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The '''Pickands–Balkema–de Haan theorem''' is often called the second theorem in [[extreme value theory]]. It gives the asymptotic [[tail distribution]] of a [[random variable]] ''X'', when the true distribution ''F'' of ''X'' is unknown. Unlike the first theorem (the [[Fisher–Tippett–Gnedenko theorem]]) in extreme value theory, the interest here is the values above a threshold. | |||
==Conditional excess distribution function== | |||
If we consider an unknown distribution function <math>F</math> of a random variable <math>X</math>, we are interested in estimating the conditional distribution function <math>F_u</math> of the variable <math>X</math> above a certain threshold <math>u</math>. This is the so-called conditional excess distribution function, defined as | |||
: <math>F_u(y) = P(X-u \leq y | X>u) = \frac{F(u+y)-F(u)}{1-F(u)} \, </math> | |||
for <math>0 \leq y \leq x_F-u</math>, where <math>x_F</math> is either the finite or infinite right endpoint of the underlying distribution <math>F</math>. The function <math>F_u</math> describes the distribution of the excess value over a threshold <math>u</math>, given that the threshold is exceeded. | |||
==Statement== | |||
Let <math>(X_1,X_2,\ldots)</math> be a sequence of [[independent and identically-distributed random variables]], and let <math>F_u</math> be their conditional excess distribution function. Pickands (1975), Balkema and de Haan (1974) posed that for a large class of underlying distribution functions <math>F</math>, and large <math>u</math>, <math>F_u</math> is well approximated by the [[generalized Pareto distribution]]. That is: | |||
: <math>F_u(y) \rightarrow G_{k, \sigma} (y),\text{ as }u \rightarrow \infty</math> | |||
where | |||
*<math>G_{k, \sigma} (y)= 1-(1+ky/\sigma)^{-1/k} </math>, if <math>k \neq 0</math> | |||
*<math>G_{k, \sigma} (y)= 1-e^{-y/\sigma} </math>, if <math>k = 0.</math> | |||
Here ''σ'' > 0, and ''y'' ≥ 0 when ''k'' ≥ 0 and 0 ≤ ''y'' ≤ −''σ''/''k'' when ''k'' < 0. | |||
==Special cases of generalized Pareto distribution== | |||
* [[Exponential distribution]] with [[expected value|mean]] <math>\sigma</math>, if ''k'' = 0. | |||
* [[Uniform distribution (continuous)|Uniform distribution]] on <math>[0,\sigma]</math>, if k = 1. | |||
* [[Pareto distribution]], if ''k'' < 0. | |||
{{primary sources|date=July 2012}} | |||
==References== | |||
* Balkema, A., and [[Laurens de Haan|de Haan, L.]] (1974). "Residual life time at great age", ''Annals of Probability'', '''2''', 792–804. | |||
* Pickands, J. (1975). "Statistical inference using extreme order statistics", ''Annals of Statistics'', '''3''', 119–131. | |||
{{DEFAULTSORT:Pickands-Balkema-de Haan theorem}} | |||
[[Category:Probability theorems]] | |||
[[Category:Extreme value data]] | |||
[[Category:Tails of probability distributions]] |
Revision as of 20:12, 19 October 2013
The Pickands–Balkema–de Haan theorem is often called the second theorem in extreme value theory. It gives the asymptotic tail distribution of a random variable X, when the true distribution F of X is unknown. Unlike the first theorem (the Fisher–Tippett–Gnedenko theorem) in extreme value theory, the interest here is the values above a threshold.
Conditional excess distribution function
If we consider an unknown distribution function of a random variable , we are interested in estimating the conditional distribution function of the variable above a certain threshold . This is the so-called conditional excess distribution function, defined as
for , where is either the finite or infinite right endpoint of the underlying distribution . The function describes the distribution of the excess value over a threshold , given that the threshold is exceeded.
Statement
Let be a sequence of independent and identically-distributed random variables, and let be their conditional excess distribution function. Pickands (1975), Balkema and de Haan (1974) posed that for a large class of underlying distribution functions , and large , is well approximated by the generalized Pareto distribution. That is:
where
Here σ > 0, and y ≥ 0 when k ≥ 0 and 0 ≤ y ≤ −σ/k when k < 0.
Special cases of generalized Pareto distribution
- Exponential distribution with mean , if k = 0.
- Uniform distribution on , if k = 1.
- Pareto distribution, if k < 0.
References
- Balkema, A., and de Haan, L. (1974). "Residual life time at great age", Annals of Probability, 2, 792–804.
- Pickands, J. (1975). "Statistical inference using extreme order statistics", Annals of Statistics, 3, 119–131.