Expansions for the distribution of M-estimates with applications to the Multi-Tone problem

¹ Applied Mathematics Group, Industrial Research Limited, Lower Hutt, New Zealand.
² School of Mathematics, University of Manchester, Manchester M13 9PL, UK; mbbsssn2@manchester.ac.uk

Received: 13 November 2008
Revised: 9 March 2009
Revised: 4 June 2009

Abstract

We give a stochastic expansion for estimates that minimise the arithmetic mean of (typically independent) random functions of a known parameter θ. Examples include least squares estimates, maximum likelihood estimates and more generally M-estimates. This is used to obtain leading cumulant coefficients of needed for the Edgeworth expansions for the distribution and density n^1/2 ( of − θ⁰) to magnitude n^−3/2 (or to n⁻² for the symmetric case), where θ⁰ is the true parameter value and n is typically the sample size. Applications are given to least squares estimates for both real and complex models. An alternative approach is given when the linear parameters of the model are nuisance parameters. The methods are illustrated with the problem of estimating the frequencies when the signal consists of the sum of sinusoids of unknown amplitudes.