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An Empirical Examination of the Asymptotic Normality of the kth Order Statistic

Received: 29 October 2018     Accepted: 15 November 2018     Published: 29 January 2019
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Abstract

In this paper, the equivalence of the sample pth quantile of a distribution and the kth order statistic of a random sample obtained from the distribution is reviewed. Based on the review, a new corollary on the almost sure convergence of the kth order statistic to the pth quantile was obtained without proof. Through an extensive Monte Carlo simulation, the extreme as well as the central kth order statistics of five different continuous distributions were obtained at different sample sizes and the asymptotic normality of the order statistics were investigated with the use of the Anderson – Darling (AD) statistic for normality test. The result showed among other things that asymptotic normality holds only for the central order statistics.

Published in International Journal of Statistical Distributions and Applications (Volume 4, Issue 4)
DOI 10.11648/j.ijsd.20180404.11
Page(s) 68-73
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Asymptotic Normality, Inverse Distribution Function, kth Order Statistic, Monte Carlo Simulation, pth Quantile, Test for Normality

References
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[3] T. A. Severini, Elements of Distribution Theory. Cambridge: Cambridge University Press, 2005.
[4] M. J. R. Healy, Multivariate Normal Plotting,Appl. Statist., vol. 17 no. 2, pp. 157-161, 1968.
[5] N. J. H. Small, Plotting Squared Radii,Biometrika, vol. 65, no. 3, pp. 657-658, 1978.
[6] L. Scrucca, Assessing Multivariate Normality through Interactive Dynamic Graphics,Quaderni di statistica, vol. 2, pp. 221-240, 2000.
[7] S. K. Ahn, F-Probability Plot and Its Application to Multivariate Normality, Commun. Statist. Theory Methods, vol. 21, no. 4, pp. 997-1023, 1992.
[8] A. Singh, Omnibus Robust Procedures for Assessment of Multivariate Normality and Detection of Multivariate Outliers,Multivariate Environmental Statistics, G.P. Patil and C.R. Rao eds, Amsterdam: North Holland, 1993.
[9] T. Hwu, C. Han, and K. J. Rogers, The Combination Test for Multivariate Normality, J. Statist. Comput. Simul., vol. 72, no. 5, pp. 379-390, 2002.
[10] R. R. Bahadur, A Note on Quantiles in Large Samples, Ann. Math. Statist., vol. 37, pp. 577-580, 1960.
[11] R. J. Serfling, Approximation Theorems of Mathematical Statistics, New York: John Wiley and Sons Inc., pp. 74-89, 1980.
[12] G. T. Babu, A Note on Bootstrapping the Variance of Sample Quantile, Ann. Instit. Statist. Math., vol. 38 part A, pp. 439-443, 1986.
[13] Y. Miao, Y. Chen and S. Xu, Asymptotic Properties of the Deviation Between Order Statistics and p-Quantile, Commun. Statist. Theory Methods, vol. 40, no. 1, 8-14, 2011.
[14] A. M. Mood, F. A. Graybill and D. C. Boes, Introduction to the Theory of Statistics, New York: McGraw-Hill Inc, pp. 256-258, 1974.
[15] H. A. David and H. N. Nagaraja, Order Statistics, New York: John Wiley and Sons Inc., pp. 79-80, 2003.
[16] A. W. van der Vaart, Asymptotic Statistics. New York: Cambridge University Press, pp. 304-305, 1998.
[17] V. I. Pagurova, On the asymptotic simultaneous distribution of randomly indexed order statistics, Moscow University Computational Mathematics and Cybernetics 34 (2), 87-90, 2010.
[18] A. Dembinska, Asymptotic normality of numbers of observations in random regions determined by order statistics. Statistics 48 (3), 508-523, 2014.
[19] K. Jasinski, Asymptotic normality of numbers of observations near order statistics from stationary processes. Statistics and Probability Letters 119, 259-263, 2016.
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  • APA Style

    Mbanefo Solomon Madukaife. (2019). An Empirical Examination of the Asymptotic Normality of the kth Order Statistic. International Journal of Statistical Distributions and Applications, 4(4), 68-73. https://doi.org/10.11648/j.ijsd.20180404.11

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    ACS Style

    Mbanefo Solomon Madukaife. An Empirical Examination of the Asymptotic Normality of the kth Order Statistic. Int. J. Stat. Distrib. Appl. 2019, 4(4), 68-73. doi: 10.11648/j.ijsd.20180404.11

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    AMA Style

    Mbanefo Solomon Madukaife. An Empirical Examination of the Asymptotic Normality of the kth Order Statistic. Int J Stat Distrib Appl. 2019;4(4):68-73. doi: 10.11648/j.ijsd.20180404.11

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  • @article{10.11648/j.ijsd.20180404.11,
      author = {Mbanefo Solomon Madukaife},
      title = {An Empirical Examination of the Asymptotic Normality of the kth Order Statistic},
      journal = {International Journal of Statistical Distributions and Applications},
      volume = {4},
      number = {4},
      pages = {68-73},
      doi = {10.11648/j.ijsd.20180404.11},
      url = {https://doi.org/10.11648/j.ijsd.20180404.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsd.20180404.11},
      abstract = {In this paper, the equivalence of the sample pth quantile of a distribution and the kth order statistic of a random sample obtained from the distribution is reviewed. Based on the review, a new corollary on the almost sure convergence of the kth order statistic to the pth quantile was obtained without proof. Through an extensive Monte Carlo simulation, the extreme as well as the central kth order statistics of five different continuous distributions were obtained at different sample sizes and the asymptotic normality of the order statistics were investigated with the use of the Anderson – Darling (AD) statistic for normality test. The result showed among other things that asymptotic normality holds only for the central order statistics.},
     year = {2019}
    }
    

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    T1  - An Empirical Examination of the Asymptotic Normality of the kth Order Statistic
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    AB  - In this paper, the equivalence of the sample pth quantile of a distribution and the kth order statistic of a random sample obtained from the distribution is reviewed. Based on the review, a new corollary on the almost sure convergence of the kth order statistic to the pth quantile was obtained without proof. Through an extensive Monte Carlo simulation, the extreme as well as the central kth order statistics of five different continuous distributions were obtained at different sample sizes and the asymptotic normality of the order statistics were investigated with the use of the Anderson – Darling (AD) statistic for normality test. The result showed among other things that asymptotic normality holds only for the central order statistics.
    VL  - 4
    IS  - 4
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Author Information
  • Department of Statistics, University of Nigeria, Nsukka, Nigeria

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