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October, 1968 Asymptotic Normality of Sample Quantiles for $m$-Dependent Processes
Pranab Kumar Sen
Ann. Math. Statist. 39(5): 1724-1730 (October, 1968). DOI: 10.1214/aoms/1177698155


The usual technique of deriving the asymptotic normality of a quantile of a sample in which the random variables are all independent and identically distributed [cf. Cramer (1946), pp. 367-369] fails to provide the same result for an $m$-dependent (and possibly non-stationary) process, where the successive observations are not independent and the (marginal) distributions are not necessarily all identical. For this reason, the derivation of the asymptotic normality is approached here indirectly. It is shown that under certain mild restrictions, the asymptotic almost sure equivalence of the standardized forms of a sample quantile and the empirical distribution function at the corresponding population quantile, studied by Bahadur (1966) [see also Kiefer (1967)] for a stationary independent process, extends to an $m$-dependent process, not necessarily stationary. Conclusions about the asymptotic normality of sample quantiles then follow by utilizing this equivalence in conjunction with the asymptotic normality of the empirical distribution function under suitable restrictions. For this purpose, the results of Hoeffding (1963) and Hoeffding and Robbins (1948) are extensively used. Useful applications of the derived results are also indicated.


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Pranab Kumar Sen. "Asymptotic Normality of Sample Quantiles for $m$-Dependent Processes." Ann. Math. Statist. 39 (5) 1724 - 1730, October, 1968.


Published: October, 1968
First available in Project Euclid: 27 April 2007

zbMATH: 0197.16102
MathSciNet: MR232522
Digital Object Identifier: 10.1214/aoms/1177698155

Rights: Copyright © 1968 Institute of Mathematical Statistics

Vol.39 • No. 5 • October, 1968
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