On Some Fourier Methods for Inference

Abstract

Common statistical procedures such as maximum likelihood and M-estimation admit generalized representations in the Fourier domain. The Fourier domain provides fertile ground for approaching a number of difficult problems in inference. In particular, the empirical characteristic function and its extension for stationary time series are shown to be fundamental tools which support numerically simple inference procedures having arbitrarily high asymptotic efficiency and certain robustness features as well. A numerical illustration involving the symmetric stable laws is given.

Key Words:

• Fourier transforms
• Maximum likelihood
• Empirical characteristic functions
• Asymptotic efficiency
• Fisher information
• Robustness