On estimating a survival function under the generalized proportional hazards model
Autor: | Jian-Lun Xu |
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Rok vydání: | 2007 |
Předmět: |
Statistics and Probability
Independent and identically distributed random variables Generalized function Continuous function (set theory) Applied Mathematics Negative binomial distribution Estimator Combinatorics Survival function Statistics Statistics Probability and Uncertainty Likelihood function Random variable Mathematics |
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 |
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