Time-efficient algorithms for robust estimators of location, scale, symmetry, and tail heaviness
Wouter Gelade
University of Namur
Centre of Research in the Economics of Development (CRED)
Namur, Belgium
[email protected]
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Vincenzo Verardi
University of Namur
Centre of Research in the Economics of Development (CRED)
Namur, Belgium
and Université libre de Bruxelles
ECARES and iCite
Brussels, Belgium
[email protected]
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Catherine Vermandele
Université libre de Bruxelles
Laboratoire de Méthodologie du Traitement des Données (LMTD)
Brussels, Belgium
[email protected]
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Abstract. The analysis of the empirical distribution of univariate data often includes
the computation of location, scale, skewness, and tail-heaviness measures,
which are estimates of specific parameters of the underlying population
distribution. Several measures are available, but they differ by Gaussian
efficiency, robustness regarding outliers, and meaning in the case of
asymmetric distributions. In this article, we briefly compare, for each type of
parameter (location, scale, skewness, and tail heaviness), the “classical”
estimator based on (centered) moments of the empirical distribution, an
estimator based on specific quantiles of the distribution, and an estimator
based on pairwise comparisons of the observations. This last one always
performs better than the other estimators, particularly in terms of robustness,
but it requires a heavy computation time of an order of n^2.
Fortunately, as explained in Croux and Rousseeuw (1992, Computational
Statistics 1: 411–428), the algorithm of Johnson and Mizoguchi (1978,
SIAM Journal of Scientific Computing 7: 147–153) allows one to
substantially reduce the computation time to an order of n log n
and, hence, allows the use of robust estimators based on pairwise comparisons,
even in very large datasets. This has motivated us to program this algorithm
for Stata. In this article, we describe the algorithm and the associated
commands. We also illustrate the computation of these robust estimators by
involving them in a normality test of Jarque–Bera form (Jarque and Bera
1980, Economics Letters 6: 255–259; Brys, Hubert, and Struyf,
2008, Computational Statistics 23: 429–442) using real data.
View all articles by these authors:
Wouter Gelade, Vincenzo Verardi, Catherine Vermandele
View all articles with these keywords:
mhl, sqn, medcouple, robjb, location, scale, symmetry, tail heaviness, mean, median, skewness, kurtosis, robust estimation
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