Generalized Pareto distribution
In statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions. It is often used to model the tails of another distribution. It is specified by three parameters: location formula_7, scale formula_8, and shape formula_9. Sometimes it is specified by only scale and shape and sometimes only by its shape parameter. Some references give the shape parameter as formula_10. Definition. The standard cumulative distribution function (cdf) of the GPD is defined by where the support is formula_12 for formula_13 and formula_14 for formula_15. The corresponding probability density function (pdf) is Characterization. The related location -scale family of distributions is obtained by replacing the argument "z" by formula_17 and adjusting the support accordingly. The cumulative distribution function of formula_18 (formula_19, formula_20, and formula_21) is where the support of formula_23 is formula_24 when formula_25, and formula_26 when formula_15. The probability density function (pdf) of formula_18 is again, for formula_24 when formula_31, and formula_26 when formula_15. The pdf is a solution of the following differential equation: Generating generalized Pareto random variables. Generating GPD random variables. If "U" is uniformly distributed on (0, 1<nowiki>]</nowiki>, then and Both formulas are obtained by inversion of the cdf . In Matlab Statistics Toolbox, you can easily use "gprnd" command to generate generalized Pareto random numbers. GPD as an Exponential-Gamma Mixture. A GPD random variable can also be expressed as an exponential random variable, with a Gamma distributed rate parameter. and then Notice however, that since the parameters for the Gamma distribution must be greater than zero, we obtain the additional restrictions that:formula_9 must be positive. Exponentiated generalized Pareto distribution. The exponentiated generalized Pareto distribution (exGPD). If formula_56 formula_44formula_45, formula_8, formula_9 formula_48, then formula_62 is distributed according to the exponentiated generalized Pareto distribution, denoted by formula_63 formula_42 formula_65  formula_44formula_8, formula_9 formula_48. The probability density function(pdf) of formula_70 formula_42 formula_65 formula_44formula_8, formula_9 formula_76 is where the support is formula_78 for formula_79, and formula_80 for formula_81. For all formula_9, the formula_83 becomes the location parameter. See the right panel for the pdf when the shape formula_9 is positive. The exGPD has finite moments of all orders for all formula_20 and formula_86. The moment-generating function of formula_87 is where formula_89 and formula_90 denote the beta function and gamma function, respectively. The expected value of formula_70 formula_42 formula_65 formula_44formula_8, formula_9 formula_48 depends on the scale formula_98 and shape formula_99 parameters, while  the formula_99 participates through the digamma function: Note that for a fixed value for the formula_102, the formula_103 plays as the location parameter under the exponentiated generalized Pareto distribution. The variance of formula_70 formula_42 formula_65 formula_44formula_8, formula_9 formula_48 depends on the shape parameter formula_99 only through the polygamma function of order 1 (also called the trigamma function): See the right panel for the variance as a function of formula_9. Note that formula_114. Note that the roles of the scale parameter formula_8 and the shape parameter formula_9 under formula_117 are separably interpretable, which may lead to a robust efficient estimation for  the formula_9 than using the formula_119 . The roles of the two parameters are associated each other under formula_120 (at least up to the second central moment); see the formula of variance formula_121 wherein both parameters are participated. The Hill's estimator. Assume that formula_122 are formula_123 observations (not need to be i.i.d.) from an unknown heavy-tailed distribution formula_124 such that its tail distribution is regularly varying with the tail-index formula_125 (hence, the corresponding shape parameter is formula_126). To be specific, the tail distribution is described as It is of a particular interest in the extreme value theory to  estimate the shape parameter formula_9, especially when formula_9 is positive (so called the heavy-tailed distribution). Let formula_130 be their conditional excess distribution function. Pickands–Balkema–de Haan theorem (Pickands, 1975; Balkema and de Haan, 1974) states that for a large class of underlying distribution functions formula_131, and large formula_132, formula_130 is well approximated by the generalized Pareto distribution (GPD), which motivated Peak Over Threshold (POT) methods to estimate formula_9: the GPD plays the key role in POT approach. A renowned estimator using the POT methodology is the Hill's estimator. Technical formulation of the Hill's estimator is as  follows. For formula_135, write formula_136 for the formula_137-th largest value of formula_138. Then, with this notation, the Hill's estimator (see page 190 of Reference 5 by Embrechts et al ) based on the formula_139 upper order statistics is defined as In practice, the Hill estimator is used as follows. First, calculate the estimator formula_141 at each integer formula_142, and then plot the ordered pairs formula_143. Then, select from the set of Hill estimators formula_144 which are roughly constant with respect to formula_139: these stable values are regarded as reasonable estimates for the shape parameter formula_9. If formula_138 are i.i.d ., then the Hill's estimator is a consistent estimator for the shape parameter formula_9 . Note that the Hill estimator formula_141 makes a use of the log-transformation for the observations formula_122. (The Pickand's estimator formula_151 also employed the log-transformation, but in a slightly different way