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

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

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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 cole dactuariat,

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

Statistical Modeling of Loss Distributions Using actuar

Vincent Goulet

École d’actuariat, Université Laval Québec, Canada

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

actuar

Provides additional Actuarial Science functionality to R Current version covers

Loss distribution modeling Risk theory (including ruin theory) Simulation of compound hierarchical models Credibility theory

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

Summary

1

Probability Laws

2

Grouped Data

3

Minimum Distance Estimation

4

Censored Data

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

Summary

1

Probability Laws

2

Grouped Data

3

Minimum Distance Estimation

4

Censored Data

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

At a Glance

Support for 18 probability laws not in base R Mostly positive, heavy tail distributions New utility functions in addition to dfoo, pfoo, qfoo, rfoo

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

Supported Distributions

Transformed Beta Family

9 special cases (including Burr and Pareto)

Transformed Gamma Family

5 special cases (including inverse distributions)

Loggamma Single parameter Pareto Generalized Beta Phase-type distributions

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

New Utility Functions

mfoo to compute theoretical raw moments mk = E[Xk] levfoo to compute theoretical limited moments E[(X ∧ )k] = E[min(X, )k] mgffoo to compute the moment generating function MX(t) = E[etX] when it exists Also support for: beta, exponential, chi-square, gamma, lognormal, normal (no lev), uniform, Weibull, inverse Gaussian

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

New Utility Functions

mfoo to compute theoretical raw moments mk = E[Xk] levfoo to compute theoretical limited moments E[(X ∧ )k] = E[min(X, )k] mgffoo to compute the moment generating function MX(t) = E[etX] when it exists Also support for: beta, exponential, chi-square, gamma, lognormal, normal (no lev), uniform, Weibull, inverse Gaussian

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

Summary

1

Probability Laws

2

Grouped Data

3

Minimum Distance Estimation

4

Censored Data

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

Definition and Rationale

Data presented in an interval-frequency manner: Group Line 1 Line 2 (0, 25] 30 26 (25, 50] 31 33 (50, 100] 57 31 Need for a “standard” storage method Useful for minimum distance estimation

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

Creation and Manipulation of Objects

> x <- grouped.data(Group = c(0, 25, + 50, 100), Line.1 = c(30, 31, 57), + Line.2 = c(26, 33, 31)) > x Group Line.1 Line.2 1 (0, 25] 30 26 2 (25, 50] 31 33 3 (50, 100] 57 31

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

Calculation of Empirical Moments

> mean(x) Line.1 Line.2 49.25847 43.19444 > emm(x, 2) Line.1 Line.2 3253.884 2604.167 > E <- elev(x[, -3]) > E(c(25, 50)) [1] 21.82203 37.18220

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

Plot of the Histogram and Ogive

> hist(x[, -3])

Histogram of x[, −3]

x[, −3] Density 20 40 60 80 100 0.000 0.004 0.008

> plot(ogive(x[, -3]))

  • 20

40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0

  • give(x[, −3])

x F(x)

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

Summary

1

Probability Laws

2

Grouped Data

3

Minimum Distance Estimation

4

Censored Data

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

mde() Supports Three Distance Measures

1 Cramér-von Mises

d(θ) =

n

  • j=1

j[F(j; θ) − Fn(j; θ)]2

2 Modified chi-square

d(θ) =

r

  • j=1

j[n(F(cj; θ) − F(cj−1; θ)) − nj]2,

3 Layer average severity

d(θ) =

r

  • j=1

j[LAS(cj−1, cj; θ) − ˜ LASn(cj−1, cj; θ)]2, where LAS(, y) = E[min(X, y)] − E[min(X, )]

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

mde() Supports Three Distance Measures

1 Cramér-von Mises

d(θ) =

n

  • j=1

j[F(j; θ) − Fn(j; θ)]2

2 Modified chi-square

d(θ) =

r

  • j=1

j[n(F(cj; θ) − F(cj−1; θ)) − nj]2,

3 Layer average severity

d(θ) =

r

  • j=1

j[LAS(cj−1, cj; θ) − ˜ LASn(cj−1, cj; θ)]2, where LAS(, y) = E[min(X, y)] − E[min(X, )]

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

mde() Supports Three Distance Measures

1 Cramér-von Mises

d(θ) =

n

  • j=1

j[F(j; θ) − Fn(j; θ)]2

2 Modified chi-square

d(θ) =

r

  • j=1

j[n(F(cj; θ) − F(cj−1; θ)) − nj]2,

3 Layer average severity

d(θ) =

r

  • j=1

j[LAS(cj−1, cj; θ) − ˜ LASn(cj−1, cj; θ)]2, where LAS(, y) = E[min(X, y)] − E[min(X, )]

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

Summary

1

Probability Laws

2

Grouped Data

3

Minimum Distance Estimation

4

Censored Data

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

Context

Common in statistical and actuarial applications to work with censored data Actuarial terminology: left censoring ⇔ (ordinary) deductible right censoring ⇔ policy limit

5 10 15 0.00 0.04 0.08 0.12

Left Censoring

5 10 15 0.00 0.02 0.04 0.06 0.08 0.10 0.12

Right Censoring

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

A Different Approach

Package survival has extensive support for censored distributions Our approach is different coverage() returns pdf or cdf of censored distribution (with many options) function can be used in fitting as usual (fitdistr(), mde(), ...)

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

Example With Left and Right Censoring

> f <- coverage(pdf = dgamma, cdf = pgamma, + deductible = 1, limit = 10) > fitdistr(y, f, start = list(shape = 2, + rate = 0.5)) shape rate 4.5822202 0.8634705 (0.7672822) (0.1518537)

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

Example With Left and Right Censoring

> f <- coverage(pdf = dgamma, cdf = pgamma, + deductible = 1, limit = 10) > fitdistr(y, f, start = list(shape = 2, + rate = 0.5)) shape rate 4.5822202 0.8634705 (0.7672822) (0.1518537)

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Statistical Modeling of Loss Distributions Using actuar Vincent Goulet Probability Laws Grouped Data Minimum Distance Estimation Censored Data

More Information

Project’s web site http://www.actuar-project.org Package vignettes actuar Introduction to actuar coverage Complete formulas used by coverage credibility Risk theory features lossdist Loss distributions modeling features risk Risk theory features Demo files