On Survival Function Estimation in Dependent Partially Informative Random Censorship

In such areas as bio-medicine, engineering and insurance researchers are interested in positive variables, which are expressed as a time until a certain event. But observed data may be incomplete, because it is censored. Moreover, the random variables of interest (lifetimes) and censoring times can be influenced by other variable, often called prognostic factor or covariate. The basic problem is the estimation of survival function of lifetime. In this article we propose three asymptotical equivalent estimators of survival function in partially informative competing risks model. This paper deals with the estimation of a survival function with random right censoring and dependent censoring mechanism through covariate. We extend exponential – hazard, product limit and relative risk power estimators of survival functions in partially informative censoring model in which conditional on a covariate, the survival and censoring times are assumed to be independent. In this model, each observation is the minimum of one lifetime and two censoring times. The survival function of one of these censoring times is a power of the survival function of the lifetime. The distribution of the other censoring time has no relation with the distribution of the lifetime (non-informative censoring). For estimators we show their uniform strong consistency and convergence to same Gaussian process. Comparisons of estimators with the Jensen-Wiedmann’s estimator are included.


Introduction
In survival analysis it is often faced with censored lifetime data, i.e. with the only partially observable lifetimes. Let { , 1} i X i ≥ be independent and identically distributed (i.i.d.  rather than the PL -estimator [8]. Namely, the estimation of survival, hazard, quantile, mean and percentile residual life, total time on test, Lorenz, density and hazard rate functions is considered, and boot strapping in this model is sketched [8]. Testing the model is also discussed (see, also [10,[12][13][14]). However, it has been pointed out by many authors that the assumptions of PHM are much restrictive. Therefore, it is desirable to develop generalization of the PHM that are more appropriate for practical situations. Abdushukurov [1,2,6], Abdushukurov, Makhmudova [4], Gather and Pawlitschko [9] have proposed some generalizations of PHM, where only a part of the censored observations is supposed to be informative.
In some cases it is not reasonable to assume independence between the lifetime and censoring variables. The dependence may be due to a covariate. For example, in competing risks situation, where some technical system fails due to one or more competing causes, only one observes the time to failure of the system and the corresponding failure cause. If a system with two failure causes 1 A and 2 A fails due to cause 1 A , then the failure time of cause 2 A is randomly censored and vice versa. Since the failure times due to both causes are affected by the same stress and operating environment described by a covariate, it is likely the failure times that are positively correlated. In medical trials the survival and censoring times may be affected by a set of patients' covariates as age, blood pressure, cholesterol.
Here in the considered model we assume that the survival times and the censoring times are conditionally independent in a given covariate. Take partially informative completing risks modelin the presence of covariate [4]. Let We also suppose that the censoring by r.v.-s 1i Y for a given covariate is informative, i.e. the pairs ( ) Here β is some fixed but unknown censoring parameter. This kind of partially informative random censoring model with nuisance parameter 2 ( , ) G β in lack of covariate i Z was considered by authors [1,4,9,15].
Adapting some of ideas from [4] here we propose three asymptotical equivalent estimators of ( ) ( ) / F t z by exponentialhazard, product -limit and relative -risk power estimators using data from sample ( ) * n C .

Estimators of Survival Function
In order to construct the estimators of ( ), F t we need the following conditional sub distribution function: From (2) and (3), we get These functions can be estimated by statistics In order to estimate the conditional d.f. ( ) following from (1). For ( ) we use the following exponential hazard type estimator of Altschuler -Breslow, PL -estimator of Kaplan -Meier and relative -risk power type estimator of Abdushukurov (see, [1][2][3][4][5][6]): According to (9) using estimators (4) and (10) we get corresponding estimators of ( ) Finally, using statistics (11) we construct estimators of ( )

Asymptotic Properties of Estimators of Survival Function
In order to investigate the asymptotic properties of estimators (12) we need the following conditions. Moreover, from (I.6) we also have a chain of inequalities where (15) is consequence of (13), (14) and triangular inequality, (16) follows from (13) and inequality log log , ≤ It is easy to see that statistics (4) is strong consistent and asymptotically unbiased estimator of γ . From Adapting characterization of simple proportional hazard model under independent random censoring from the right (see, [1, 3-6, 8, 9]) we get following property of considered conditionally partially informative competing risks model.  In order to formulate these results we introduce Condition II. Condition II: n n n a n n a n → → → ∞ As Jensen and Wiedman [11], instead of estimators (12), we can consider following CIM-estimator (Conditional Independence Model) for ( ) 1 F t − without using informativeness of considered competing risks model:   (12) and (20) respectively. The next theorem shows that the estimators (12)

Discussion
We have introduced three new estimators (12) for the averaged unconditional d.f.  τ Therefore, we can consider the following truncated version of (23):

Conclusion
In this paper, we have proposed class of semi parametric estimators of exponential-hazard, product-limit and relativerisk power types of survival function in partially informative censoring model. Considered model is competing risks model in which each observation is the minimum of one lifetime and two censoring times in the presence also of covariate. The survival function of one of these censoring times is a power of the survival function of the lifetime (this is the informative censoring) and the distribution of the other censoring time has no relation with the distribution of the lifetime (non-informative censoring). We have showed that proposed estimators are uniformly strong consistent and they converges to same Gaussian process. They are also asymptotical efficient than the Jensen-Wiedmanns CIM