# Talk:Generalized Pareto distribution

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## Special cases

It would be very useful with a table of various special cases of the generalized Pareto distribution. I have collected some information from other articles, but I have no reference to compare it with. Can someone help out here? Isheden (talk) 19:56, 19 December 2011 (UTC)

Distribution Parameters location μ (θ) shape ξ (k) scale σ
Exponential distribution λ 0 0 $\tfrac{1}{\lambda}$
Pareto distribution xm, α xm $\tfrac{1}{\alpha}$ $\tfrac{x_m}{\alpha}$
Lomax distribution λ, α 0 $\tfrac{1}{\alpha}$ $\tfrac{\lambda}{\alpha}$
q-exponential distribution q, λ 0 $\tfrac{q-1}{2-q}$ $\tfrac{1}{\lambda (2-q)}$
Bounded Pareto distribution L, H, α L  ?  ?
This table is excellent. As for a reference, it will be a challenge to find. I created the q-exponential and lomax pages and did out the math for the mappings between distributions. I have not seen this published, but I think it counts as a Simple Derivation. Purple Post-its (talk) 16:21, 16 March 2012 (UTC)
Before inserting a table like this however, I think it is necessary to clarify from what reference the properties of the distribution on this page were taken in the first place. The original distribution suggested by Pickands only had two parameters, and at some later point the location parameter seems to have been added. The relationship to Pareto IV and/or to Feller-Pareto should also be clarified. Since I don't have access to all the references cited, I don't know how to sort these things out. Isheden (talk) 16:42, 16 March 2012 (UTC)

## CDF

The CDF investigated by Hosking and Wallis (1987) and others is given in Continuous Univariate Distributions Volume I, p. 614, equation (20.153) as

$F_X (x; c, k) = 1 - (1 - cx/k)^{1/c}.$

On the same page (not entirely clear) that this formulation was originally due to Pickands (1975). Special cases c=0 and c=1 correspond to exponential and uniform distributions, respectively. Negative values of c correspond to Pareto distributions. The pdf is given on page 615 as

$p_X(x) = k^{-1}(1-cx/k)^{c^{-1}-1}, \qquad (c \neq 0),$
$p_X(x) = k^{-1}\exp(-x/k), \qquad (c = 0).$

The support is x>0 for c ≤ 0 an 0 < x < k/c for c > 0. Mathstat (talk) 18:18, 17 March 2012 (UTC)

So it seems almost all of the references refer to a distribution with two parameters. The question is where the location parameter μ in the article comes from? Isheden (talk) 20:38, 17 March 2012 (UTC)
Yes, exactly. The article begins "The location-scale family of generalized Pareto distributions (GPD) has three parameters $\mu,\sigma \,$ and $\xi \,$" [1][2][3] (three references given). Having Pickands (1975) and Hosking and Wallis (1987) PDF's at hand, it is clear that neither of them define a distribution with three parameters. Citing these references to support the first statement is incorrect. Mathstat (talk) 22:28, 17 March 2012 (UTC)
I found a reference that defines the standard GPD with only the shape parameter $\xi$. It mentions that we can introduce a location-scale family GPD by replacing x by $\tfrac{x-\mu}{\sigma}$. I inserted this reference to support the cdf in the article. Isheden (talk) 22:37, 17 March 2012 (UTC)
This does not seem to follow the majority of the literature. The article should be expanded to indicate what are the definitions that have been used the most in the literature of the past 30 years or so, and just for clarity for anyone who is trying to use the article as a reference. More than likely they are looking for the definitions we easily found earlier. This is one good example for my students why one should always check references and facts. Mathstat (talk) 22:47, 17 March 2012 (UTC)
Agreed. Isheden (talk) 08:00, 18 March 2012 (UTC)

## Really minor quibble z-x

Could somebody who has proper references (or a strong opinion) make the symbol for the independent variable either x or z? Rrogers314 (talk) 15:40, 23 September 2013 (UTC)

Which one would you prefer? The variable substitution is $z=\frac{x-\mu}{\sigma}$, so with z the parameters μ and σ are not contained explicitly in the formulas. Isheden (talk) 18:25, 23 September 2013 (UTC)