1、Ch1. Loss Size Distribution,General sample distributions Distribution classify Parametric distributions Scale distributions Mixture distributions Data-dependent distribution Creating new distributions Extreme value distribution,Ch1.1 General Sample Distributions,Distribution function and its charact

2、eristic Normal Distribution (Lognormal) Exponential Distribution (Inverse Exponential) Gamma Distribution Weibull Distribution Pareto Distribution Burr Distribution,Ch1.1 General Sample Distributions,Ch1.2 Distribution Classify,Parametric distribution Parametric distribution family Scale distributio

3、n Mixture distribution Data-dependent distribution,Ch1.2.1 Parametric Distribution,Parametric distribution A set of distribution with the same type of distribution function but with different value for parameters. finite and fix number of parameter Example,Ch1.2.2 Parametric Distribution Family,Para

4、metric distribution family A set of parametric distributions that are related in some meaningful way. Parametric dis. Parametric dis. family Specify some parameters in parametric dis. family can make parameter dis. Example,Ch1.2.3 Scale Distribution,Background Loss modified by inflation index Loss v

5、alued 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,Ch1.2.3 Scale Distribution,Example: Exponential dis. is a scale dis. Note: p

6、arameter 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 multiplied by the same constant the other parameters unchanged Example parameter in is a scale paramet

7、er,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.,Ch1.2.4 Mixture Distribution,Background Traffic accident caused by: Drive after drinking Brake fail Bad weather . Loss caused by different reasons

8、 follows different distribution rule Traffic accident loss distribution can be a mixture of some different distributions,Ch1.2.4 Mixture Distribution,K-point mixture A random variable with its cdf given by cdfs of k random variables as Function: new distribution generating, given 20 random variables

9、, we can have more than 200New random variables by 2-point mixture.,Ch1.2.4 Mixture Distribution,Claim in Liability insurance Mixed with 2 Pareto dis. Then the cdf of claim in liability insurance Bimodal distribution 50%-50% mixture of Gamma distribution,Ch1.2.4 Mixture Distribution,Variable compone

10、nt mixture distribution Variable component mixture distribution is semi-parameter model Example: Variable mixture of Exponential dis.,Ch1.2.5 Data-dependent Distribution,Parameter model: given dis. with finite and fixed number of unknown parameters Semi-parameter model: given dis. but number of para

11、meters is not fixed Non-parameter model: unknown dis. Data-dependent Distribution A distribution determined by sample data, and the number of parameters increases with the sample data or amount of knowledge,Ch1.2.5 Data-dependent Distribution,Example Empirical model: Assign probability 1/n to each d

12、ata sample evenly Kernel smoothing model: Based on empirical model, assign probability 1/n to an area of each data sample kernel function satisfies,Ch1.2.5 Data-dependent Distribution,Note: Empirical model and kernel smoothing model can be written as mixture distribution 1 3 5 7 9 11 13 Mixture of u

13、niform distribution ? number of components phenomenon/r.v.s number of components sample size,Ch1.3 Creating New Dis.,Multiplication by a constant Raising to a power Exponentiation Mixing Frailty models Splicing,Ch1.3.1 Multiplication by constant and Raising to a power,Multiplication by constant Rais

14、ing to a power Inverse Transformed Inverse transormed,Ch1.3.2 Raising to a power,Example Inverse Inverse exponential dis. Transformed Weibull distribution Inverse transformed inverse Weibull dis.,Ch1.3.3 Exponentiation,Exponentiation Example,Ch1.3.4 Mixing,Mixing: parameters of parameter dis. change

15、d to random variables, i.e. for random variable , when random variable then pdf of X Example:,Ch1.3.4 Mixing,In Valuation of warranties on automobiles Number of miles varies from driver to driver and from year to year distribution for number of miles of a selected driver in a selected year? For a pa

16、rticular driver, number of miles has the inverse Weibull dis. For different year, the scale parameter of the inverse Weibull dis. has the transformed Gamma distribution. Inverse Burr,Ch1.3.5 Splicing,Background: for different interval of loss amount, loss follows different dis. K-component Splicing

17、dis.: a dis. has a density function can be expressed as follows Example: Exp(q ) and Pareto(a,g ) spliced model,Exercise,A random sample of observations is taken from a shifted exponential distribution with probability density function: the sample mean and median are 300 and 240, estimate the shift. You are given the distortion function: Calculate the distortion risk measure for losses that follow the Pareto distribution with q = 1000 and a = 4.,Exercise,You are given th