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?
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$\begingroup$ We need a finite mean (first moment) to compute an L-estimator (don't we?). Levy distributed r.v. do not come with such niceties. Correct me if i'm wrong. $\endgroup$– user603Commented Sep 18, 2010 at 22:02
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$\begingroup$ My understanding is that you need a finite mean to for a population L-moment to be defined, but not for the estimands corresponding to all other L-estimators. For example, the sample median is an L-estimator, though it isn't an L-moment. $\endgroup$– onestopCommented Oct 18, 2010 at 20:59
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1 Answer
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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
