Semiparametric models: a generalized self‐consistency approach

In semiparametric models, the dimension d of the maximum likelihood problem is potentially unlimited. Conventional estimation methods generally behave like O(d(3)). A new O(d) estimation procedure is proposed for a large class of semiparametric models. Potentially unlimited dimension is handled in a numerically efficient way through a Nelson-Aalen-like estimator. Discussion of the new method is put in the context of recently developed minorization-maximization algorithms based on surrogate objective functions. The procedure for semiparametric models is used to demonstrate three methods to construct a surrogate objective function: using the difference of two concave functions, the EM way and the new quasi-EM (QEM) approach. The QEM approach is based on a generalization of the EM-like construction of the surrogate objective function so it does not depend on the missing data representation of the model. Like the EM algorithm, the QEM method has a dual interpretation, a result of merging the idea of surrogate maximization with the idea of imputation and self-consistency. The new approach is compared with other possible approaches by using simulations and analysis of real data. The proportional odds model is used as an example throughout the paper.

[1]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[2]  David R. Cox,et al.  Regression models and life tables (with discussion , 1972 .

[3]  Philip Hougaard,et al.  Life table methods for heterogeneous populations: Distributions describing the heterogeneity , 1984 .

[4]  M. Moeschberger,et al.  A bivariate survival model with modified gamma frailty for assessing the impact of interventions. , 1993, Statistics in medicine.

[5]  T R Fleming,et al.  Survival Analysis in Clinical Trials: Past Developments and Future Directions , 2000, Biometrics.

[6]  Susan A. Murphy,et al.  Maximum Likelihood Estimation in the Proportional Odds Model , 1997 .

[7]  Zhiliang Ying,et al.  Predicting Survival Probabilities with Semiparametric Transformation Models , 1997 .

[8]  A. V. D. Vaart,et al.  Asymptotic Statistics: Frontmatter , 1998 .

[9]  Niels Keiding,et al.  Statistical Models Based on Counting Processes , 1993 .

[10]  J P Klein,et al.  Semiparametric estimation of random effects using the Cox model based on the EM algorithm. , 1992, Biometrics.

[11]  D. Hunter,et al.  Optimization Transfer Using Surrogate Objective Functions , 2000 .

[12]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[13]  Susan A. Murphy,et al.  Asymptotic Theory for the Frailty Model , 1995 .

[14]  Susan A. Murphy,et al.  Consistency in a Proportional Hazards Model Incorporating a Random Effect , 1994 .

[15]  New York Dover,et al.  ON THE CONVERGENCE PROPERTIES OF THE EM ALGORITHM , 1983 .

[16]  Susan A. Murphy,et al.  Semiparametric likelihood ratio inference , 1997 .

[17]  D. Oakes,et al.  Bivariate survival models induced by frailties , 1989 .

[18]  D. Clayton,et al.  Multivariate generalizations of the proportional hazards model , 1985 .

[19]  Z. Ying,et al.  Analysis of transformation models with censored data , 1995 .

[20]  A. W. van der Vaart,et al.  On Profile Likelihood , 2000 .

[21]  Richard D. Gill,et al.  A counting process approach to maximum likelihood estimation in frailty models , 1992 .

[22]  Erik T. Parner,et al.  Asymptotic theory for the correlated gamma-frailty model , 1998 .