Abstract
In this paper an alternative measure for the excess, called standard archα s , is introduced. It is only an affine transformation of the classical kurtosis, but has many advantages. It can be defined as the double relative asymptotic variance of the standard deviation and can be generalized as the double relative asymptotic variance of any other scale estimator. The inequalities between skewness and kurtosis given inTeuscher andGuiard (1995) are transformed to the corresponding inequalities between skewness and standard arch.
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References
Bachmaier M. (1998):Klassische, robuste und nichtparametrische Bartlett-Tests. Habilitationsschrift, Technische Universität München
Bachmaier M. (1999):Efficiency comparison of M-estimates for scale at t-distributions. Statistical Papers,the paper afterwoods
Bartlett, M.S. (1937):Properties of Sufficiency and Statistical Tests. Proceedings of the Royal Society, A 160,268–282
Box, G.E.P. (1953):Non-Normality and Tests on Variances. Biometrika 40,318–335
Box, G.E.P., Andersen S.L. (1955):Permutation theory in the derivation of robust criteria and the study of departures from assumptions. J.R.S.S. Ser. B 17,1–34
Guiard V. (1980):Robustheit I. Probleme der angewandten Statistik, Heft 4, Forschungszentrum der Tierproduktion, Dummerstorf Rostock.
Guiard V. (1981):Robustheit II. Probleme der angewandten Statistik, Heft 5, Forschungszentrum der Tierproduktion, Dummerstorf Rostock
Guiard V., Rasch D. (1987):Robustheit statistischer Verfahren. Probleme der angewandten Statistik, Heft 20, Forschungszentrum der Tierproduktion, Dummerstorf Rostock
Huber, P.J. (1981):Robust Statistics. John Wiley & Sons, Inc., New York
Johnson, N.L., Rogers C.A. (1951):The moment problem for unimodal distributions. Ann. Math. Statist. 22,433–439
Kendall M.G., Stuart A. (1969):The Advanced Theory of Statistics. 3rd three-volume edition, vol. 1, Griffin & Co. London
Miller R.G. jr. (1968):Jackknifing Variances. Annals of Mathematical Statistics 39,567–582
Pearson K. (1916):Mathematical contributions of the theory of evolution, XIX, second supplement to a memoir on skew variation. Philos. Trans. Roy. Soc. London Ser. A 216,432
Rasch D. (1987):Biometrisches Wörterbuch. 3. Auflage, Deutscher Landwirtschaftsverlag, Berlin
Rasch D., Herrendörfer G., Bock J., Victor N., Guiard V. (1996):Verfahrensbibliothek Versuchsplanung und -auswertung Band I, Oldenbourg, München
Rasch D., Tiku M.L., Sumpf D. (1994):Elsevier's Dictionary of Biometry. Elsevier Amsterdam
Simpson J.A., Welch B.L. (1960):Table of the bounds of the probability integral when the first four moments are given. Biometrika 47,399–410
Teuscher F., Guiard V. (1995):Sharp inequalities between skewness and kurtosis for unimodal distributions. Statistics & Probability Letters 22,257–260
Wilkins J.E. (1944):A note on skewness and kurtosis. Ann. Math. Statist. 15,333–335
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Bachmaier, M., Guiard, V. An alternative and generalized excess measure and its advantages. Statistical Papers 41, 37–52 (2000). https://doi.org/10.1007/BF02925675
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DOI: https://doi.org/10.1007/BF02925675