Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T05:10:17.518Z Has data issue: false hasContentIssue false

Sample Path Large Deviations for Order Statistics

Published online by Cambridge University Press:  14 July 2016

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: [email protected]
∗∗Postal address: Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, Via della Ricerca Scientifica, I-00133 Rome, Italy. Email address: [email protected]
∗∗∗Postal address: Istituto per le Applicazioni del Calcolo ‘Mauro Picone’, Consiglio Nazionale delle Ricerche (CNR), Via dei Taurini 19, 00185, Rome, Italy. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Aban, I. B., Meerschaert, M. M. and Panorska, A. K. (2006). {Parameter estimation for the truncated Pareto distribution}. J. Amer. Statist. Assoc. 101, 270277.Google Scholar
[2] Aleshkyavichene, A. K. (1991). {Large and moderate deviations for L-statistics}. Litovsk. Mat. Sb. 31, 227241.Google Scholar
[3] Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.Google Scholar
[4] Boistard, H. (2007). {Large deviations for L-statistics}. Statist. Decisions 25, 89125.Google Scholar
[5] Callaert, H., Vandemaele, M. and Veraverbeke, N. (1982). {A Cramér type large deviation theorem for trimmed linear combinations of order statistics}. Commun. Statist. Theory Meth. 11, 26892698.CrossRefGoogle Scholar
[6] De Haan, L. and Resnick, S. (1998). {On asymptotic normality of the Hill estimator}. Commun. Statist. Stoch. Models 14, 849866.Google Scholar
[7] Dembo, A. and Zeitouni, O. (1998). Large Deviation Techniques and Applications. Springer, Berlin.CrossRefGoogle Scholar
[8] Drees, H., de Haan, L. and Resnick, S. (2000). {How to make a Hill plot}. Ann. Statist. 28, 254274.Google Scholar
[9] Fristedt, B. and Gray, L. (1997). A Modern Approach to Probability Theory. Birkhäuser, Boston, MA.Google Scholar
[10] Groeneboom, P. and Shorack, G. R. (1981). {Large deviations of goodness of fit statistics and linear combinations of order statistics}. Ann. Prob. 9, 971987.Google Scholar
[11] Groeneboom, P., Oosterhoff, J. and Ruymgaart, F. H. (1979). {Large deviation theorems for empirical probability measures}. Ann. Prob. 7, 553586.Google Scholar
[12] Haeusler, E. and Segers, J. (2007). {Assessing confidence intervals for the tail index by Edgeworth expansions for the Hill estimator}. Bernoulli 13, 175194.Google Scholar
[13] Hill, B. M. (1975). {A simple general approach to inference about the tail of a distribution}. Ann. Statist. 3, 11631174.Google Scholar
[14] Koponen, I. (1995). {Analytic approach to the problem of convergence of truncated {Lévy} flights towards the Gaussian stochastic process}. Phys. Rev. E 52, 11971199.CrossRefGoogle Scholar
[15] Léonard, C. and Najim, J. (2002). {An extension of Sanov's theorem: application to the Gibbs conditioning principle}. Bernoulli 8, 721743.Google Scholar
[16] Lewis, J. T., Pfister, C.-E. and Sullivan, W. G. (1995). {Entropy, concentration of probability and conditional limit theorems}. Markov Process. Relat. Fields 1, 319386.Google Scholar
[17] Lynch, J. and Sethuraman, J. (1987). {Large deviations for processes with independent increments}. Ann. Prob. 15, 610627.Google Scholar
[18] Mantegna, R. N. and Stanley, H. E. (1994). {Stochastic process with ultraslow convergence to a Gaussian: the truncated Lévy flight}. Phys. Rev. Lett. 73, 29462949.CrossRefGoogle ScholarPubMed
[19] Mogul'ski{ı˘, A. A.} (1976). {Large deviations for trajectories of multi-dimensional random walks}. Theoret. Prob. Appl. 21, 300315.Google Scholar
[20] Mogul'ski{ı˘, A. A.} (1993). {Large deviations for processes with independent increments}. Ann. Prob. 21, 202215.Google Scholar
[21] Najim, J. (2002). {A Cramér type theorem for weighted random variables}. Electron. J. Prob. 7, 32 pp.CrossRefGoogle Scholar
[22] Puhalskii, A. (1995). {Large deviation analysis of the single server queue}. Queueing Systems 21, 566. (Correction: 23 (1996), 337.)Google Scholar
[23] Puhalskii, A. (2001). Large Deviations and Idempotent Probability (Monogr. Surveys Pure Appl. Math. 119). Chapman and Hall/CRC, Boca Raton, FL.Google Scholar
[24] Puhalskii, A. and Whitt, W. (1997). {Functional large deviation principles for first-passage-time processes}. Ann. Appl. Prob. 7, 362381.Google Scholar
[25] Puhalskii, A. A. and Whitt, W. (1998). {Functional large deviation principles for waiting and departure processes}. Prob. Eng. Inf. Sci. 12, 479507.CrossRefGoogle Scholar
[26] Resnick, S. and Stărică, C. (1998). {Tail index estimation for dependent data}. Ann. Appl. Prob. 8, 11561183.Google Scholar
[27] Sanov, I. N. (1957). {On the probability of large deviations of random magnitudes}. Mat. Sb. N. S. 42, 1144.Google Scholar
[28] Segers, J. (2002). {Abelian and Tauberian theorems on the bias of the Hill estimator}. Scand. J. Statist. 29, 461483.CrossRefGoogle Scholar
[29] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. John Wiley, New York.Google Scholar
[30] Skorokhod, A. V. (1956). {Limit theorems for stochastic processes}. Teor. Veroyat. Primen. 1, 289319.Google Scholar
[31] Stigler, S. M. (1974). {Linear functions of order statistics with smooth weight functions}. Ann. Statist. 2, 676693.Google Scholar
[32] Vandemaele, M. and Veraverbeke, N. (1982). {Cramér type large deviations for linear combinations of order statistics}. Ann. Prob. 10, 423434.Google Scholar
[33] Varadhan, S. R. S. (1966). {Asymptotic probabilities and differential equations}. Commun. Pure Appl. Math. 19, 261286.Google Scholar
[34] Vol'pert, A. I. (1967). Spaces BV and quasilinear equations}. Mat. Sb. N. S. 73, 255302.Google Scholar
[35] Whitt, W. (1980). {Some useful functions for functional limit theorems}. Math. Operat. Res. 5, 6785.CrossRefGoogle Scholar
[36] Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.Google Scholar