library(readxl)
library(evmix)
## Loading required package: MASS
## Loading required package: splines
## Loading required package: gsl
## Loading required package: SparseM
##
## Attaching package: 'SparseM'
## The following object is masked from 'package:base':
##
## backsolve
library(eva)
##
## Attaching package: 'eva'
## The following objects are masked from 'package:evmix':
##
## dgpd, pgpd, qgpd, rgpd
setwd("C:/Users/LENOVO/OneDrive/Documents/NATIONAL LIFE/Retensi")
data.ajn = read_excel("Data Klaim Meninggal 2020-2023.xlsx",
col_types = c("numeric", "text", "text", "numeric",
"numeric", "date", "date", "date", "date",
"numeric", "numeric", "numeric", "numeric",
"numeric", "numeric", "numeric", "numeric",
"date", "date", "numeric", "text", "date",
"numeric", "numeric"))
## New names:
## • `Approved Date` -> `Approved Date...19`
## • `Approved Date` -> `Approved Date...22`
data.ajn = data.ajn[order(data.ajn$`Claim Date`), ]
head(data.ajn, 1500)
## # A tibble: 1,500 × 24
## NO `JENIS PRODUK` `NOMOR KLAIM` NO.Polis `Certificate No`
## <dbl> <chr> <chr> <dbl> <dbl>
## 1 160 PJKN CLD200400000029 1.80e14 180204072190
## 2 2 PJKN CLD200200000006 1.90e14 190295237834
## 3 26 PJKN CLD200200000076 1.90e14 190295075607
## 4 8 PJKN CLD200100000097 1.90e14 190295250485
## 5 6 PJKN CLD200100000093 1.90e14 190295297584
## 6 7 PJKN CLD200100000094 1.90e14 190295275166
## 7 252 PJKN CLD200600000042 1.90e14 190295311582
## 8 319 PJKN CLD200600000176 1.80e14 180204116601
## 9 13 PJKN CLD200200000009 1.90e14 190295233417
## 10 31 PJKN CLD200200000082 1.90e14 190295186133
## # ℹ 1,490 more rows
## # ℹ 19 more variables: `Birth Date` <dttm>, `Start Date` <dttm>,
## # `End Date` <dttm>, `Claim Date` <dttm>, `Sum Insured ND` <dbl>,
## # `Usia Masuk` <dbl>, `Usia Meninggal (thn)` <dbl>, `usia Polis(BLN)` <dbl>,
## # Premium <dbl>, `Submission Claim` <dbl>, `Claim Approved Amount` <dbl>,
## # `Claim Paid Amount` <dbl>, `Submission Date` <dttm>,
## # `Approved Date...19` <dttm>, `Claim NAS` <dbl>, Status <chr>, …
Sum Insured ND
plot(x = data.ajn$`Claim Date`, y = data.ajn$`Sum Insured ND`, type = "h",
xlab = "Years", ylab = "Loss Amount")
insured.max = aggregate(data.ajn$`Sum Insured ND`,
by = list(as.numeric(substr(data.ajn$`Claim Date`, 1, 4))),
max, na.rm = TRUE)
names(insured.max) = c("Year", "Max_Insured_Value")
max_insured_rows = data.frame()
for (i in 1:nrow(insured.max)) {
rows = data.ajn[data.ajn$`Sum Insured ND` == insured.max[i, "Max_Insured_Value"] &
substr(data.ajn$`Claim Date`, 1, 4) == insured.max[i, "Year"],
c("Claim Date", "Sum Insured ND")]
max_insured_rows <- rbind(max_insured_rows, rows)
}
print(max_insured_rows)
## # A tibble: 4 × 2
## `Claim Date` `Sum Insured ND`
## <dttm> <dbl>
## 1 2020-07-24 00:00:00 500000000
## 2 2021-02-13 00:00:00 500000000
## 3 2022-08-08 00:00:00 1500000000
## 4 2023-11-04 00:00:00 726644000
plot(data.ajn$`Claim Date`, data.ajn$`Sum Insured ND`, type = "h",
xlab = "Years", ylab = "Annual maxima loss amount", col = "grey")
points(max_insured_rows, pch = 15, cex = 1, col = "black")
#Visualizing Tails
Ins.Amt = data.ajn$`Sum Insured ND`
# Threshold, sumber dari https://www.jstor.org/stable/1269343
a = 1/2*mean(Ins.Amt)*((mean(Ins.Amt))^2/(var(Ins.Amt))^2 + 1)
k = 0.5*((mean(Ins.Amt))^2/(var(Ins.Amt))^2 - 1)
u = a / (k + 1)
u
## [1] 53008246
u == mean(Ins.Amt)
## [1] TRUE
round(var(Ins.Amt), 15)
## [1] 5.282369e+15
a
## [1] 26504123
k
## [1] -0.5
InsAmt.exc = Ins.Amt[Ins.Amt > u]
n.u = length(InsAmt.exc)
surv.prob = 1 - (rank(InsAmt.exc))/(n.u + 1)
plot(InsAmt.exc, surv.prob, log = "xy", xlab = "Exceedances",
ylab = "Survival probability", col="ivory4",
main="Survival Probability Plot for Insured Amount Exceedances")
alpha = - cov(log(InsAmt.exc), log(surv.prob)) / var(log(InsAmt.exc))
alpha
## [1] 1.813302
#plot
plot(InsAmt.exc, surv.prob, log = "xy", xlab = "Exceedances",
ylab = "Survival probability", col="ivory4",
main="Survival Probability Plot for Insured Amount Exceedances")
x = seq(u, max(InsAmt.exc), length = 100)
y = (x/u)^(-alpha)
lines(x, y, col="firebrick4", lwd=2)
# CDF
prob = rank(InsAmt.exc) / (n.u + 1)
plot(InsAmt.exc, prob, log = "x", xlab= "Exceedances",
ylab = "Probability of non exceedance")
y = 1 - (x / u)^(-alpha)
lines(x, y)
library(dplyr)
##
## Attaching package: 'dplyr'
## The following object is masked from 'package:MASS':
##
## select
## The following objects are masked from 'package:stats':
##
## filter, lag
## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union
# Estimation
# 1) GEV Dist
nllik.gev = function(par, data) {
mu = par[1]
sigma = par[2]
xi = par[3]
if((sigma <= 0) | (xi <= -1))
return(1e6)
n = length(data)
if(xi == 0)
n * log(sigma) + sum((data - mu) / sigma) + sum(exp(-(data - mu) / sigma))
else {
if(any((1 + xi * (data - mu) / sigma) <= 0))
return(1e6)
n * log(sigma) + (1 + 1/ xi) *
sum(log(1 + xi * (data - mu) / sigma)) +
sum((1 + xi * (data - mu) / sigma)^(-1/xi))
}
}
years = as.numeric(substr(data.ajn$`Claim Date`, 1, 4))
Amt.max = aggregate(Ins.Amt ~ `Sum Insured ND` + years, data = data.ajn, FUN = max, na.rm = TRUE)
Amt.max
## Sum Insured ND years Ins.Amt
## 1 1125000 2020 1125000
## 2 1187500 2020 1187500
## 3 1375000 2020 1375000
## 4 1500000 2020 1500000
## 5 1525375 2020 1525375
## 6 1687500 2020 1687500
## 7 1800000 2020 1800000
## 8 1875000 2020 1875000
## 9 1968750 2020 1968750
## 10 2025000 2020 2025000
## 11 2062500 2020 2062500
## 12 2125000 2020 2125000
## 13 2156250 2020 2156250
## 14 2250000 2020 2250000
## 15 2375000 2020 2375000
## 16 2500000 2020 2500000
## 17 2533509 2020 2533509
## 18 2579145 2020 2579145
## 19 2583333 2020 2583333
## 20 2800000 2020 2800000
## 21 2812500 2020 2812500
## 22 2875000 2020 2875000
## 23 2916667 2020 2916667
## 24 2916668 2020 2916668
## 25 3000000 2020 3000000
## 26 3100000 2020 3100000
## 27 3125000 2020 3125000
## 28 3281250 2020 3281250
## 29 3300000 2020 3300000
## 30 3360000 2020 3360000
## 31 3437500 2020 3437500
## 32 3541667 2020 3541667
## 33 3593750 2020 3593750
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## 36 3900000 2020 3900000
## 37 3937500 2020 3937500
## 38 4000000 2020 4000000
## 39 4125000 2020 4125000
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## 1894 255000000 2023 255000000
## 1895 256176000 2023 256176000
## 1896 259800000 2023 259800000
## 1897 260000000 2023 260000000
## 1898 262000000 2023 262000000
## 1899 262704000 2023 262704000
## 1900 270000000 2023 270000000
## 1901 274023912 2023 274023912
## 1902 275000000 2023 275000000
## 1903 275712000 2023 275712000
## 1904 277000000 2023 277000000
## 1905 279800000 2023 279800000
## 1906 282600000 2023 282600000
## 1907 285000000 2023 285000000
## 1908 286740000 2023 286740000
## 1909 288000000 2023 288000000
## 1910 290000000 2023 290000000
## 1911 290500000 2023 290500000
## 1912 293000000 2023 293000000
## 1913 296100000 2023 296100000
## 1914 298560000 2023 298560000
## 1915 298800000 2023 298800000
## 1916 299300000 2023 299300000
## 1917 300000000 2023 300000000
## 1918 306210000 2023 306210000
## 1919 306328000 2023 306328000
## 1920 311880000 2023 311880000
## 1921 312000000 2023 312000000
## 1922 314500000 2023 314500000
## 1923 318777000 2023 318777000
## 1924 320600000 2023 320600000
## 1925 324000000 2023 324000000
## 1926 325740000 2023 325740000
## 1927 335220000 2023 335220000
## 1928 341005000 2023 341005000
## 1929 344760000 2023 344760000
## 1930 360000000 2023 360000000
## 1931 362160000 2023 362160000
## 1932 369324000 2023 369324000
## 1933 375000000 2023 375000000
## 1934 377000000 2023 377000000
## 1935 377482000 2023 377482000
## 1936 378209000 2023 378209000
## 1937 382620000 2023 382620000
## 1938 405000000 2023 405000000
## 1939 455000000 2023 455000000
## 1940 456084000 2023 456084000
## 1941 477552000 2023 477552000
## 1942 480204000 2023 480204000
## 1943 495504000 2023 495504000
## 1944 500000000 2023 500000000
## 1945 601824000 2023 601824000
## 1946 726644000 2023 726644000
Amt.max = Amt.max$Ins.Amt
sigma.start = sqrt(6) * sd(Amt.max) / pi
mu.start = mean(Amt.max) + digamma(1) * sigma.start
fit.gev = nlm(nllik.gev, c(mu.start, sigma.start, 0),
hessian = TRUE, data = Amt.max)
fit.gev
## $minimum
## [1] 37971.08
##
## $estimate
## [1] 4.257783e+07 7.749759e+07 1.772316e+00
##
## $gradient
## [1] -1.469866e-05 1.114520e-05 0.000000e+00
##
## $hessian
## [,1] [,2] [,3]
## [1,] 1.734628e-11 -8.898694e-12 2.828038e-04
## [2,] -8.898694e-12 4.608022e-12 -1.535275e-04
## [3,] 2.828038e-04 -1.535275e-04 5.146953e+03
##
## $code
## [1] 2
##
## $iterations
## [1] 17
ridge = 1e-8 # Small positive constant
hessian = fit.gev$hessian + ridge * diag(nrow(fit.gev$hessian))
sqrt(diag(solve(hessian)))
## [1] 9999.0964461 9999.9857712 0.0139528
# 2) Poisson-Generalized Pareto Model
nllik.gp = function(par, u, data) {
tau = par[1]
xi = par[2]
if((tau <= 0) | (xi < -1))
return(1e6)
m = length(data)
if(xi == 0)
m*log(tau) + sum(data - u) / tau
else {
if(any((1 + xi*(data - u) / tau) <= 0))
return(1e6)
m * log(tau) + (1 + 1/xi) * sum(log(1+xi*(data-u)/tau))
}
}
tau.start = mean(InsAmt.exc) - u
fit.gp = nlm(nllik.gp, c(tau.start, 0), u = u, hessian = TRUE,
data = InsAmt.exc)
fit.gp
## $minimum
## [1] 30740.34
##
## $estimate
## [1] 8.426417e+07 2.206233e-02
##
## $gradient
## [1] 4.214434e-07 2.728484e-06
##
## $hessian
## [,1] [,2]
## [1,] 2.055952e-13 1.713284e-05
## [2,] 1.713284e-05 3.763736e+03
##
## $code
## [1] 2
##
## $iterations
## [1] 6
# Derive confidence intervals (Continued)
prof.nllik.gp = function(par, xi, u, data) {
nllik.gp(c(par, xi), u, data)
}
prof.fit.gp = function(x) {
-nlm(prof.nllik.gp, tau.start, xi = x, u = u, hessian = TRUE, data = InsAmt.exc)$minimum
}
vxi = seq(0, 1.8, by = 0.025)
prof.lik = Vectorize(prof.fit.gp)(vxi)
opt = optimize(f = prof.fit.gp, interval = c(0, 3), maximum = TRUE)
opt
## $maximum
## [1] 0.02206716
##
## $objective
## [1] -30740.34
up = opt$objective
I = which(prof.lik >= up-qchisq(p = 0.95, df = 1))
plot(vxi, prof.lik, type = "l", xlab = expression(xi), ylab = "Profile log-likelihood")
abline(h = up, lty=2)
abline(h = up-qchisq(p = 0.95, df = 1), col = "grey")
lines(vxi[I], rep(up-qchisq(p = 0.95, df = 1), length(I)),
lwd = 5, col = "grey")
abline(v = range(vxi[I]), col = "grey", lty = 2)
abline(v = opt$maximum, col="grey")
# 3) Point process
nllik.pp <- function(par, u, data, n.b) {
mu <- par[1]
sigma <- par[2]
xi <- par[3]
if ((sigma <= 0) | (xi <= -1))
return(1e6)
if (xi == 0)
poiss.meas <- n.b * exp(-(u - mu) / sigma)
else
poiss.meas <- n.b * max(0, 1 + xi * (u - mu) / sigma)^(-1/xi)
exc <- data[data > u]
m <- length(exc)
if (xi == 0)
poiss.meas + m * log(sigma) + sum((exc - mu) / sigma)
else {
if (any((1 + xi * (exc - mu) / sigma) <= 0))
return(1e6)
poiss.meas + m * log(sigma) + (1 + 1 / xi) *
sum(log(1 + xi * (exc - mu) / sigma))
}
}
n.b = 2024 - 2020
sigma.start = sqrt(6) * sd(InsAmt.exc) / pi
mu.start = mean(InsAmt.exc) + (log(n.b) + digamma(1))*sigma.start
fit.pp = nlm(nllik.pp, c(mu.start, sigma.start, 0), u = u,
hessian = TRUE, data = Ins.Amt, n.b = n.b)
fit.pp
## $minimum
## [1] 25591.17
##
## $estimate
## [1] 1.938152e+08 6.988539e+07 4.360273e-01
##
## $gradient
## [1] -2.676544e-05 2.549023e-05 -1.455192e-04
##
## $hessian
## [,1] [,2] [,3]
## [1,] 7.985419e-12 -1.628630e-11 1.965974e-03
## [2,] -1.628630e-11 3.378291e-11 -4.099131e-03
## [3,] 1.965974e-03 -4.099131e-03 4.956769e+05
##
## $code
## [1] 2
##
## $iterations
## [1] 12
# 4) Other tail Index Estimates
logXs = log(sort(Ins.Amt, decreasing=TRUE))
n = length(logXs)
xi = 1/1:n * cumsum(logXs) - logXs
ci.up = xi + qnorm(0.975) * xi / sqrt(1:n)
ci.low = xi - qnorm(0.975) * xi / sqrt(1:n)
matplot(1:n, cbind(ci.low, xi, ci.up),lty = 1, type = "l",
col = c("grey", "black", "grey"), ylab = expression(xi),
xlab = "Number of extreme values")
alpha = 1 / xi
alpha.std.err = alpha / sqrt(1:n)
ci.up = alpha + qnorm(0.975) * alpha / sqrt(1:n)
ci.low = alpha - qnorm(0.975) * alpha / sqrt(1:n)
matplot(1:n, cbind(ci.low, alpha, ci.up), lty = 1, type = "l",
col = c("grey", "black", "grey"), ylab = expression(alpha),
xlab = "Number of extreme values")
# Checking for the Asymptotic Regime Assumption
# 1) Mean Excess Plot
meanExcessPlot = function(data, u.range = range(data), n.u = length(InsAmt.exc)) {
mean.excess = ci.up <- ci.low <- rep(NA, n.u)
all.u = seq(u.range[1], u.range[2], length = n.u)
for (i in 1:n.u) {
u = all.u[i]
excess = data[data > u] - u
n.u = length(excess)
mean.excess[i] = mean(excess)
var.mean.excess = var(excess)
ci.up[i] = mean.excess[i] + qnorm(0.975) *
sqrt(var.mean.excess / n.u)
ci.low[i] = mean.excess[i] - qnorm(0.975) *
sqrt(var.mean.excess / n.u)
}
matplot(all.u, cbind(ci.low, mean.excess, ci.up), col = 1,
lty = c(2, 1, 2), type = "l", xlab = "u",
ylab = "Mean excess")
}
meanExcessPlot(InsAmt.exc)
# 2) Paremeter Stability
tcplot(Ins.Amt, tlim = NULL, nt = min(200, length(Ins.Amt)), p.or.n = FALSE,
alpha = 0.05, ylim.xi = NULL, ylim.sigmau = NULL,
legend.loc = "bottomright", try.thresh=NULL)
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
#Model Checking
# Q-Q Plot
qqgpd = function(data, u, tau, xi){
excess = data[data > u]
m = length(excess)
prob = 1:m / (m + 1)
x.hat = u + tau / xi * ((1 - prob)^-xi - 1)
ylim <- xlim <- range(x.hat, excess)
plot(sort(excess), x.hat, xlab = "Sample Quantiles",
ylab = "Fitted Quantiles", xlim = xlim, ylim = ylim, main = "Q-Q Plot")
abline(0, 1, col = "grey", lwd = 2)
}
qqgpd(Ins.Amt, u, fit.gp$estimate[1], fit.gp$estimate[2])
# P-P Plot
ppgpd = function(data, u, tau, xi){
excess = data[data > u]
m = length(excess)
emp.prob = 1:m / (m + 1)
prob.hat = 1 - (1 + xi * (sort(excess) - u) / tau)^(-1 / xi)
plot(emp.prob, prob.hat, xlab = "Empirical Probabilities", ylab = "Fitted Probabilities", xlim = c(0, 1), ylim = c(0, 1), main = "P-P Plot")
abline(0, 1, col = "grey", lwd = 2)
}
ppgpd(Ins.Amt, u, fit.gp$estimate[1], fit.gp$estimate[2])
# Return Level Plot
rlgpd = function(data, u, tau, xi, nb) {
excess = data[data > u]
n = length(data)
m = length(excess)
zeta.u <- m / n
rl = function(T) {
prob = 1 - 1 / (nb * T)
u + tau / xi * (((1 - prob) / zeta.u)^-xi - 1)
}
plot(rl, from = 1 / nb + 0.1, to = 100, log = "x",
xlab = "Return period", ylab = "Return level")
points((m+1) / (nb * zeta.u * m:1), sort(excess))
}
days = 365*length(Ins.Amt)/as.numeric(difftime(max(data.ajn$`Claim Date`), min(data.ajn$`Claim Date`), units = "days"))
days
## [1] 1412.413
rlgpd(Ins.Amt, u, fit.gp$estimate[1], fit.gp$estimate[2], days)
# Premium pricing
par_estimate = gpdFit(Ins.Amt, threshold = u, method = c("mle"))
par_estimate$par.ests
## Scale (Intercept) Shape (Intercept)
## 8.107001e+07 3.730889e-02
shape = par_estimate$par.ests[2]
location = u
scale = par_estimate$par.ests[1]
attachment = quantile(Ins.Amt, 0.95)
integration = function(f, lower_bound, upper_bound) {
result = integrate(f, lower_bound, upper_bound)
return(result)
}
# GPD pdf function
f_x = function(x) {
(x-attachment)*(1 / scale) * (1 + shape*(x - location)/scale)^(-1/shape-1)
}
# expected loss exceeding attachment, but less than highest data value
loss1 = integration(f_x, attachment, quantile(Ins.Amt, 1))
loss1
## 14876438 with absolute error < 7.7
# expected loss exceeding highest empirical value
probability_exceed = function(x) {
(1 / scale) * (1 +shape*(x-location)/scale)^(-1/shape-1)
}
prob = integration(probability_exceed, location, quantile(Ins.Amt, 1))
loss2 = (1-prob$value)*quantile(Ins.Amt, 1)
loss2
## 100%
## 1718.221
prob
## 0.9999989 with absolute error < 5.3e-07
expected_total_premium = loss1$value + loss2
expected_total_premium
## 100%
## 14878156
quantile(Ins.Amt, 0.95)
## 95%
## 2.04e+08
quantile(Ins.Amt, 1)
## 100%
## 1.5e+09
max(Ins.Amt)
## [1] 1.5e+09
Submission Claim
plot(x = data.ajn$`Claim Date`, y = data.ajn$`Submission Claim`, type = "h",
xlab = "Years", ylab = "Loss amount")
submission.max = aggregate(data.ajn$`Submission Claim`, by = list(as.numeric(substr(data.ajn$`Claim Date`, 1, 4))), max, na.rm = TRUE)
names(submission.max) = c("Year", "Max_Submissioned_Value")
max_submission_rows = data.frame()
for (i in 1:nrow(submission.max)) {
rows = data.ajn[data.ajn$`Submission Claim` == submission.max[i, "Max_Submissioned_Value"] &
substr(data.ajn$`Claim Date`, 1, 4) == submission.max[i, "Year"],
c("Claim Date", "Submission Claim")]
max_submission_rows <- rbind(max_submission_rows, rows)
}
print(max_submission_rows)
## # A tibble: 4 × 2
## `Claim Date` `Submission Claim`
## <dttm> <dbl>
## 1 2020-07-24 00:00:00 500000000
## 2 2021-02-13 00:00:00 486810000
## 3 2022-08-08 00:00:00 780477120
## 4 2023-11-04 00:00:00 646354536
plot(data.ajn$`Claim Date`, data.ajn$`Submission Claim`, type = "h",
xlab = "Years", ylab = "Annual maxima loss amount", col = "grey")
points(max_submission_rows, pch = 15, cex = 1, col = "black")
# Visualizing Tails
Sub.Amt = data.ajn$`Submission Claim`
# Threshold, sumber dari https://www.jstor.org/stable/1269343
a = 1/2*mean(Sub.Amt)*((mean(Sub.Amt))^2/(var(Sub.Amt))^2 + 1)
k = 0.5*((mean(Sub.Amt))^2/(var(Sub.Amt))^2 - 1)
u = round(a / (k + 1), 0)
u
## [1] 39864910
u == round(mean(Sub.Amt),0)
## [1] TRUE
mean(Sub.Amt)
## [1] 39864910
SubAmt.exc = Sub.Amt[Sub.Amt > u]
n.u = length(SubAmt.exc)
surv.prob = 1 - (rank(SubAmt.exc))/(n.u + 1)
plot(SubAmt.exc, surv.prob, log = "xy", xlab = "Exceedances",
ylab = "Survival probability", col="ivory4",
main="Survival Probability Plot for Submission Claim Exceedances")
alpha = - cov(log(SubAmt.exc), log(surv.prob)) / var(log(SubAmt.exc))
alpha
## [1] 1.69936
#plot
plot(SubAmt.exc, surv.prob, log = "xy", xlab = "Exceedances",
ylab = "Survival probability", col="ivory4",
main="Survival Probability Plot for Submission Claim Exceedances")
x = seq(u, max(SubAmt.exc), length = 100)
y = (x/u)^(-alpha)
lines(x, y, col="firebrick4", lwd=2)
# CDF
prob = rank(SubAmt.exc) / (n.u + 1)
plot(SubAmt.exc, prob, log = "x", xlab= "Exceedances",
ylab = "Probability of non exceedance")
y = 1 - (x / u)^(-alpha)
lines(x, y)
library(dplyr)
# Estimation
# 1) GEV Dist
nllik.gev = function(par, data) {
mu = par[1]
sigma = par[2]
xi = par[3]
if((sigma <= 0) | (xi <= -1))
return(1e6)
n = length(data)
if(xi == 0)
n * log(sigma) + sum((data - mu) / sigma) + sum(exp(-(data - mu) / sigma))
else {
if(any((1 + xi * (data - mu) / sigma) <= 0))
return(1e6)
n * log(sigma) + (1 + 1/ xi) *
sum(log(1 + xi * (data - mu) / sigma)) +
sum((1 + xi * (data - mu) / sigma)^(-1/xi))
}
}
Amt.max <- aggregate(Sub.Amt ~ `Submission Claim` + years, data = data.ajn, FUN = max, na.rm = TRUE)
Amt.max
## Submission Claim years Sub.Amt
## 1 134307.0 2020 134307.0
## 2 166665.0 2020 166665.0
## 3 176217.0 2020 176217.0
## 4 202259.0 2020 202259.0
## 5 235772.0 2020 235772.0
## 6 266999.0 2020 266999.0
## 7 269643.0 2020 269643.0
## 8 281250.0 2020 281250.0
## 9 312499.0 2020 312499.0
## 10 312501.0 2020 312501.0
## 11 318295.0 2020 318295.0
## 12 332879.0 2020 332879.0
## 13 357112.0 2020 357112.0
## 14 375000.0 2020 375000.0
## 15 400318.0 2020 400318.0
## 16 419613.0 2020 419613.0
## 17 423610.0 2020 423610.0
## 18 428972.0 2020 428972.0
## 19 432743.0 2020 432743.0
## 20 437499.0 2020 437499.0
## 21 437668.0 2020 437668.0
## 22 468750.0 2020 468750.0
## 23 484270.5 2020 484270.5
## 24 488333.3 2020 488333.3
## 25 500000.0 2020 500000.0
## 26 500513.0 2020 500513.0
## 27 507703.0 2020 507703.0
## 28 519343.0 2020 519343.0
## 29 524901.0 2020 524901.0
## 30 544121.0 2020 544121.0
## 31 562500.0 2020 562500.0
## 32 588471.6 2020 588471.6
## 33 588647.0 2020 588647.0
## 34 592340.0 2020 592340.0
## 35 592714.4 2020 592714.4
## 36 597447.0 2020 597447.0
## 37 624999.0 2020 624999.0
## 38 625001.0 2020 625001.0
## 39 625002.0 2020 625002.0
## 40 642008.0 2020 642008.0
## 41 660575.0 2020 660575.0
## 42 662807.0 2020 662807.0
## 43 669504.0 2020 669504.0
## 44 681642.0 2020 681642.0
## 45 683347.0 2020 683347.0
## 46 688146.0 2020 688146.0
## 47 699170.0 2020 699170.0
## 48 722458.0 2020 722458.0
## 49 748840.0 2020 748840.0
## 50 749999.0 2020 749999.0
## 51 761221.0 2020 761221.0
## 52 770417.0 2020 770417.0
## 53 776490.0 2020 776490.0
## 54 798465.0 2020 798465.0
## 55 812560.0 2020 812560.0
## 56 831506.0 2020 831506.0
## 57 833333.3 2020 833333.3
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## 63 897000.6 2020 897000.6
## 64 898344.2 2020 898344.2
## 65 902850.0 2020 902850.0
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## 67 937499.0 2020 937499.0
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## 75 1017063.1 2020 1017063.1
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## 101 1207070.0 2020 1207070.0
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## 5243 223170000.0 2023 223170000.0
## 5244 223256000.0 2023 223256000.0
## 5245 225226659.0 2023 225226659.0
## 5246 226835601.0 2023 226835601.0
## 5247 230000000.0 2023 230000000.0
## 5248 232439335.0 2023 232439335.0
## 5249 233297285.0 2023 233297285.0
## 5250 233960076.0 2023 233960076.0
## 5251 234060416.0 2023 234060416.0
## 5252 234899640.0 2023 234899640.0
## 5253 235963113.0 2023 235963113.0
## 5254 240299954.0 2023 240299954.0
## 5255 240924652.0 2023 240924652.0
## 5256 241644685.0 2023 241644685.0
## 5257 244290929.0 2023 244290929.0
## 5258 245109906.0 2023 245109906.0
## 5259 249566450.0 2023 249566450.0
## 5260 252731430.0 2023 252731430.0
## 5261 255386668.0 2023 255386668.0
## 5262 255693469.0 2023 255693469.0
## 5263 271620000.0 2023 271620000.0
## 5264 272804000.0 2023 272804000.0
## 5265 273052941.0 2023 273052941.0
## 5266 273703226.0 2023 273703226.0
## 5267 275023499.0 2023 275023499.0
## 5268 276679290.0 2023 276679290.0
## 5269 279800000.0 2023 279800000.0
## 5270 280877690.0 2023 280877690.0
## 5271 280985876.0 2023 280985876.0
## 5272 281473228.0 2023 281473228.0
## 5273 282286291.0 2023 282286291.0
## 5274 283475849.0 2023 283475849.0
## 5275 287160296.0 2023 287160296.0
## 5276 287489614.7 2023 287489614.7
## 5277 287737000.0 2023 287737000.0
## 5278 288140517.0 2023 288140517.0
## 5279 288664618.0 2023 288664618.0
## 5280 290000000.0 2023 290000000.0
## 5281 290524000.0 2023 290524000.0
## 5282 293001938.0 2023 293001938.0
## 5283 294000000.0 2023 294000000.0
## 5284 294807688.0 2023 294807688.0
## 5285 296375030.0 2023 296375030.0
## 5286 297666498.0 2023 297666498.0
## 5287 297739000.0 2023 297739000.0
## 5288 298800000.0 2023 298800000.0
## 5289 300000000.0 2023 300000000.0
## 5290 300441043.0 2023 300441043.0
## 5291 302050000.0 2023 302050000.0
## 5292 305037666.0 2023 305037666.0
## 5293 306493464.0 2023 306493464.0
## 5294 307971087.0 2023 307971087.0
## 5295 311170209.0 2023 311170209.0
## 5296 319691525.0 2023 319691525.0
## 5297 340558610.0 2023 340558610.0
## 5298 352227141.0 2023 352227141.0
## 5299 368400752.0 2023 368400752.0
## 5300 371791080.0 2023 371791080.0
## 5301 377000000.0 2023 377000000.0
## 5302 428492607.0 2023 428492607.0
## 5303 500000000.0 2023 500000000.0
## 5304 558583654.0 2023 558583654.0
## 5305 646354536.0 2023 646354536.0
Amt.max = Amt.max$Sub.Amt
sigma.start = sqrt(6) * sd(Amt.max) / pi
mu.start = mean(Amt.max) + digamma(1) * sigma.start
fit.gev = nlm(nllik.gev, c(mu.start, sigma.start, 0),
hessian = TRUE, data = Amt.max)
fit.gev
## $minimum
## [1] 100975.5
##
## $estimate
## [1] 1.371382e+07 4.735151e+07 2.261532e-01
##
## $gradient
## [1] -3.490596e-05 6.716773e-05 -7.275958e-08
##
## $hessian
## [,1] [,2] [,3]
## [1,] 1.805642e-12 5.853282e-13 -6.774827e-05
## [2,] 5.853282e-13 -5.187296e-13 3.655155e-05
## [3,] -6.774827e-05 3.655155e-05 2.354966e+03
##
## $code
## [1] 2
##
## $iterations
## [1] 8
ridge = 1e-8 # Small positive constant
hessian = fit.gev$hessian + ridge * diag(nrow(fit.gev$hessian))
sqrt(diag(solve(hessian)))
## [1] 1.000007e+04 1.000054e+04 2.060926e-02
# 2) Poisson-Generalized Pareto Model
nllik.gp = function(par, u, data) {
tau = par[1]
xi = par[2]
if((tau <= 0) | (xi < -1))
return(1e6)
m = length(data)
if(xi == 0)
m*log(tau) + sum(data - u) / tau
else {
if(any((1 + xi*(data - u) / tau) <= 0))
return(1e6)
m * log(tau) + (1 + 1/xi) * sum(log(1+xi*(data-u)/tau))
}
}
tau.start = mean(SubAmt.exc) - u
fit.gp = nlm(nllik.gp, c(tau.start, 0), u = u, hessian = TRUE,
data = SubAmt.exc)
fit.gp
## $minimum
## [1] 30558.19
##
## $estimate
## [1] 6.769015e+07 5.332681e-02
##
## $gradient
## [1] 1.267280e-06 5.456968e-08
##
## $hessian
## [,1] [,2]
## [1,] 2.810277e-13 1.919033e-05
## [2,] 1.919033e-05 2.689338e+03
##
## $code
## [1] 2
##
## $iterations
## [1] 6
# Derive confidence intervals (Continued)
prof.nllik.gp = function(par, xi, u, data) {
nllik.gp(c(par, xi), u, data)
}
prof.fit.gp = function(x) {
-nlm(prof.nllik.gp, tau.start, xi = x, u = u, hessian = TRUE, data = SubAmt.exc)$minimum
}
vxi = seq(0, 1.8, by = 0.025)
prof.lik = Vectorize(prof.fit.gp)(vxi)
opt = optimize(f = prof.fit.gp, interval = c(0, 3), maximum = TRUE)
opt
## $maximum
## [1] 0.05331371
##
## $objective
## [1] -30558.19
up = opt$objective
I = which(prof.lik >= up-qchisq(p = 0.95, df = 1))
plot(vxi, prof.lik, type = "l", xlab = expression(xi), ylab = "Profile log-likelihood")
abline(h = up, lty=2)
abline(h = up-qchisq(p = 0.95, df = 1), col = "grey")
lines(vxi[I], rep(up-qchisq(p = 0.95, df = 1), length(I)),
lwd = 5, col = "grey")
abline(v = range(vxi[I]), col = "grey", lty = 2)
abline(v = opt$maximum, col="grey")
# 3) Point process
nllik.pp <- function(par, u, data, n.b) {
mu <- par[1]
sigma <- par[2]
xi <- par[3]
if ((sigma <= 0) | (xi <= -1))
return(1e6)
if (xi == 0)
poiss.meas <- n.b * exp(-(u - mu) / sigma)
else
poiss.meas <- n.b * max(0, 1 + xi * (u - mu) / sigma)^(-1/xi)
exc <- data[data > u]
m <- length(exc)
if (xi == 0)
poiss.meas + m * log(sigma) + sum((exc - mu) / sigma)
else {
if (any((1 + xi * (exc - mu) / sigma) <= 0))
return(1e6)
poiss.meas + m * log(sigma) + (1 + 1 / xi) *
sum(log(1 + xi * (exc - mu) / sigma))
}
}
n.b = 4 - 0
sigma.start = sqrt(6) * sd(SubAmt.exc) / pi
mu.start = mean(SubAmt.exc) + (log(n.b) + digamma(1))*sigma.start
fit.pp = nlm(nllik.pp, c(mu.start, sigma.start, 0), u = u,
hessian = TRUE, data = Sub.Amt, n.b = n.b)
fit.pp
## $minimum
## [1] 25216.29
##
## $estimate
## [1] 1.552437e+08 5.894190e+07 4.562354e-01
##
## $gradient
## [1] -3.173926e-05 2.983128e-05 7.057679e-04
##
## $hessian
## [,1] [,2] [,3]
## [1,] 1.564893e-11 -3.095792e-11 3.092561e-03
## [2,] -3.095792e-11 6.199260e-11 -6.218487e-03
## [3,] 3.092561e-03 -6.218487e-03 6.224640e+05
##
## $code
## [1] 2
##
## $iterations
## [1] 11
# 4) Other tail Index Estimates
logXs = log(sort(Sub.Amt, decreasing=TRUE))
n = length(logXs)
xi = 1/1:n * cumsum(logXs) - logXs
ci.up = xi + qnorm(0.975) * xi / sqrt(1:n)
ci.low = xi - qnorm(0.975) * xi / sqrt(1:n)
matplot(1:n, cbind(ci.low, xi, ci.up),lty = 1, type = "l",
col = c("grey", "black", "grey"), ylab = expression(xi),
xlab = "Number of extreme values")
alpha = 1 / xi
alpha.std.err = alpha / sqrt(1:n)
ci.up = alpha + qnorm(0.975) * alpha / sqrt(1:n)
ci.low = alpha - qnorm(0.975) * alpha / sqrt(1:n)
matplot(1:n, cbind(ci.low, alpha, ci.up), lty = 1, type = "l",
col = c("grey", "black", "grey"), ylab = expression(alpha),
xlab = "Number of extreme values")
# Checking for the Asymptotic Regime Assumption
# 1) Mean Excess Plot
meanExcessPlot = function(data, u.range = range(data), n.u = length(SubAmt.exc)) {
mean.excess = ci.up <- ci.low <- rep(NA, n.u)
all.u = seq(u.range[1], u.range[2], length = n.u)
for (i in 1:n.u) {
u = all.u[i]
excess = data[data > u] - u
n.u = length(excess)
mean.excess[i] = mean(excess)
var.mean.excess = var(excess)
ci.up[i] = mean.excess[i] + qnorm(0.975) *
sqrt(var.mean.excess / n.u)
ci.low[i] = mean.excess[i] - qnorm(0.975) *
sqrt(var.mean.excess / n.u)
}
matplot(all.u, cbind(ci.low, mean.excess, ci.up), col = 1,
lty = c(2, 1, 2), type = "l", xlab = "u",
ylab = "Mean excess")
}
meanExcessPlot(SubAmt.exc)
# 2) Paremeter Stability
tcplot(Sub.Amt, tlim = NULL, nt = min(200, length(Sub.Amt)), p.or.n = FALSE,
alpha = 0.05, ylim.xi = NULL, ylim.sigmau = NULL,
legend.loc = "bottomright", try.thresh=NULL)
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): check convergence
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
## Warning in fgpd(x, u): observed information matrix is singular
rlgpd = function(data, u, tau, xi, nb) {
excess = data[data > u]
n = length(data)
m = length(excess)
zeta.u <- m / n
rl = function(T) {
prob = 1 - 1 / (nb * T)
u + tau / xi * (((1 - prob) / zeta.u)^-xi - 1)
}
plot(rl, from = 1 / nb + 0.1, to = 100, log = "x",
xlab = "Return period", ylab = "Return level")
points((m+1) / (nb * zeta.u * m:1), sort(excess))
}
days = 365*length(Sub.Amt)/as.numeric(difftime(max(data.ajn$`Claim Date`), min(data.ajn$`Claim Date`), units = "days"))
days
## [1] 1412.413
rlgpd(Sub.Amt, u, fit.gp$estimate[1], fit.gp$estimate[2], days)
