Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Time:2021-6-11

Link to the original text:http://tecdat.cn/?p=21425 

Extreme value theory focuses on the tail characteristics of risk loss distribution, and is usually used to analyze rare probability events. It can rely on a small number of sample data to get the change of extreme value in the population distribution when the population distribution is unknown, and has the estimation ability beyond the sample data. Therefore, the model based on GPD (generalized Pareto distribution) distribution can make more effective use of the limited information of catastrophe loss data, thus becoming the mainstream technology of extreme value theory.

Aiming at the characteristics of low frequency, high loss, lack of data and thick tail of catastrophe, the GPD model is used to build the statistical model of fire economic loss data; The shape and scale parameters are estimated. The model test shows that GPD model has good fitting effect and fitting precision for the characteristics of catastrophe risk thick tail, which provides a theoretical basis for the modeling of catastrophe risk estimation and the pricing of catastrophe bonds.

Fire loss data

The data used in this paper are collected from reinsurance companies, including 2167 fire losses from 1980 to 1990. Inflation has been adjusted. The total claim amount has been divided into building loss and profit loss.

base1=read.table( "dataunivar.txt",
 header=TRUE)
base2=read.table( "datamultiva.txt",
 header=TRUE)

Consider the first data set (so far, we are dealing with univariate extremum),

 > D=as.Date(as.character(base1$Date),"%m/%d/%Y")
> plot(D,X,type="h")

The chart is as follows:

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Then a natural idea is visualization

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

for example

 > plot(log(Xs),log((n:1)/(n+1)))

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

linear regression

The point here is in a straight line. The slope can be obtained by linear regression,

 lm(formula = Y ~ X, data = B)
lm(Y~X,data=B[(n-500):n,])
lm(formula = Y ~ X, data = B[(n - 100):n, ]) 

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Heavy tailed distribution

The slope here is related to the tail index of the distribution. Consider some heavy tailed distributions

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Because the natural estimator is order statistics, the slope of the line is opposite to the tail index  Extreme value theory of R language: analysis of fire loss distribution based on GPD model

The estimated slope is (only the largest observation is considered)

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Hill estimate

Hill’s estimate is based on the assumption that the denominator above is almost 1.

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Then we can get the convergence hypothesis. further

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Based on this (asymptotic) distribution, a (asymptotic) confidence interval can be obtained  Extreme value theory of R language: analysis of fire loss distribution based on GPD model

> xi=1/(1:n)*cumsum(logXs)-logXs
> xise=1.96/sqrt(1:n)*xi

> polygon(c(1:n,n:1),c(xi+xise,rev(xi-xise)), 

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Incremental method

Similarly (there are also additional assumptions about the rate of convergence)

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

(obtained using the incremental method). Similarly, we can use this result to obtain (asymptotic) confidence intervals

 > alphase=1.96/sqrt(1:n)/xi
> polygon(c(1:n,n:1),c(alpha+alphase,rev(alpha-alphase)), 

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Deckers einmal de Haan estimator

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Then (considering the condition of convergence rate again, i.e,

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Pickends estimation

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

  because  Extreme value theory of R language: analysis of fire loss distribution based on GPD model

​,

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

code

> xi=1/log(2)*log( (Xs[seq(1,length=trunc(n/4),by=1)]-
+ Xs[seq(2,length=trunc(n/4),by=2)])/

> xise=1.96/sqrt(seq(1,length=trunc(n/4),by=1))*
+sqrt( xi^2*(2^(xi+1)+1)/((2*(2^xi-1)*log(2))^2))

> polygon(c(seq(1,length=trunc(n/4),by=1),rev(seq(1, 

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Fitting GPD distribution

The maximum likelihood method can also be used to fit the GPD distribution at high threshold.

 > gpd
$n
[1] 2167

$threshold
[1] 5

$p.less.thresh
[1] 0.8827873

$n.exceed
[1] 254

$method
[1] "ml"

$par.ests
xi      beta
0.6320499 3.8074817

$par.ses
xi      beta
0.1117143 0.4637270

$varcov
[,1]        [,2]
[1,]  0.01248007 -0.03203283
[2,] -0.03203283  0.21504269

$information
[1] "observed"

$converged
[1] 0

$nllh.final
[1] 754.1115

attr(,"class")
[1] "gpd"

Or equivalently

> gpd.fit
$threshold
[1] 5

$nexc
[1] 254

$conv
[1] 0

$nllh
[1] 754.1115

$mle
[1] 3.8078632 0.6315749

$rate
[1] 0.1172127

$se
[1] 0.4636270 0.1116136

It can visualize the profile likelihood of tail index,

> gpd.prof

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

perhaps

> gpd.prof

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Therefore, the maximum likelihood estimator of tail index can be drawn as a function of threshold (including confidence interval),

Vectorize(function(u){gpd(X,u)$par.ests[1]})

plot(u,XI,ylim=c(0,2))
segments(u,XI-1.96*XIS,u,XI+ 

Extreme value theory of R language: analysis of fire loss distribution based on GPD model

Finally, you can use the block maximum technique.

gev.fit
$conv
[1] 0

$nllh
[1] 3392.418

$mle
[1] 1.4833484 0.5930190 0.9168128

$se
[1] 0.01507776 0.01866719 0.03035380

The estimate of the tail index is the last coefficient here.


Extreme value theory of R language: analysis of fire loss distribution based on GPD model

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