A Quasi-Newton Algorithm for Efficient Computation of Gehan Estimates

The analysis of lifetime data is an important research area in statistics, particularly among econometricians and biostatisticians. The two most popular semi-parametric models are the proportional hazards model and the accelerated failure time (AFT) model. The proportional hazards model is computationally advantageous over virtually any other competing semi-parametric model because the ubiquitous maximum partial likelihood estimator is computed using ordinary Newton methods. Rank-based estimation in the semi-parametric AFT model is computationally more challenging, for example, because the Hessian matrix is not directly estimable without nonparametric smoothing. Recently, authors showed that the rank-based estimators may be written as the solution to a linear programming problem. Unfortunately, the size of the linear programming problem is O(n) subject to n linear constraints, where n denotes sample size. Thus, the linear programming solution to rank-based estimation is restricted by well-known computational limitations of linear programming and impractical for many applications. In this paper, we describe quasi-Newton methods for rank-based estimation in the semi-parametric AFT model. The algorithm converges super-linearly and is computationally efficient. Thus, the computational cost of our algorithm remains low for even large data sets. We illustrate the algorithm through speed trials and three real data sets.