Abstract
Extreme values are a major concern in various areas including finance, insurance, hydrology, etc. Often, natural phenomena are explained by mixture distributions, for example, distribution of wind speed is the mixture of distribution of wind speed under normal circumstances and extreme circumstances. In this article, we consider the extreme behaviour of a mixture distribution and study the max domains of attraction of mixture distributions. A few illustrative examples are also discussed.
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References
Al-Hussaini EK, El-Adll ME (2004) Asymptotic distribution of normalized maximum under finite mixture models. Stat Probab Lett 70:109–117
Balkema AA (2013) New power limits for extremes. Extremes 16:457–485
Bhati D, Ravi S (2018) Diagnostic plots for identifying max domains of attraction under power normalization. J Appl Stat. https://doi.org/10.1080/02664763.2017.1421915
Embrechts P, Klüeppelberg C, Mikosch T (1997) Modelling extremal events for insurance and finance. Springer, Heidelberg
Falk M, Hüsler J, Reiss R-D (2010) Laws of small numbers, 3rd edn. Birkhaüser, Basel
Galambos J (1978) The asymptotic theory of extreme order statistics, 2nd edn. Wiley, Hoboken
Gwak W, Goo YH, Ahn JY (2016) Extreme value theory in mixture distributions and a statistical method to control the possible bias. J Korean Stat Soc 45:581–594
Kale BK, Sebastian G (1995) On a class of symmetric non-normal distributions with a kurtosis of three. In: Nagaraja HN et al (eds) Statistical theory and applications, in honour of David. H.A. Springer, New York, pp 55–63
Mohan NR, Ravi S (1993) Max domains of attraction of univariate and multivariate p-max stable laws. Theory Probab Appl 37(4):632–643 (Translated from Russian Journal)
Pancheva E (1984) Limit theorems for extreme order statistics under non-linear normalization. Lecture notes in mathematics, vol 1155. Springer, Berlin, pp 284–309
Ravi S, Praveena AS (2010) A note on tail behaviour of distributions in the max domain of attraction of the Fréchet/Weibull law under power normalization. ProbStat Forum 03:01–10
Resnick SI (1987) Extreme values, regular variation, and point processes. Springer, Berlin
Sreehari M, Ravi S (2010) On extremes of mixtures of distributions. Metrika 71:117–123
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Research work of the first author supported by post-doctoral fellowship for women of the University Grants Commission (UGC), New Delhi, vide Ref: No.F.15-1/2011-12/PDFWM-2011-12-GE-KAR-2333(SA-II) dated 01 Nov 2013. Research work of the second and the third authors supported by UGC Major Research Project F.No.43-541/2014(SR) dated 16.10.2015 with the second author as project fellow and the third author as the principal investigator.
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Praveena, A.S., Srinath, S. & Ravi, S. On Mixture Distributions and Their Max Domains. J Indian Soc Probab Stat 20, 173–183 (2019). https://doi.org/10.1007/s41096-019-00063-w
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DOI: https://doi.org/10.1007/s41096-019-00063-w
Keywords
- Extreme value distribution
- l-Max stable law
- p-Max stable law
- Max domains of attraction
- Mixture distribution