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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 M1 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
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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