Abstract
In this article, we obtain the limit distribution of the maxima and minima of independent observations from tempered stable and bilateral gamma distributions, and provide simple closed form expressions for the associated norming constants. We explore the associated near-maxima and near-minima random variables. We also discuss some applications.
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References
Arnold BC, Balakrishnan N, Nagaraja HN (2008) A first course in order statistics. SIAM. Wiley, Philadelphia
Balakrishnan N, Stepanov A (2008) Asymptotic properties of number of near-minimum observations under progressive type-II censoring. J Stat Plann Inference 138:1010–1020
Castillo E, Hadi AS, Balakrishnan N, Sarabia JM (2004) Extreme value and related models with applications in engineering and science. John Wiley, New York
David HA, Nagaraja HN (2003) Order statistics, 3rd edn. John Wiley, New York
Edit Rroji, Lorenzo Mercuri (2015) Mixed tempered stable distribution. Quant Fin 15:1559–1569
Embrechts P, Klüppelberg C, Mikosch T (1997) Modelling extremal events for insurance and finance. Springer, Berlin
Falk M, Husler J, Reiss RD (1994) Laws of small numbers: extremes and rare events. Birkhauser, Boston, USA
Galambos J (1978) The asymptotic theory of extreme order statistics. John Wiley, New York
Grabchak M (2010) Tempered stable distributions: stochastic models for multiscale processes. Springer Briefs in Mathematics. Springer, New York
Hu Z, Su C (2003) Limit theorems for the number and sum of near-maxima for medium tails. Stat Probab Lett 63:229–237
Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1, 2nd edn. John Wiley, New York
Kim YS, Rachev ST, Chung DM, Bianchi ML (2009) The modified tempered stable distributions, GARCH models and option pricing. Probab Math Stat 29:91–117
Kleiber C, Kotz S (2003) Statistical size distributions in economics and actuarial sciences. John Wiley, New York
Koponen J (1995) Analytic approach to the problem of convergence of truncated Levy flights towards the Gaussian stochastic process. Phys Rev E 52:1197–1199
Küchler U, Tappe S (2008) Bilateral Gamma distributions and processes in financial mathematics. Stoch Process Appl 118:261–283
Küchler U, Tappe S (2008) On the shapes of bilateral Gamma densities. Stat Probab Lett 78:2478–2484
Küchler U, Tappe S (2013) Tempered stable distributions and processes. Stoch Process Appl 123:4256–4293
Li Y (1999) A note on the number of records near the maximum. Stat Probab Lett 43:153–158
Mantegna RN, Stanley HE (1994) Stochastic process with ultra slow convergence to a Gaussian: the truncated Levy flights. Phys Rev Lett 73(22):2946–2949
Nagaraja HN, Karthik B, Zhang F (2015) Spacings around an order statistic. Ann Inst Stat Math 67:515–540
Pakes AG, Steutel F (1997) On the number of records near the maximum. Aust J Stat 39:179–192
Pakes AG (2000) The number and sum of near-maxima for thin tailed populations. Adv Appl Probab 32:1100–1116
Resnick SI (1987) Extreme values, regular variation, and point processes. Springer-Verlag, New York
Rosinski J (2007) Tempering stable processes. Stoch Process Appl 117:677–707
Sato K, Yamazato M (1978) On distribution functions of class L. Zeitschrift fur Wahrscheinlichkeitheorie und verwandte Gebiete 43:273–308
Vasudeva R, Vasantha Kumari J, Ravi S (2014) On the asymptotic behaviour of extremes and near-maxima of random observations from the General error distributions. J Appl Probab 51:528–541
Weissman I (1978) Estimation of parameters and larger quantiles based on the \(k\) largest obsertions. J Am Stat Assoc 73:812–815
Acknowledgements
This work is a part of the Emeritus fellowship project of the first author. He is grateful to the University Grants commission, New Delhi, India, for the support.
The authors thank the referee for his valuable suggestions.
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Vasudeva, R., Nagaraja, H.N. On the Asymptotic Behaviour of Extremes of Observations from a Tempered Stable Distribution. J Indian Soc Probab Stat 23, 359–374 (2022). https://doi.org/10.1007/s41096-022-00118-5
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DOI: https://doi.org/10.1007/s41096-022-00118-5