In a satisfying way, it turns out that there are only so many shapes that probability densities can assume as they head off towards infinity. Extreme value theory makes this notion precise and gives us some tools to work with them. Important application: understanding the kinds of heavy tailed variables we can observe in nature.
See also densities and intensities, survival analysis.
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Tail limit theorems
The main result of use to our ends from EVT is the Pickands-Balkema-de Haan theorem (Balkema and de Haan 1974; Pickands III 1975).
This tells us that we can find a function such that if (and only if) is in the maximal domain of attraction of the extreme value distribution with parameter for some .
This maximal domain of attraction was introduced in the Fisher-Tippett theorem (Fisher and Tippett 1928), and is analysed in the EVT literature (e.g. Embrechts, Kluppelberg, and Mikosch 1997). It is pretty hard to find a distribution that does not fit in the MDA. I should try.
Practically, this means that for many purposes, the tails of a random variable may as well be assumed to be a GPD, Then for and assuming that the survival probability over an interval is
Generalized Pareto Distribution
Best intro from Hosking and Wallis (1987):
The generalized Pareto distribution is the distribution of a random variable defined by where is a random variable with the standard exponential distribution. The generalized Pareto distribution has distribution function
and density function
the range of is for and for The parameters of the distribution are the scale parameter, and the shape parameter. The special cases and yield, respectively, the exponential distribution with mean and the uniform distribution on Pareto distributions are obtained when
and
- The failure rate is given by and is monotonic in decreasing if constant if and increasing if
- If the random variable has a generalized Pareto distribution, then the conditional distribution of given is also generalized Pareto, with the same value of
- Let where the are independent and identically distributed as (1) and has a Poisson distribution. Then has, essentially, a generalized extreme value (GEV) distribution as defined by Jenkinson (1955); that is, there exist quantities and independent of such that
furthermore, that is, the shape parameters of the GEV and the GPD are equal.
Generalized Extreme Value distributions
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References
Balkema, and de Haan. 1974.
“Residual Life Time at Great Age.” The Annals of Probability.
Castillo, and Hadi. 1997.
“Fitting the Generalized Pareto Distribution to Data.” Journal of the American Statistical Association.
Charpentier, and Flachaire. 2019.
“Pareto Models for Risk Management.” arXiv:1912.11736 [Econ, Stat].
Dargahi-Noubary. 1989.
“On Tail Estimation: An Improved Method.” Mathematical Geology.
Davison. 1984.
“Modelling Excesses over High Thresholds, with an Application.” In
Statistical Extremes and Applications. NATO ASI Series.
Embrechts, Kluppelberg, and Mikosch. 1997. Extremal Events in Finance and Insurance.
Embrechts, Klüppelberg, and Mikosch. 1997.
“Risk Theory.” In
Modelling Extremal Events. Applications of Mathematics 33.
Fisher, and Tippett. 1928.
“Limiting Forms of the Frequency Distribution of the Largest or Smallest Member of a Sample.” Mathematical Proceedings of the Cambridge Philosophical Society.
Ghitany, Gómez-Déniz, and Nadarajah. 2018.
“A New Generalization of the Pareto Distribution and Its Application to Insurance Data.” Journal of Risk and Financial Management.
Giesbrecht, and Kempthorne. 1976.
“Maximum Likelihood Estimation in the Three-Parameter Lognormal Distribution.” Journal of the Royal Statistical Society: Series B (Methodological).
Makarov. 2006.
“Extreme Value Theory and High Quantile Convergence.” The Journal of Operational Risk.
McNeil, Alexander J, Frey, and Embrechts. 2005. Quantitative Risk Management : Concepts, Techniques and Tools.
Naveau, Hannart, and Ribes. 2020.
“Statistical Methods for Extreme Event Attribution in Climate Science.” Annual Review of Statistics and Its Application.
Nolde, and Zhou. 2021.
“Extreme Value Analysis for Financial Risk Management.” Annual Review of Statistics and Its Application.
Vajda. 1951.
“Analytical Studies in Stop-Loss Reinsurance.” Scandinavian Actuarial Journal.
Wong, and Li. 2006.
“A Note on the Estimation of Extreme Value Distributions Using Maximum Product of Spacings.” In
Institute of Mathematical Statistics Lecture Notes - Monograph Series.