On the asymptotic distribution of an alternative measure of kurtosis

  • Authors

    • Kagba Suaray California State University Long Beach
    2015-08-07
    https://doi.org/10.14419/ijasp.v3i2.5007
  • Kurtosis, Quantile Estimator, Ratio of Normal Variables, Spread Function.

  • Pearson defined the fourth standardized moment of a symmetric distribution as its kurtosis. There has been much discussion in the literature concerning both the meaning of kurtosis, as well as the effectiveness of the classical sample kurtosis as a reliable measure of peakedness and tail weight. In this paper, we consider an alternative measure, developed by Crow and Siddiqui, used to describe kurtosis. Its value is calculated for a number of common distributions, and a derivation of its asymptotic distribution is given. Simulations follow, which reveal an interesting connection to the literature on the ratio of normal random variables.

  • References

    1. [1] E. Crow and M. Siddiqui, Robust Estimation of Location, Journal of the American Statistical Association, 62, 318 (1967) 353-389.

      [2] K. Balanda and H.L. MacGillivray, Kurtosis: A Critical review, The American Statistician, 42, 2, (1988) 111-119.

      [3] S. Stigler, Studies in the History of Probability and Statistics. XXXII: Laplace, Fisher and the Discovery of the Concept of Sufficiency, Biometrika, 60, 3 (1973) 439-445.

      [4] T. Kim and H. White, On More Robust Estimation of Skewness and Kurtosis, Finance Research Letters, 1, 1 (2004) 56-73.

      [5] R. Serfling, Approximation Theorems of Mathematical Statistics, John Wiley, New York, 1980.

      [6] R. Hogg, Adaptive Robust Procedures: A partial review and some suggestions for future applications and theory, Journal of the American Statistical Association, 69, 348 (1974) 909-923.

      [7] K. Balanda and H.L. MacGillivray, Kurtosis and Spread, The Canadian Journal of Statistics, 18, 1, (1990) 17-30.

      [8] DasGupta, Asymptotic Theory of Statistics and Probability, Springer, New York, 2008.

      [9] R. Hyndman and Y. Fan, Sample Quantiles in Statistical Packages, The American Statistician, 50, 4, (1996) 361-365.

      [10] S. Yang, A Smooth Nonparametric Estimator of a Quantile Function, Journal of the American Statistical Association, 80, 392 (1985) 1004-1011.

      [11] G. Marsaglia, Ratios of Normal Variables and Ratios of Sums of Uniform Variables, Journal of the American Statistical Association, 60, (1965) 193-204.

      [12] G. Marsaglia, Ratios of Normal Variables, Journal of Statistical Software, 16, 4 (2006) 1-10.

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