Archive
Special Issues

Volume 4, Issue 4, December 2018, Page: 68-73
An Empirical Examination of the Asymptotic Normality of the kth Order Statistic
Mbanefo Solomon Madukaife, Department of Statistics, University of Nigeria, Nsukka, Nigeria
Received: Oct. 29, 2018;       Accepted: Nov. 15, 2018;       Published: Jan. 29, 2019
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.
Keywords
Asymptotic Normality, Inverse Distribution Function, kth Order Statistic, Monte Carlo Simulation, pth Quantile, Test for Normality
Mbanefo Solomon Madukaife, An Empirical Examination of the Asymptotic Normality of the kth Order Statistic, International Journal of Statistical Distributions and Applications. Vol. 4, No. 4, 2018, pp. 68-73. doi: 10.11648/j.ijsd.20180404.11
Reference
[1]
C. M. Jones, Estimating Densities, Quantiles, Quantile Densities and Density Quantiles,Ann. Inst. Statist. Math, vol. 44 no. 4, pp. 721-727, 1992.
[2]
S. Xu and Y. Miao, Limit Behaviors of the Deviation Between the Sample Quantiles and the Quantile,Filomat, vol. 25 no. 2, pp. 197-206, 2011.
[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.