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.