On estimating a survival function under the generalized proportional hazards model

Let X and Y be two competing lifetimes with continuous survival functions F¯(t) and G¯(t), respectively, and let β be a nonnegative random variable with B(b)=P[β⩽b]. A generalized proportional hazards model proposed by Peña and Rohatgi [1989. Survival function estimation for a generalized proportional hazards model of random censorship. J. Statist. Plann. Inference 22, 371-389] assumed that X and (Y,β) are independent and G¯(t)=EB[F¯(t)]β. When B(b) is uniquely determined by an unknown parameter θ with a mild condition, they used n independent and identically distributed (iid) observations (Zi,δi)i=1n with Zi=min(Xi,Yi) and δi=I(Xi⩽Yi) to find the maximum likelihood estimators (MLEs) of F¯(t) and θ and their large sample properties. In this paper we point out that the estimators of F¯(t) and θ proposed by Peña and Rohatgi (1989) are not MLEs in general. Specifically, we prove that Peña and Rohatgi's estimators (PREs) of F¯(t) and θ are MLEs if and only if the random variable β is degenerate at a constant b>0 which was studied by Cheng and Lin [1987. Maximum likelihood estimation of survival function under the Koziol-Green proportional hazards model. Statist. Probab. Lett. 5, 75-80]. It is also shown that Z and δ are always positively correlated under the generalized proportional hazards model. A modification of the generalized proportional hazards model which can be used to fit a data set showing the negative correlation between Z and δ is proposed. A conditionally proportional odds model which assumes β has a negative binomial distribution with size two and parameter θ is introduced and used to conduct simulation studies when sample size is small or moderate. In terms of variance of simulation studies, the MLE is better than the product-limit estimator and PRE.