International Statistical Review

On Testing for the Nullity of Some Skewness Coefficients

Joseph Ngatchou-Wandji

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Three tests for the skewness of an unknown distribution are derived for iid data. They are based on suitable normalization of estimators of some usual skewness coefficients. Their asymptotic null distributions are derived. The tests are next shown to be consistent and their power under some sequences of local alternatives is investigated. Their finite sample properties are also studied through a simulation experiment, and compared to those of the \sqrt{b1}-test.

Article information

Source
Internat. Statist. Rev., Volume 74, Number 1 (2006), 47-65.

Dates
First available in Project Euclid: 29 March 2006

Permanent link to this document
https://projecteuclid.org/euclid.isr/1143654386

Zentralblatt MATH identifier
1142.62347

Keywords
Empirical quantiles Kernel estimator Mode estimation Nonparametric testing Skewness \linebreak Symmetry

Citation

Ngatchou-Wandji, Joseph. On Testing for the Nullity of Some Skewness Coefficients. Internat. Statist. Rev. 74 (2006), no. 1, 47--65. https://projecteuclid.org/euclid.isr/1143654386


Export citation

References

  • [1] Abdous, B., Ghoudi, K. & Rémillard, B. (2003). Nonparametric weighted symmetry tests. The Canadian J. Statist, 31, 357-381.
  • [2] Ahmad, I.A. & Li, Q. (1997). Testing symmetry of an unknown density function by kernel method. J. Nonparametr. Statist., 7, 279-293.
  • [3] Aki, S. (1981). Asymptotic distribution of a Cramér-von Mises type statistic for testing symmetry when the center is estimated. Ann. Inst. Statist. Math., 33, Part A, 1-14.
  • [4] Antille, A., Kersting, G. & Zuccini, W. (1982). Testing symmetry. J. Amer. Statist. Assoc., 77, 639-646.
  • [5] Azzalini, A. & Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika, 83, 715-726.
  • [6] Badrinath, S.G. & Chatterjee, S. (1988). On measuring skewness and elongation in common stock return distributions: The case of the Market Index. J. of Business, 61, 451-472.
  • [7] Badrinath, S.G. & Chatterjee, S. (1991). A data-analytic look at skewness and elongation in common-stock-return distributions. J. of Business & Econ. Statist., 9, 223-233.
  • [8] Bai, J. & Ng, S. (2005). Tests for skewness, kurtosis, and normality for time series data. J. Business Econ. Statist., 23, 49-61.
  • [9] Bhattacharya, P.K., Gastwirth, J.L. & Wright, A.L. (1982). Two modified Wilcoxon tests for symmetry about an unknown location parameter. Biometrika, 69, 377-382.
  • [10] Billingsley P. (1968). Convergence of Probability Measures. Academic Press.
  • [11] Boos, D. (1982). A test for asymmetry associated with the Hodges-Lehmann estimator. J. Amer. Statist. Assoc., 77, 647-651.
  • [12] Bowman, K.O. & Shenton, L.R. (1973). Notes on the distribution of \sqrt{b_1} in sampling from Pearson distribution. Biometrika, 60, 155-167.
  • [13] D'Agostino, R.B. & Tietjen, G.L. (1973). Approaches to the null distribution of \sqrt{b_1}. Biometrika, 60, 169-173.
  • [14] D'Agostino, R. & Pearson, E. S. (1973). Tests for departure from normality. Empirical results for the distribution of b_2 and \sqrt{b_1}. Biometrika, 60, 613-622.
  • [15] David, F.N. & Johnson, N.L. (1956). Some tests of significance with ordered variables. J. Roy. Statist. Soc. B, 18, 1-20.
  • [16] David, H.A. & Nagaraja, H.N. (2003). Order Statistics. Third edition. New York: Wiley.
  • [17] Doksum K.A., Fenstad, G. & Aaberge, R. (1977). Plots and tests for symmetry. Biometrika, 64, 473-487. \pagebreak
  • [18] Dutta, K.K. & Babbel, D.F. (2003). On measuring skewness and kurtosis in short rate distributions: The case of the US dollar London Inter Bank Offer Rates. Working paper 02-25 of the Wharton School, University of Pennsylvania.
  • [19] Eddy, W.F. (1980). Optimum kernel estimators of the mode. Ann. Statist., 8, 870-882.
  • [20] Ellison, B. (1964). Two theorems for inferences about the normal distribution with applications in acceptance sampling. J. Amer. Statist. Assoc., 59, 89-95.
  • [21] Ghosh, J.K. (1971). A new proof of the Bahadur representation of quantiles and an application. Ann. Math. Statist., 42, 1957-1961.
  • [22] Groeneveld, R.A. & Meeden, G. (1984). Measuring skewness and kurtosis. The Statistician, 33, 391-399.
  • [23] Gupta, M.K. (1967). An asymptotically nonparametric test of symmetry. Ann. Math. Statist., 38, 849-866.
  • [24] Hall, P. & Huang, L.S. (2002). Unimodal density estimation using kernel methods. Statist. Sinica, 12, 965-990.
  • [25] Kendall, M. & Stuart, A. (1969). The Advanced Theory of Statistics. Mcgraw Hill.
  • [26] Kim, T.H. & White, H. (2004). On more robust estimation of skewness and kurtosis: Simulation and application to the S& P500 Index. Finance Research Letters, 1, 56-70.
  • [27] Koziol, J.A. (1985). A note on testing symmetry with estimated parameters. Statist. and Probab. Letters, 3, 227-230.
  • [28] Koutrouvelis, I.A. (1985). Distribution-free procedures for location and symmetry inference problems based on the empirical characteristic function. Scandinavian J. Statist., 12, 257-269.
  • [29] Mammen, E., Marron, J.S. & Fisher, N.I. (1992). Some asymptotics for multimodality tests based on kernel density estimates. Theory Probab Rel. Fields, 91, 115-132.
  • [30] Meyer, M.C. (2001). An alternative unimodal density estimator with consistent estimate of the mode. Statist. Sinica, 11, 1159-1174.
  • [31] Meyer, M.C. & Woodroofe, M. (2004). Consistent maximum likelihood estimation of a unimodal density using shape restrictions. Canadian J. Statist., 32, 85-100.
  • [32] Mulholland, H.P. (1977). On the null distribution of \sqrt{b_1} for samples of size at most 25, with tables. Biometrika, 64, 401-409.
  • [33] Parzen, E. (1962). On the estimation of a probability density function and its mode. Ann. Math. Statist., 33, 1065-1076.
  • [34] Randles, R.H., Fligner, M.A., Policello II, G.E. & Wolfe, D.A. (1980). An asymptotically distribution-free test for symmetry versus asymmetry. J. Amer. Statist. Assoc., 75, 168-172.
  • [35] Shuster, E.F. (1969). Estimation of a probability density function and its derivatives. Ann. Math. Statist., 40, 1187-1195.
  • [36] Silverman, B.W. (1978). Weak and strong uniform consistency of the kernel estimate of a density and its derivatives. Ann. Statist., 6, 177-184.
  • [37] Silverman, B.W. (1981). Using kernel density estimates to investigate the multimodality. J. Roy. Statist. Soc. B, 43, 97-99.
  • [38] Silverman, B.W. (1983). Some properties of a test for unimodality based on Kernel density estimates. In Probability Analysis and Statistics, Eds. J.F.C. Kingsman and G.E. Reuter. IMS Lecture Notes, 79, 248-259.
  • [39] Singh, R.S. (1976). Nonparametric estimation of mixed partial derivatives of a multivariate density. J. Multivariate Anal., 6, 111-122.
  • [40] Singh, R.S. (1979). On necessary and sufficient conditions for uniform strong consistency of estimators of a density and its derivatives. J. Multivariate Anal., 9, 157-164.
  • [41] Singh, R.S. (1981). On the exact asymptotic behavior of estimators of a density and its derivatives. Ann. Statist., 9, 453-456.
  • [42] Tassi, P. (1992). Méthodes Statistiques. Economica, deuxième édition.