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

• Volume: 15, page 139-167
• ISSN: 1292-8100

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## Abstract

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We give a stochastic expansion for estimates $\stackrel{^}{\theta }$ that minimise the arithmetic mean of (typically independent) random functions of a known parameterθ. Examples include least squares estimates, maximum likelihood estimates and more generally M-estimates. This is used to obtain leading cumulant coefficients of $\stackrel{^}{\theta }$ needed for the Edgeworth expansions for the distribution and densityn1/2θ0) 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.

## How to cite

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Withers, Christopher S., and Nadarajah, Saralees. "Expansions for the distribution of M-estimates with applications to the Multi-Tone problem." ESAIM: Probability and Statistics 15 (2011): 139-167. <http://eudml.org/doc/277156>.

@article{Withers2011,
abstract = {We give a stochastic expansion for estimates $\widehat\{\theta \}$ that minimise the arithmetic mean of (typically independent) random functions of a known parameterθ. Examples include least squares estimates, maximum likelihood estimates and more generally M-estimates. This is used to obtain leading cumulant coefficients of $\widehat\{\theta \}$ needed for the Edgeworth expansions for the distribution and densityn1/2θ0) 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.},
author = {Withers, Christopher S., Nadarajah, Saralees},
journal = {ESAIM: Probability and Statistics},
keywords = {bias; edgeworth; maximum likelihood; M-estimates; skewness; Edgeworth; -estimates},
language = {eng},
pages = {139-167},
publisher = {EDP-Sciences},
title = {Expansions for the distribution of M-estimates with applications to the Multi-Tone problem},
url = {http://eudml.org/doc/277156},
volume = {15},
year = {2011},
}

TY - JOUR
AU - Withers, Christopher S.
TI - Expansions for the distribution of M-estimates with applications to the Multi-Tone problem
JO - ESAIM: Probability and Statistics
PY - 2011
PB - EDP-Sciences
VL - 15
SP - 139
EP - 167
AB - We give a stochastic expansion for estimates $\widehat{\theta }$ that minimise the arithmetic mean of (typically independent) random functions of a known parameterθ. Examples include least squares estimates, maximum likelihood estimates and more generally M-estimates. This is used to obtain leading cumulant coefficients of $\widehat{\theta }$ needed for the Edgeworth expansions for the distribution and densityn1/2θ0) 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.
LA - eng
KW - bias; edgeworth; maximum likelihood; M-estimates; skewness; Edgeworth; -estimates
UR - http://eudml.org/doc/277156
ER -

## References

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1. [1] J.G. Booth, P. Hall and A.T.A. Wood, On the validity of Edgeworth and saddlepoint approximations. Journal of Multivariate Analysis51 (1994) 121–138. Zbl0807.62015MR1309372
2. [2] L. Comtet, Advanced Combinatorics. Reidel, Dordrecht, Holland (1974). Zbl0283.05001MR460128
3. [3] R. Gatto and E. Ronchetti, General saddlepoint approximations of marginal densities and tail probabilities. Journal of the American Statistical Association91 (1996) 666–673. Zbl0869.62017MR1395734
4. [4] Y. Kakizawa and M. Taniguchi, Higher order asymptotic relation between Edgeworth approximation and saddlepoint approximation. Journal of the Japan Statistical Society24 (1994) 109–119. Zbl0818.62014MR1326522
5. [5] A.C. Monti, A new look at the relationship between Edgeworth expansion and saddlepoint approximation. Statistics and Probability Letters17 (1993) 49–52. Zbl0765.62025MR1225364
6. [6] I.S. Reed, On a moment theorem for complex Gaussian processes. IRE Transactions on information theory IT-8 (1962) 194–195 Zbl0102.34903MR137145
7. [7] C.S. Withers and S. Nadrajah, The bias and skewness of (univariate) M-estimates in regression. Technical Report, Applied Mathematics Group, Industrial Research Ltd., Lower Hutt, New Zealand (2007).
8. [8] C.S. Withers and S. Nadarajah, Tilted Edgeworth expansions for asymptotically normal vectors. Annals of the Institute of Statistical Mathematics, doi: 10.1007/s10463-008-0206-0 (2008).

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