非寿险精算LossSizeDistribution

? Specify some parameters in parametric dis. family can make parameter dis.
Example
? ?1 Loglogistic(? ,? )
Burr(? ,? ,?)
? ??
Paralogistic (? ,? )
Ch1.2.3 Scale Distribution
Background Traffic accident caused by: Drive after drinking Brake fail Bad weather ......
? Loss caused by different reasons follows different distribution rule
非寿险精算 Loss Size Distribution
Ch1.1 General Sample Distributions
Distribution function and its characteristic ? Normal Distribution (Lognormal) ? Exponential Distribution (Inverse Exponential) ? Gamma Distribution ? Weibull Distribution ? Pareto Distribution ? Burr Distribution
? Traffic accident loss distribution can be a mixture of some different distributions
? ? 1 In-Pareto (? ,? )
Tr ? Beta(? ,? ,? ,? ) ? ? 1 Burr(? ,? ,? ) ? ? 1
? ??
? ?1
? ?1 ? ?1
In ? Burr(? ,?,? )
???
Loglogistic (? ,? ) Paralogistic (? ,? ) In-Paralogistic (? ,? )
Ch1.2.3 Scale Distribution
Example: Exponential dis. is a scale dis. Let X : Exp(? ) ~ F (x) ? 1? exp(?x / ? ) ? ? c ? 0, Y ? cX : Exp(c? )
FY (y) ? Pr(Y ? y) ? Pr(X ? y / c) ? FX (y / c) ? 1? exp(? (y / c) / ? ) ? 1 ? exp(?y / (c? ))
? multiplied by the same constant ? the other parameters unchanged Example
parameter ? in Exp(? ) is a scale parameter
Ch1.2.3 Scale Distribution
Demonstrate a parametric dis. is a scale dis.
and check out its scale parameter
? Gamma dis. is a scale dis. with scale para. ?
Let X : Gamma(? ,? ) ~ FX (x) ? Γ(? ;x / ? )
? c ? 0, Y ? cX
FY (y) ? FX (y / c) ? Γ(? ;(y / c) / ? ) ? Γ(? ;y / (c? )) ~ Gamma(? ,c? )
? Y ~ Exp(c? )
Note:
parameter ? is multiplied by a constant c
Ch1.2.3 Scale Distribution
Scale parameter A parameter for a scale distribution that meets two conditions when the random variable is multiplied by a constant
Ch1.2.2 Parametric Distribution Family
Parametric distribution family
? A set of parametric distributions that are related in someis. ＜ Parametric dis. family
Background
? Loss modified by inflation index ? Loss valued by different measures
＝＞ Loss distribution remains the same Scale distribution
? A parametric distribution in which each random variable is multiplied by a positive constant and remains in the same parametric distributions
Ch1.1 General Sample Distributions
? ?1
Tr ? Gamma (? ,? ,? )
? ?1
Gamma (? ,? ) Weibull (? ,? )
? ?1
Exp(? )
? ?1
? ? 1 Pareto (? ,? )
Ge ? Pareto(? ,? ,? )
? ?1
X
: Gamma(?
,? )
~
fX
(x)
?
x e ? ?1 ? x/? ? ? Γ(? )
,?
c
?
0,Y
? cX
fY
(y) ?
fX
(y c
/
c)
?
(y / c)? e ?1 ? (y/c)/? c? ? Γ(? )
?
y e ? ?1 ? y/(c?
?c? ?? Γ(?
)
)
~
Gamma(?
,c?
)
Ch1.2.4 Mixture Distribution