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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>
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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.
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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>
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*<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 ''&sigma;''&nbsp;>&nbsp;0, and ''y''&nbsp;≥&nbsp;0 when ''k''&nbsp;≥&nbsp;0 and 0&nbsp;≤&nbsp;''y''&nbsp;≤&nbsp;&minus;''&sigma;''/''k'' when ''k''&nbsp;<&nbsp;0.
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==Special cases of generalized Pareto distribution==
* [[Exponential distribution]] with [[expected value|mean]] <math>\sigma</math>,  if ''k''&nbsp;=&nbsp;0.
* [[Uniform distribution (continuous)|Uniform distribution]] on <math>[0,\sigma]</math>,  if k = 1.
* [[Pareto distribution]], if ''k''&nbsp;<&nbsp;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]]

Latest revision as of 17:19, 27 December 2014


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