Nonparametric estimation of survival function in the presence of information through functionals

We are motivated by the reference [5] on nonparametric maximum likelihood estimate (NPMLE) of a lifetime cumulative distribution function (cdf) F on the basis of two independent samples, one of size m from F and the second of size n from G(x), a length biased distribution of F. One can obtain NPMLE of S(x) = 1 - F(x), the Survival Function when G is a suitable functional of F. We obtain NPMLE of 5, when G(t) is known a positive power of F. When power is unknown we propose estimators of the power and F. Based on extensive simulations, performance of estimators, using sup, L₁ and L₂ norms have been studied.