Abstract
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.
Highlights
In survival analysis it is often faced with censored lifetime data, i.e. with the only partially observable lifetimes
In the theorem we show that all three statistics
We have proposed class of semi parametric estimators of exponential-hazard, product-limit and relative risk power types of survival function in partially informative censoring model
Summary
In survival analysis it is often faced with censored lifetime data, i.e. with the only partially observable lifetimes. In the considered model we assume that the survival times and the censoring times are conditionally independent in a given covariate. In C*(n) the r.v.–s of interest Yki ’s are censored from the right by r.v.–s min (Y1i ,Y1i ) and observable partially only in case of δi = 1 .The estimation of d.f. F and its functionals from sample C*(n) is one of the main goals in survival analysis. Β is some fixed but unknown censoring parameter This kind of partially informative random censoring model with nuisance parameter (β ,G2 ) in lack of covariate Zi was considered by authors [1, 4, 9, 15]. F (t / z) by exponential – hazard, product – limit and relative – risk power estimators using data from sample C*(n)
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