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Sample Path Large Deviations for Order Statistics

Part of: Nonparametric inference Limit theorems

Published online by Cambridge University Press:  14 July 2016

Ken R. Duffy ,
Claudio Macci  and
Giovanni Luca Torrisi
Ken R. Duffy*
Affiliation:
National University of Ireland, Maynooth
Claudio Macci*
Affiliation:
Università di Roma ‘Tor Vergata’
Giovanni Luca Torrisi*
Affiliation:
Consiglio Nazionale delle Ricerche
*
∗ Postal address: Hamilton Institute, National University of Ireland Maynooth, Co. Kildare, Ireland. Email address: ken.duffy@nuim.ie
∗∗ Postal address: Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, Via della Ricerca Scientifica, I-00133 Rome, Italy. Email address: macci@mat.uniroma2.it
∗∗∗ Postal address: Istituto per le Applicazioni del Calcolo ‘Mauro Picone’, Consiglio Nazionale delle Ricerche (CNR), Via dei Taurini 19, 00185, Rome, Italy. Email address: torrisi@iac.rm.cnr.it
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Abstract

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We consider the sample paths of the order statistics of independent and identically distributed random variables with common distribution function F. If F is strictly increasing but possibly having discontinuities, we prove that the sample paths of the order statistics satisfy the large deviation principle in the Skorokhod M 1 topology. Sanov's theorem is deduced in the Skorokhod M'1 topology as a corollary to this result. A number of illustrative examples are presented, including applications to the sample paths of trimmed means and Hill plots.

Keywords

Large deviationorder statisticempirical lawSkorokhod topologyweak convergence

MSC classification

Secondary: 60F10: Large deviations 62G30: Order statistics; empirical distribution functions

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 1 , March 2011 , pp. 238 - 257
Copyright
Copyright © Applied Probability Trust 2011 

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