I have a sequence of multivariate data that are generated by cauchy distribution with location parameter "delta" and scale parameter "gamma". Delta is the mode and median of the data points and gamma is the "half width of half maximum" of the distribution. I like to know what efficient estimators are there for estimation of these parameters?
Thank in advance.
it has been reported that the method of moment estimation and the maximum likelihood method of estimation are not successful in estimation of the Cauchy distribution and that the Bayesian estimation is very unstable. You may want to look at the following article which proposes a new unbiased L-estimator.
http://link.springer.com/article/10.1007%2Fs00180-009-0163-y
Hi Hossein Abedi; Please pulout following link which will help in your research . Regards
http://www.sciencedirect.com/science/article/pii/037837589290088A
Dear Abedi ji please go through this article also. It will help. Regards
http://www.sciencedirect.com/science/article/pii/S0167715200000596
Dear Hossein, I am sorry, but I don't know what is the "Cauchy distribution" and so I can' t answer your question. Ask me, please, something from tectonics or geodynamics.
Good luck,
Yu. Gatinsky
.
the estimation method proposed below worked well for me :
http://www.maths.uq.edu.au/MASCOS/Talks/ASC10.pdf
and for the median, see
http://www.johndcook.com/blog/cauchy_estimation/
.
Dea Amin,
Well it is not a simulation, I am going to use some of the data generated at different generations of an evolutionary algorithm with a Cauchy distribution and then use them to get the best parameters that help me to estimate the fitness landscape with another Cauchy distribution.
Re the univariate Cauchy problem see Section 7 of Clarke, B.R. (1983) Annals of Statistics, Vol 11 , pp. 1196-1205. For the multivariate maybe you can check conditions A in that paper are satisfied by the mle?
The most general form for the Cauchy distribution has two parameter: one is the median or centre, and the other characterizes the width. Physicists call this the Breit-Wigner distribution. As usual, the maximum-likelihood estimator is the optimal estimator for both parameters. If you require a non-parametric estimator, then it is well known that the sample mean does not converge because the mean of the distribution is not defined and the variance is infinite, so the sample median is a much better estimator of the location parameter..
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