On estimating a survival function under the generalized proportional hazards model

Autor: Jian-Lun Xu
Rok vydání: 2007
Předmět:
Zdroj: Journal of Statistical Planning and Inference. 137:1161-1172
ISSN: 0378-3758
DOI: 10.1016/j.jspi.2006.04.003
Popis: 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 Pena 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 ) = E B [ 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 ( Z i , δ i ) i = 1 n with Z i = min ( X i , Y i ) and δ i = I ( X i ⩽ Y i ) 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 Pena and Rohatgi (1989) are not MLEs in general. Specifically, we prove that Pena 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.
Databáze: OpenAIRE