Open Access
June, 1982 Nonparametric Maximum Likelihood Estimation by the Method of Sieves
Stuart Geman, Chii-Ruey Hwang
Ann. Statist. 10(2): 401-414 (June, 1982). DOI: 10.1214/aos/1176345782


Maximum likelihood estimation often fails when the parameter takes values in an infinite dimensional space. For example, the maximum likelihood method cannot be applied to the completely nonparametric estimation of a density function from an $\operatorname{iid}$ sample; the maximum of the likelihood is not attained by any density. In this example, as in many other examples, the parameter space (positive functions with area one) is too big. But the likelihood method can often be salvaged if we first maximize over a constrained subspace of the parameter space and then relax the constraint as the sample size grows. This is Grenander's "method of sieves." Application of the method sometimes leads to new estimators for familiar problems, or to a new motivation for an already well-studied technique. We will establish some general consistency results for the method, and then we will focus on three applications.


Download Citation

Stuart Geman. Chii-Ruey Hwang. "Nonparametric Maximum Likelihood Estimation by the Method of Sieves." Ann. Statist. 10 (2) 401 - 414, June, 1982.


Published: June, 1982
First available in Project Euclid: 12 April 2007

zbMATH: 0494.62041
MathSciNet: MR653512
Digital Object Identifier: 10.1214/aos/1176345782

Primary: 62A10
Secondary: 62G05

Keywords: Density estimation , maximum likelihood , Method of sieves , nonparametric estimation , regression

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 2 • June, 1982
Back to Top