Abstract
The asymptotic distribution of sample quantiles in the classical definition is well-known to be normal for absolutely continuous distributions. However, this is no longer true for discrete distributions or samples with ties. We show that the definition of sample quantiles based on mid-distribution functions resolves this issue and provides a unified framework for asymptotic properties of sample quantiles from absolutely continuous and from discrete distributions. We demonstrate that the same asymptotic normal distribution result as for the classical sample quantiles holds at differentiable points, whereas a more general form arises for distributions whose cumulative distribution function has only one-sided differentiability. For discrete distributions with finite support, the same type of asymptotics holds and the sample quantiles based on mid-distribution functions either follow a normal or a two-piece normal distribution. We also calculate the exact distribution of these sample quantiles for the binomial and Poisson distributions. We illustrate the asymptotic results with simulations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berry G., Armitage P. (1995) Mid-P confidence intervals: a brief review. The Statistician 44: 417–423
David H.A., Nagaraja H.N. (2003) Order statistics (3rd ed). Wiley, Hoboken
Galton F. (1889) Natural inheritance. Macmillan, London
Genest C., Nešlehová J. (2007) A primer on copulas for count data. The Astin Bulletin 37: 475–515
Genton M.G., Ma Y., Parzen E. (2006) Discussion of “Sur une limitation très générale de la dispersion de la médiane” by M. Fréchet, 1940. Journal de la Société Française de Statistique 147: 51–60
Gibbons J.F., Mylroie S. (1973) Estimation of impurity profiles in ion-implanted amorphous targets using joined half-Gaussian distributions. Applied Physics Letters 22: 568–569
González-Barrios J., Rueda R. (2001) On convergence theorems for quantiles. Communications in Statistics Theory and Methods 30: 943–955
Hald A. (1998) A history of mathematical statistics from 1750 to 1930. Wiley, New York
John S. (1982) The three-parameter two-piece Normal family of distributions and its fitting. Communications in Statistics Theory and Methods 11: 879–885
Lancaster H.O. (1961) Significance tests in discrete distributions. Journal of the American Statistical Association 56: 223–234
Larocque D., Randles R.H. (2008) Confidence intervals for a discrete population median. The American Statistician 62: 32–39
Machado J., Santos Silva J. (2005) Quantiles for counts. Journal of the American Statistical Association 100: 1226–1237
Parzen E. (1954) On uniform convergence of families of sequences of random variables. California Publications in Statistics, University of California Press, Berkeley
Parzen E. (1997) Concrete statistics. In: Ghosh S., Schucany W.R., Smith W.B. (eds) Statistics in quality. Marcel Dekker, New York, pp 309–332
Parzen E. (2004) Quantile probability and statistical data modeling. Statistical Science 19: 652–662
Smirnov N.V. (1952) Limit distributions for the terms of a variational series 1. American Mathematical Society, Ser., Translation Number 67: 1–64
Walker A.M. (1968) A note on the asymptotic distribution of sample quantiles. Journal of the Royal Statistical Society Series B 30: 570–575
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Ma, Y., Genton, M.G. & Parzen, E. Asymptotic properties of sample quantiles of discrete distributions. Ann Inst Stat Math 63, 227–243 (2011). https://doi.org/10.1007/s10463-008-0215-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-008-0215-z