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

Doksum, Kjell A.

Ann. Stat. 15, 325-345 (1987).

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

MSC:

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

Keywords:

power transformation model; asymptotic results; marginal likelihood; rank likelihood; semiparametric transformation models; linear transformation model; intercept; Cox’s proportional hazard model; time independent covariates; resampling scheme; likelihood sampler; maximum partial likelihood estimates; Monte Carlo