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/
[1] , and , On the validity of Edgeworth and saddlepoint approximations. Journal of Multivariate Analysis 51 (1994) 121-138. | MR | Zbl
[2] , Advanced Combinatorics. Reidel, Dordrecht, Holland (1974). | MR | Zbl
[3] and , General saddlepoint approximations of marginal densities and tail probabilities. Journal of the American Statistical Association 91 (1996) 666-673. | MR | Zbl
[4] and , Higher order asymptotic relation between Edgeworth approximation and saddlepoint approximation. Journal of the Japan Statistical Society 24 (1994) 109-119. | MR | Zbl
[5] , A new look at the relationship between Edgeworth expansion and saddlepoint approximation. Statistics and Probability Letters 17 (1993) 49-52. | MR | Zbl
[6] , On a moment theorem for complex Gaussian processes. IRE Transactions on information theory IT-8 (1962) 194-195 | MR | Zbl
[7] and , The bias and skewness of (univariate) M-estimates in regression. Technical Report, Applied Mathematics Group, Industrial Research Ltd., Lower Hutt, New Zealand (2007).
[8] and , Tilted Edgeworth expansions for asymptotically normal vectors. Annals of the Institute of Statistical Mathematics, doi: 10.1007/s10463-008-0206-0 (2008). | MR | Zbl
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