Fast computation of fully automated log-density and log-hazard estimators
SIAM Journal on Scientific and Statistical Computing
IEEE Transactions on Image Processing
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This paper considers simultaneous estimation of the regression coefficients and baseline hazard in proportional hazard models using the maximum penalized likelihood (MPL) method where a penalty function is used to smooth the baseline hazard estimate. Although MPL methods exist to fit proportional hazard models, they suffer from the following deficiencies: (i) the positivity constraint on the baseline hazard estimate is either avoided or poorly treated leading to efficiency loss, (ii) the asymptotic properties of the MPL estimator are lacking, and (iii) simulation studies comparing the performance of MPL to that of the partial likelihood have not been conducted. In this paper we propose a new approach and aim to address these issues. We first model baseline hazard using basis functions, then estimate this approximate baseline hazard and the regression coefficients simultaneously. The penalty function included in the likelihood is quite general but typically assumes prior knowledge about the smoothness of the baseline hazard. A new iterative optimization algorithm, which combines Newton's method and a multiplicative iterative algorithm, is developed and its convergence properties studied. We show that if the smoothing parameter tends to zero sufficiently fast, the new estimator is consistent, asymptotically normal and retains full efficiency under independent censoring. A simulation study reveals that this method can be more efficient than the partial likelihood method, particularly for small to moderate samples. In addition, our simulation shows that the new estimator is substantially less biased under informative censoring.