Estimating parameters of sum-stable RV via L-estimators One of the purported uses of L-estimators is the ability to 'robustly' estimate the parameters of a random variable drawn from a given class. One of the downsides of using Levy $\alpha$-stable distributions is that it is difficult to estimate the parameters given a sample of observations drawn from the class. Has there been any work in estimating parameters of a Levy RV using L-estimators? There is an obvious difficulty in the fact that the PDF and CDF of the Levy distribution do not have a closed form, but perhaps this could be overcome by some trickery. Any hints?
 A: The Levy distribution has 4 parameter. Each of them has a quantile-based sample equivalent:


*

*$\mu$, the location parameter, can be estimated by the median. This is a high efficiency alternative (ARE$\approx 0.85$).

*$\gamma$, the scale parameter, can be estimated by the median absolute deviation (or more efficiently yet by the Qn estimator (1) with ARE similar to that of the median)

*$\beta$, the skew parameter, can be estimated by the $S_k$ estimator, with 
$S_k=(Q_x(\frac{3}{4})-2Q_x(\frac{1}{2})+Q_x(\frac{1}{4}))(Q_x(\frac{3}{4})-Q_x(\frac{1}{4}))^{-1}$ where $Q_x(\tau)$ is the $\tau$^th quantile of $x$.

*$\alpha$, the tail parameter, can be estimated by Moors's quantile based kurtosis estimator (2).


List of references:


*

*P.J. Rousseeuw, C. Croux (1993) Alternatives to the Median
 Absolute Deviation, JASA, 88, 1273-1283.

*J. J. A. Moors, (1988) A Quantile Alternative for Kurtosis 
Journal of the Royal Statistical Society. Series D (The Statistician)
Vol. 37, No. 1, pp. 25-32

