Estimation of Time Transformation Models with Bernstein Polynomials

Time transformation models assume that the survival time is linearly related to covariates through an unknown monotonic transformation function and an error term with known distribution. In this paper the sieve method of maximum likelihood is used to estimate the unknown monotonic transformation of survival time. More specifically a suitable class of Bernstein polynomials is used to estimate the transformation function, that preserve the monotonicity and smoothness. This estimation method is less parametrically intensive than current time transformation methods. Furthermore, our method produces a smooth estimate of the time transformation and hence the survival function. We discuss the selection of the number of parameters for the polynomial asymptotically, and for practical sample sizes. The asymptotic properties of the estimators are shown, including the asymptotic normality and efficiency of the regression ∗A. McLain ([email protected]) is a VIGRE Postdoctoral Fellow, Department of Statistics, North Carolina State University, Raleigh, NC. †S. Ghosh is Professor, Department of Statistics, North Carolina State University, Raleigh, NC. 1 coefficient. Simulation studies illustrate that our estimator has reasonably good empirical properties in practical sample sizes. The method is demonstrated on two data sets and compared to previous similar works.