An Extension of Partial Likelihood Methods for Proportional Hazard Models to General Transformation Models

Abstract

Estimates of the linear model parameters in a linear transformation model with unknown increasing transformation are obtained by maximizing a partial likelihood. A resampling scheme (likelihood sampler) is used to compute the maximum partial likelihood estimates. It is shown that for a certain "local" parameter set where the "signal to noise ratio" is small, it is asymptotically possible to estimate the linear model parameters using the partial likelihood as well as if the transformation were known. In the case of the power transformation model with symmetric error distribution, this result is shown to also hold when the distribution of the error in the transformed linear model is unknown and is estimated. Monte Carlo results are used to show that for moderate sample size and small to moderate signal to noise ratio, the asymptotic results are approximately in effect and thus the partial likelihood estimates perform very well. Estimates of the transformation are introduced and it is shown that the estimates, when centered at the transformation and multiplied by $\sqrt{n}$, converge weakly to Gaussian processes.