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 n1/2 ( of \widehat{\theta}$ - 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.
Mots clés : bias, edgeworth, maximum likelihood, M-estimates, skewness
@article{PS_2011__15__139_0,
author = {Withers, Christopher S. and Nadarajah, Saralees},
title = {Expansions for the distribution of $M$-estimates with applications to the {Multi-Tone} problem},
journal = {ESAIM: Probability and Statistics},
pages = {139--167},
publisher = {EDP-Sciences},
volume = {15},
year = {2011},
doi = {10.1051/ps/2009012},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2009012/}
}
TY - JOUR AU - Withers, Christopher S. AU - Nadarajah, Saralees TI - Expansions for the distribution of $M$-estimates with applications to the Multi-Tone problem JO - ESAIM: Probability and Statistics PY - 2011 SP - 139 EP - 167 VL - 15 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2009012/ DO - 10.1051/ps/2009012 LA - en ID - PS_2011__15__139_0 ER -
%0 Journal Article %A Withers, Christopher S. %A Nadarajah, Saralees %T Expansions for the distribution of $M$-estimates with applications to the Multi-Tone problem %J ESAIM: Probability and Statistics %D 2011 %P 139-167 %V 15 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2009012/ %R 10.1051/ps/2009012 %G en %F PS_2011__15__139_0
Withers, Christopher S.; Nadarajah, Saralees. Expansions for the distribution of $M$-estimates with applications to the Multi-Tone problem. ESAIM: Probability and Statistics, Tome 15 (2011), pp. 139-167. doi : 10.1051/ps/2009012. http://www.numdam.org/articles/10.1051/ps/2009012/
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