Am Freitag, 08. März 2002, spricht Prof. Ludger Rüschendorf über "Adaptives Schätzen mit Schätzern vom neuronalen Netztyp".

Termin: Freitag, 08. März 2002, 11:15 Uhr

Ort: Technische Universität Wien 1040 Wien, Wiedner Hauptstraße 8-10 Freihaus, Turm C (grüner Bereich), 6. Stock, Seminarraum 107

WWW: http://www.fam.tuwien.ac.at/schedule

Abstract:

We obtain consistency results and determine convergence rates for neural nets type estimators. In detail we consider the estimation of the log-hazard function in random censoring models with covariates. Our results are based on a general approach to sieved maximum likelihood estimators (or minimum contrast estimators) including an adaptive version of the estimators based on the method of structural risk estimation. A related approach was developed recently in Birge Massart(1998) and Barron Birge Massart(1999). In comparison we obtain upper bounds for the estimation error involving more simple covering numbers. We discuss two types of applications of the general results. For smoothness classes we establish an adaptive version of the tensor product spline estimator as introduced in Kooperberg Stone Truong(1995a). The minimax optimal rate of convergence is not achieved for the standard sigmoidal neural net estimator but is attained approximatively for some other activation functions as e.g. for the threshold function. Assuming the existence of a certain integral representation of the log-hazard function which is related to some smoothness conditions, we obtain improved convergence rates for net sieves type estimators as neural nets, radial basis function nets and wavelet nets. Similar convergence rate results have been established before for regression estimation in Barron(1993) and for density estimation in Modha Masry(1996). Our improvement of the convergence rate is based on an improved approximation result for functions of this type by finite net classes.