Some large sample properties of an estimator of the hazard function from randomly censored data (Q1895525)

In many lifetime studies, some of the subjects under study are censored on the right by a prior censoring time. We observe only the censored data. It is important to estimate the life characteristics of the subjects. Let \(T_ 1\), \(T_ 2, \dots, T_ n\) be independent and identically distributed nonnegative random variables with common continuous distribution function \(F\) and continuous probability density function \(f\). In the model of right random censoring, associated with each \(T_ i\) there is an independent nonnegative censoring time \(C_ i\) and \(C_ 1\), \(C_ 2, \dots, C_ n\) are assumed to be iid random variables with continuous distribution function \(G\). In this model we can observe only the pairs \((Z_ i, \delta_ i)\), where \(Z_ i = \min (T_ i, C_ i)\), \(\delta_ i = I[T_ i \leq C_ i]\), \(i = 1,2, \dots, n\), and \(I[\cdot]\) is the indicator function of a certain event. Clearly \(Z_ 1\), \(Z_ 2, \dots, Z_ n\) are iid with continuous distribution function \(H = 1 - (1 - F) (1 - G)\). The problem considered is to estimate the hazard function \(\lambda (t) = f(t)/(1 - F(t))\) by using the above censoring data observed and investigate the large sample properties.