Smooth estimation of survival functions under mean residual life ordering

Abstract

This paper deals with smooth estimation of two survival functions (SFs), S 1 and S 2 subject to the mean residual life (MRL) ordering, i.e. subject to M 1 ⩽ M 2 , where M 1 and M 2 denote the MRLs corresponding to S 1 and S 2 , respectively. This is achieved by smooth estimation of the two MRLs, M 1 and M 2 subject to M 1 ⩽ M 2 and obtaining the corresponding SFs. Two estimators are provided: one using the (non-smooth) estimator of Hu et al. [2002. Estimation of two ordered mean residual life functions. J. Statist. Plann. Inference 107, 321–341] and another utilizing the “uniform stochastic ordering” (USO) of ∫ x ∞ S i ( u ) d u , i = 1 , 2 , and the technique of the (non-smooth) estimation of Rojo and Samaniego [1993. On estimating a survival curve subject to a uniform stochastic ordering constraint. J. Amer. Statist. Assoc. 88, 566–572] of the two SFs ordered by USO. The smooth estimators are based on two earlier papers, one by Chaubey and Sen [1996. On smooth estimation of survival and density functions. Statist. Decisions 14, 1–22] on smooth estimation of a SF and another by Chaubey and Sen [1999. On smooth estimation of mean residual life. J. Statist. Plann. Inference 75, 223–236] on smooth estimation of an MRL function. This paper proves the strong consistency of the resulting estimators. A simulation study comparing the proposed estimators is also presented.