of X₁, ⋯, Xₙ) censoring times with G(y) = P Y₁ > y. Based on observing only Zᵢ = min(Xᵢ, Yᵢ) and which observations are censored (i.e., Xᵢ > Yᵢ), we give a class of estimators of the mean survival time μ = ∫^∞₀F(x) dx. The estimators are of the form μ̂ = ∫Mₙ₀ F̂(x)dx, where Mₙ ↑ ∞ as n ↑ ∞ and F̂ is an estimator of F depending on the Zᵢ's and the censoring pattern. Conditions of $F, G$ and ₙ\ for the asymptotic normality of μ̂ are stated and proved in Section 2 based on approximations detailed in Section 3. Section 4 gives conditions for strong consistency of μ̂ with rates, while Section 5 examines the meaning of the conditions for the case of the negative exponential distributions for F and $G.$