An extension of partial likelihood methods for proportional hazard models to general transformation models. (English) Zbl 0639.62026

Independent rv ’s \(Y_ 1,...,Y_ n\) are said to follow a linear transformation model if for some increasing transformation h \[ h(Y_ i)=\alpha +\beta x_ i+\sigma \epsilon_ i,\quad i=1,...,n, \] where \(x_ i'=(x_{i1},...,x_{ip})\), \(i=1,...,n\), are known constants, \(\beta =(\beta_ 1,...,\beta_ p)\) is a vector of regression parameters, \(\alpha\) is an intercept parameter and \(\epsilon_ 1,...,\epsilon_ n\) are i.i.d. with distribution F. Primarily the nonparametric case is considered where h is continuous and increasing, but otherwise arbitrary.
Cox’s proportional hazard model with time independent covariates is a special case of such a transformation model (with \(h(y)=\log (-\log [1- F_ 0(y)])\) and \(F_ 0(y):=1-\exp (-\int^{y}_{0}r(t)dt)\) for some baseline hazard function r(t)) and since partial (marginal) likelihood methods have proven so useful in the proportional hazard model, the present paper investigates properties of partial likelihood methods in the transformation model to obtain estimates of the parameters.
In this connection a resampling scheme called the likelihood sampler is introduced to compute the partial likelihood and the maximum partial likelihood estimates. Monte Carlo techniques are used to show that the estimates perform very well for moderate sample sizes and a certain range of parameter values.
Reviewer: P.Gänßler


62G05 Nonparametric estimation
60F05 Central limit and other weak theorems
62J02 General nonlinear regression
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