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The abscissa of convergence of the Laplace transform

Part of: Harmonic analysis in one variable Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Peter Hall ,
Jozef L. Teugels  and
Ann Vanmarcke
Peter Hall*
Affiliation:
Australian National University
Jozef L. Teugels*
Affiliation:
Katholieke Universiteit Leuven
Ann Vanmarcke*
Affiliation:
Katholieke Universiteit Leuven
*
∗Postal address, Department of Statistics, Faculty of Economics and Commerce, Australian National University, GPO Box 4, Canberra ACT 2601, Australia.
∗∗Postal address: Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200 B, 3001 Leuven (Heverlee), Belgium.
∗∗Postal address: Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200 B, 3001 Leuven (Heverlee), Belgium.
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Abstract

Assume that we want to estimate – σ, the abscissa of convergence of the Laplace transform. We show that no non-parametric estimator of σ can converge at a faster rate than (log n)–1, where n is the sample size. An optimal convergence rate is achieved by an estimator of the form where xn = O(log n) and is the mean of the sample values overshooting xn. Under further parametric restrictions this (log n)–1 phenomenon is also illustrated by a weak convergence result.

Keywords

CONVERGENCE RATEDIVERGENCE POINTEMPIRICAL METHODEXTREME VALUE THEORYMINIMAX

MSC classification

Primary: 60E10: Characteristic functions; other transforms
Secondary: 42A61: Probabilistic methods 60G70: Extreme value theory; extremal processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 2 , June 1992 , pp. 353 - 362
Copyright
Copyright © Applied Probability Trust 1992 

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References

[1] Beirlant, J. and Broniatowski, M. (1991) The mean residual life function and Chernoff-type large deviation. Technical report.Google Scholar
[2] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987) Regular Variation. Cambridge University Press.Google Scholar
[3] Csörgő, S. and Mason, D. M. (1985) Central limit theorems for sums of extreme values. Math. Proc. Camb. Phil. Soc. 98, 547–558.Google Scholar
[4] Csörgő, S. and Teugels, J. L. (1990) Empirical Laplace transform and approximation of compound distributions. J. Appl. Prob. 27, 88–101.Google Scholar
[5] Davison, A. C. and Smith, R. L. (1990) Models for exceedances over high thresholds. J. R. Statist. Soc. B52, 393–442.Google Scholar
[6] Gnedenko, B. (1943) Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44, 423–453.Google Scholar
[7] Smith, R. L. (1987) Estimating tails of probability distributions. Ann. Statist. 15, 1174–1207.Google Scholar
[8] Vanmarcke, A. (1992) Doctoral Dissertation, Katholieke Universiteit Leuven.Google Scholar
[9] Widder, D. (1941) The Laplace Transform. Princeton University Press.Google Scholar