Statistical Modeling of Loss Vincent Goulet Distributions Using - PowerPoint PPT Presentation

Statistical Modeling of Loss Distributions Using actuar Statistical Modeling of Loss Vincent Goulet Distributions Using actuar Probability Laws Grouped Data Vincent Goulet Minimum Distance Estimation École d’actuariat, Université Laval Censored Québec, Canada Data

actuar Statistical Modeling of Loss Distributions Using actuar Vincent Provides additional Actuarial Science Goulet functionality to R Probability Laws Current version covers Grouped Loss distribution modeling Data Risk theory (including ruin theory) Minimum Distance Simulation of compound hierarchical models Estimation Credibility theory Censored Data

Summary Statistical Modeling of Loss Distributions Using actuar Probability Laws 1 Vincent Goulet Probability Grouped Data 2 Laws Grouped Data Minimum Minimum Distance Estimation 3 Distance Estimation Censored Data Censored Data 4

Summary Statistical Modeling of Loss Distributions Using actuar Probability Laws 1 Vincent Goulet Grouped Data 2 Probability Laws Grouped Data Minimum Distance Estimation 3 Minimum Distance Estimation Censored Data 4 Censored Data

At a Glance Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Support for 18 probability laws not in base R Probability Laws Mostly positive, heavy tail distributions Grouped New utility functions in addition to d foo , p foo , Data q foo , r foo Minimum Distance Estimation Censored Data

Supported Distributions Statistical Modeling of Loss Distributions Using actuar Transformed Beta Family Vincent 9 special cases (including Burr and Pareto) Goulet Transformed Gamma Family Probability Laws 5 special cases (including inverse distributions) Grouped Loggamma Data Minimum Single parameter Pareto Distance Estimation Generalized Beta Censored Data Phase-type distributions

New Utility Functions Statistical m foo to compute theoretical raw moments Modeling of Loss Distributions Using m k = E [ X k ] actuar Vincent Goulet lev foo to compute theoretical limited moments Probability Laws E [( X ∧  ) k ] = E [ min ( X,  ) k ] Grouped Data mgf foo to compute the moment generating Minimum Distance function Estimation M X ( t ) = E [ e tX ] Censored Data when it exists Also support for: beta, exponential, chi-square, gamma, lognormal, normal (no lev ), uniform, Weibull, inverse Gaussian

New Utility Functions Statistical m foo to compute theoretical raw moments Modeling of Loss Distributions Using m k = E [ X k ] actuar Vincent Goulet lev foo to compute theoretical limited moments Probability Laws E [( X ∧  ) k ] = E [ min ( X,  ) k ] Grouped Data mgf foo to compute the moment generating Minimum Distance function Estimation M X ( t ) = E [ e tX ] Censored Data when it exists Also support for: beta, exponential, chi-square, gamma, lognormal, normal (no lev ), uniform, Weibull, inverse Gaussian

Summary Statistical Modeling of Loss Distributions Using actuar Probability Laws 1 Vincent Goulet Grouped Data 2 Probability Laws Grouped Data Minimum Distance Estimation 3 Minimum Distance Estimation Censored Data 4 Censored Data

Definition and Rationale Statistical Modeling of Loss Distributions Using actuar Data presented in an interval-frequency manner: Vincent Goulet Group Line 1 Line 2 Probability Laws ( 0 , 25 ] 30 26 Grouped ( 25 , 50 ] 31 33 Data ( 50 , 100 ] 57 31 Minimum Distance Estimation Need for a “standard” storage method Censored Useful for minimum distance estimation Data

Creation and Manipulation of Objects Statistical Modeling of Loss Distributions Using actuar > x <- grouped.data(Group = c(0, 25, Vincent Goulet + 50, 100), Line.1 = c(30, 31, 57), + Line.2 = c(26, 33, 31)) Probability Laws > x Grouped Data Group Line.1 Line.2 Minimum 1 (0, 25] 30 26 Distance Estimation 2 (25, 50] 31 33 Censored Data 3 (50, 100] 57 31

Calculation of Empirical Moments Statistical Modeling of Loss Distributions Using > mean(x) actuar Vincent Line.1 Line.2 Goulet 49.25847 43.19444 Probability Laws > emm(x, 2) Grouped Data Line.1 Line.2 Minimum 3253.884 2604.167 Distance Estimation > E <- elev(x[, -3]) Censored Data > E(c(25, 50)) [1] 21.82203 37.18220

Plot of the Histogram and Ogive Statistical Modeling of Loss Distributions Using > hist(x[, -3]) > plot(ogive(x[, -3])) actuar Vincent Goulet Histogram of x[, −3] ogive(x[, −3]) Probability 1.0 ● Laws 0.008 0.8 Grouped 0.6 Density Data F(x) ● 0.004 0.4 Minimum ● 0.2 Distance 0.000 Estimation 0.0 ● Censored 0 20 40 60 80 100 0 20 40 60 80 100 Data x[, −3] x

Summary Statistical Modeling of Loss Distributions Using actuar Probability Laws 1 Vincent Goulet Grouped Data 2 Probability Laws Grouped Data Minimum Distance Estimation 3 Minimum Distance Estimation Censored Data 4 Censored Data

mde() Supports Three Distance Measures Statistical 1 Cramér-von Mises Modeling of Loss Distributions n Using  j [ F (  j ; θ ) − F n (  j ; θ )] 2 � actuar d ( θ ) = Vincent j = 1 Goulet 2 Modified chi-square Probability Laws r Grouped  j [ n ( F ( c j ; θ ) − F ( c j − 1 ; θ )) − n j ] 2 , Data � d ( θ ) = Minimum j = 1 Distance Estimation 3 Layer average severity Censored Data r � LAS n ( c j − 1 , c j ; θ )] 2 , ˜ d ( θ ) =  j [ LAS ( c j − 1 , c j ; θ ) − j = 1 where LAS ( , y ) = E [ min ( X, y )] − E [ min ( X,  )]

mde() Supports Three Distance Measures Statistical 1 Cramér-von Mises Modeling of Loss Distributions n Using  j [ F (  j ; θ ) − F n (  j ; θ )] 2 � actuar d ( θ ) = Vincent j = 1 Goulet 2 Modified chi-square Probability Laws r Grouped  j [ n ( F ( c j ; θ ) − F ( c j − 1 ; θ )) − n j ] 2 , Data � d ( θ ) = Minimum j = 1 Distance Estimation 3 Layer average severity Censored Data r � LAS n ( c j − 1 , c j ; θ )] 2 , ˜ d ( θ ) =  j [ LAS ( c j − 1 , c j ; θ ) − j = 1 where LAS ( , y ) = E [ min ( X, y )] − E [ min ( X,  )]

mde() Supports Three Distance Measures Statistical 1 Cramér-von Mises Modeling of Loss Distributions n Using  j [ F (  j ; θ ) − F n (  j ; θ )] 2 � actuar d ( θ ) = Vincent j = 1 Goulet 2 Modified chi-square Probability Laws r Grouped  j [ n ( F ( c j ; θ ) − F ( c j − 1 ; θ )) − n j ] 2 , Data � d ( θ ) = Minimum j = 1 Distance Estimation 3 Layer average severity Censored Data r � LAS n ( c j − 1 , c j ; θ )] 2 , ˜ d ( θ ) =  j [ LAS ( c j − 1 , c j ; θ ) − j = 1 where LAS ( , y ) = E [ min ( X, y )] − E [ min ( X,  )]

Summary Statistical Modeling of Loss Distributions Using actuar Probability Laws 1 Vincent Goulet Grouped Data 2 Probability Laws Grouped Data Minimum Distance Estimation 3 Minimum Distance Estimation Censored Data 4 Censored Data

Context Statistical Modeling of Common in statistical and actuarial applications Loss Distributions to work with censored data Using actuar Actuarial terminology: Vincent Goulet left censoring (ordinary) deductible ⇔ right censoring policy limit Probability ⇔ Laws Grouped Left Censoring Right Censoring Data 0.12 Minimum 0.10 0.12 Distance Estimation 0.08 ● Censored 0.08 0.06 Data 0.04 0.04 0.02 0.00 0.00 0 5 10 15 0 5 10 15

A Different Approach Statistical Modeling of Loss Distributions Using actuar Package survival has extensive support for Vincent Goulet censored distributions Probability Our approach is different Laws Grouped coverage() returns pdf or cdf of censored Data distribution (with many options) Minimum Distance function can be used in fitting as usual Estimation ( fitdistr() , mde() , ...) Censored Data

Example With Left and Right Censoring Statistical Modeling of Loss Distributions Using actuar > f <- coverage(pdf = dgamma, cdf = pgamma, Vincent Goulet + deductible = 1, limit = 10) Probability Laws > fitdistr(y, f, start = list(shape = 2, Grouped + rate = 0.5)) Data Minimum shape rate Distance Estimation 4.5822202 0.8634705 Censored (0.7672822) (0.1518537) Data

Example With Left and Right Censoring Statistical Modeling of Loss Distributions Using actuar > f <- coverage(pdf = dgamma, cdf = pgamma, Vincent Goulet + deductible = 1, limit = 10) Probability Laws > fitdistr(y, f, start = list(shape = 2, Grouped + rate = 0.5)) Data Minimum shape rate Distance Estimation 4.5822202 0.8634705 Censored (0.7672822) (0.1518537) Data

More Information Statistical Modeling of Loss Project’s web site Distributions Using http://www.actuar-project.org actuar Vincent Goulet Package vignettes Probability actuar Introduction to actuar Laws Grouped coverage Complete formulas used by Data coverage Minimum Distance credibility Risk theory features Estimation lossdist Loss distributions modeling Censored Data features risk Risk theory features Demo files