Smooth estimation of uniformly stochastically ordered surviaval functions

Let S and T denote two survival functions, such that the function θ(x) = S(x)/T(x) is non -T. This concept is useful in reliability and life testing as it is equivalent to to the hazard ordering. Rojo and Samaniego (1991, 1993) study consistent estimation of S under uniform stochastic ordering where as Mukerjee (1996) considered the general problem of estimating S and T along the lines considered in Rojo and Samaniego (1991). The delicate matter of the asymptotic distribution of estimators in question has been recently tackled by Arcones and Samaniego (2000). When the underlying survival functions are assumed obsolutely continuous, here is naturally an interest in finding smooth estimators, which are not available through the above estimators. Their smooth version through the popular smoothing methods do not guarantee the stochastic ordering property in the resulting estimators. In this paper we consider an adaptation of the smoothing technique introduced in Chaubey and Sen (1996) and show that the smooth estimator kind of dove tails the non-smooth estimator and as such the strong and weak convergence properties of the previous estimators are maintained.