Strong consistency under proportional censorship when covariables are present
Introduction
Censoring is a typical phenomenon in survival analysis. In a censored sample incomplete observations may occur. The general random censorship model (from the right) assumes that one observes the pair (Z,δ) where Z=min(Y,C) and δ=I(Y⩽C), Y being the lifetime with distribution function F and C being the censoring variable (independent of Y) with distribution function G. Inference under this model is carried out via Kaplan–Meier-based estimation.
The additional condition that led us to the Koziol–Green model (Koziol and Green, 1976) isWhen F and G are continuous (as we will assume throughout the text), this proportional censorship model is characterized by the independence between Z and δ (Armitage, 1959; Sethuraman, 1965). Then letting H and θ be the distribution function of Z and the mean value of δ respectively, it is easily proved thatwhere θ=1/(1+β). This relation motivates the ACL estimator F∼n(y)=1−(1−Hn(y))θn (Abdushukurov, 1984; Cheng and Lin (1984), Cheng and Lin (1987)) based on the given sample (Z1,δ1),…,(Zn,δn), where Hn is the empirical distribution function of the Z's and θn is the sample mean of the δ's. The estimator F∼n has been extensively analyzed in the literature.
Here we consider a situation in which the observed variable is the vector (X,Z,δ), where X is a covariate (in general p-dimensional) paired with the lifetime. We propose an extension of the Koziol–Green model based on the hypothesisUnder (1.3) we have that Z is independent of δ and hence (1.1) and (1.2) hold. For identifiability reasons we assume the conditionCondition (1.4) is discussed and used to get estimators under random censorship in Stute (1993). Under (1.4) we havefor a general FXY-integrable ϕ-function, FXY denoting the joint distribution function of (X,Y). ButNow we use the independence between (X,Z) and δ (and relations (1.1) and (1.2)) to concludeunder (1.3) and (1.4). Our estimator of E(ϕ(X,Y))≡S(ϕ)≡S is given bywhere is the empirical distribution function of the Z's and is the empirical distribution function of the (X,Z)'s. By considering the indicator function ϕ≡I((−∞,x]×(−∞,y]) we obtain for FXY the estimate
In this paper we prove the strong consistency of (1.6) for each FXY-integrable function ϕ using martingale theory. As a consequence, we obtain strong consistency for the extended ACL-based integralwherefor the class of ϕ-functions that satisfyThe estimator F∼XY collapses to F∼n when there is no X. Our results cover such a case. In this manner we obtain an alternative proof for a result on strong consistency of given in Stute (1992).
When no covariables are present, condition (1.4) is superfluous, and (1.3) collapses to (1.1). Under our extended Koziol–Green model (1.3)–(1.4), ACL-based estimation is still consistent. Other authors have considered (conditional) extensions for the model, by assuming the independence between Z and δ conditionally on X. See Veraverbeke and Cadarso-Suárez (1997). Such ideas require smoothing methods for the estimation of the (conditional) lifetime distribution.
Section snippets
Main results
Now we give our results for both estimators and given in Section 1. Theorem 2.1 For each FXY-integrable function ϕ we have a.s. Theorem 2.2 For each function ϕ satisfying (1.8) we have a.s. Corollary 2.3 For each function ϕ satisfying (1.8) we have a.s.
An immediate consequence of our theorems is the strong consistency of F∼XY and . As mentioned in Stute (1993), these results can be also used to get consistency of the estimated variance–covariance matrix of (X,Y). Consistent least-squares estimation in linear
Proofs
Firstly we show that, for proving Theorem 2.1, it suffices to establishwhere Z(1:n)⩽⋯⩽Z(n:n) are the ordered Z-values and X[i:n] is the ith concomitant. Certainly, by the mean-value theoremwith between θn and θ, 1⩽i⩽n. Now by the SLLN given ϵ>0 we have |θn−θ|<ϵ a.s. for big n. Hencea.s. for big n. Now write for 1⩽i⩽n
Concluding remarks
In this paper we introduce consistent estimators for quantities of interest in survival analysis when the lifetime is at risk of being censored from the right and an extended Koziol–Green model of proportional censorship holds. In order to get this, we analyze an approximated generalized ACL estimator and we prove strong consistency for , ϕ denoting any FXY-integrable function. We use this result to get consistency when dealing with the actual ACL weights. We mention several
Acknowledgements
The authors thank to a Referee and an Associate Editor for their careful reading of the paper and their suggestions.
References (13)
- et al.
Maximum likelihood estimation of a survival function under the Koziol–Green proportional hazard model
Statist. Probab. Lett.
(1987) Strong consistency under the Koziol–Green model
Statist. Probab. Lett.
(1992)Consistent estimation under random censorship when covariables are present
J. Multivariate Anal.
(1993)- Abdushukurov, A.A., 1984. On some estimates of the distribution function under random censorship, Conf. of Young...
The comparison of survival curves
Journal of the Royal Statistical Society, Series A
(1959)- Cheng, P.E., Lin, G.D., 1984. Maximum likelihood estimation of a survival function under the Koziol–Green proportional...
Cited by (0)
- 1
This work was partially supported by the project XUGA20701B96 of the Xunta de Galicia (Spain) and by the DGES grant PB95-0826.