Semiparametric alternation: convergence and efficiency

Vladimir Spokoiny (WIAS Berlin)
Wednesday, January 4, 2012 - 10:00am
Mohrenstrasse 39, Erhard-Schmidt-Raum

A general problem of semiparametric estimation is considered when the target parameter \( \theta \) describing the model structure has to be estimated under the presence of the high dimensional nuisance \( \eta \). The alternating approach assumes that the partial estimation of \( \eta \) under the given target \( \theta \) and vice versa can be done efficiently even in large dimension because the related problem is convex or even quadratic. This naturally leads to the procedure based on sequential optimization: one starts from some initial value of \( \theta \) or \( \eta \) and then sequentially estimates one parameter with the other one fixed. Unfortunately, precise theoretical results addressing the overall quality of such procedures are only available in special cases. One example is given by linear models. In this case, an alternating procedure converges and is efficient under quite simple and tractable identifiability conditions. We aim at developing a novel approach to systematic study of the quality and efficiency of such iterative procedures based on the idea of approximating the original model by a linear one in a local vicinity of the central point. This allows for extending the algorithmic properties of the procedure like geometric convergence from the linear to a general smooth case.