The shifted Pareto distribution , also known as the Lomax distribution , is a probability distribution considered in mathematical statistics , which is particularly suitable for modeling major losses, especially in industrial and reinsurance. Mathematically, this is a Pareto distribution , the distribution curve of which is shifted by a fixed parameter value , from which the name of this distribution is derived.
definition A continuous random variable is sufficient for the shifted Pareto distribution with the parameters and , if it is the probability density
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{\ displaystyle \ operatorname {Par ^ {\ star}} (a, b)}
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{\ displaystyle f (x) = {\ begin {cases} ab (1 + bx) ^ {- (a + 1)} & x \ geq 0 \\ 0 & x <0. \ end {cases}}}
owns. Here is a scale parameter of the distribution.
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{\ displaystyle {\ frac {1} {b}}}
properties Distribution function The distribution function is given by
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{\ displaystyle F (x) = P (X \ leq x) = 1- (1 + bx) ^ {- a}}
In particular thus for the survival function : .
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{\ displaystyle P (X> x) = 1-F (x) = (1 + bx) ^ {- a}}
Expected value The expected value results from:
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{\ displaystyle \ operatorname {E} (X) = {\ frac {1} {b (a-1)}}.}
Variance The variance can be given as
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{\ displaystyle \ operatorname {Var} (X) = {\ frac {1} {b ^ {2}}} \ left ({\ frac {a} {a-2}} - {\ frac {a ^ {2 }} {(a-1) ^ {2}}} \ right) = {\ frac {a} {b ^ {2} (a-1) ^ {2} (a-2)}}.}
Standard deviation The standard deviation results from the expected value and the variance
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{\ displaystyle \ sigma (X) = {\ sqrt {{\ frac {1} {b ^ {2}}} \ left ({\ frac {a} {a-2}} - {\ frac {a ^ { 2}} {(a-1) ^ {2}}} \ right)}} = {\ frac {1} {b (a-1)}} {\ sqrt {\ frac {a} {a-2} }}.}
Coefficient of variation The coefficient of variation is obtained from the expected value and the variance
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{\ displaystyle \ operatorname {VarK} (X) = {\ sqrt {\ frac {a} {a-2}}}.}
Crookedness For the skew results
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{\ displaystyle \ operatorname {v} (X) = {\ frac {\ displaystyle {\ frac {a} {a-3}} - 3 {\ frac {a ^ {2}} {(a-2) (a -1)}} + 2 {\ frac {a ^ {3}} {(a-1) ^ {3}}}} {\ displaystyle \ left ({\ frac {a} {a-2}} - { \ frac {a ^ {2}} {(a-1) ^ {2}}} \ right) ^ {\ frac {3} {2}}}}.}
Characteristic function The characteristic function cannot be specified in closed form for the shifted Pareto distribution.
Moment generating function The torque-generating function cannot be specified in closed form for the shifted Pareto distribution.
literature Klaus Jürgen Schröter: Procedure for the approximation of the total damage distribution: systematization, techniques and comparisons. Volume 1 of the Karlsruher series, contributions to insurance science, Verlag Versicherungswirtsch., 1995, ISBN 978-3-88487-471-4 , p. 35. Individual evidence
↑ Christian Hipp: Risk Theory 1 (script) ( page no longer available , search in web archives ) Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. , P 179, accessed June 17, 2011@1 @ 2 Template: Dead Link / insurance.fbv.kit.edu
Discrete univariate distributions
Continuous univariate distributions
Multivariate distributions
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