Package 'mbbefd'

Maxwell Boltzmann Bose Einstein Fermi Dirac Distribution and Destruction Rate Modelling

Description

The idea of this package emerged in 2013 from G.A. Spedicato who at this time worked in the area of quantitative risk assessment. In 2015, M. Gesmann and C. Dutang joined the project. This project is hosted at github.

This package contains the core functions for the two parametrizations of the MBBEFD distribution (distribution function, density, quantile functions, random generation, aka d, p, q, r) as well as MBBEFD exposure curve (ec) and raw moments (m).

This package also provides other distributions used for destruction rate modelling, that is the beta, the shifted truncated Pareto and the generalized beta distributions. Due to the presence of a total loss, a one-inflated version of the previous distributions is provided.

The vignette shows code snippets to fit the distribution to empirical data: Exposure rating, destruction rate models and the mbbefd package. Main functions are fitDR for fitting destruction rate modeling, bootDR for bootstrap procedures.

Author(s)

Christophe Dutang (maintainer), Giorgio Spedicato, Markus Gesmann

References

BERNEGGER, STEFAN (1997). The Swiss Re Exposure Curves And The MBBEFD Distribution Class, ASTIN Bulletin, 27(1), pp99-111, doi:10.2143/AST.27.1.563208.

See Also

See mbbefd-distr for the MBBEFD distribution;
swissRe, exposureCurve for exposure curves;
gbeta, stpareto for finite-support distributions;
oidistribution, oibeta, oigbeta, oiunif, oistpareto for one-inflated distributions.

---

Large commercial risks in Asia-Pacific

Description

A completed project by the Insurance Risk and Finance Research Centre (www.IRFRC.com) has assembled a unique dataset from Large Commercial Risk losses in Asia-Pacific (APAC) covering the period 2000-2013. The data was generously contributed by one global reinsurance company and two large Lloyd's syndicates in London. This dataset is the result of the project co-lead by Dr Milidonis (IRFRC and University of Cyprus) and Enrico Biffis (Imperial College Business School), which can be referred to as the IRFRC LCR Dataset.

As expected, the dataset is fully anonymized, as the LCR losses are aggregated along a few dimensions. First, data is categorized based on the World Bank's economic development classification. This means that losses either come from developed or developing countries. The second dimension used to aggregate the data is the time period covered. Data is grouped into (at least) two time-periods: the period before and after the 2008 crisis.

A large commercial risk (LCR) is defined as a loss caused by man-made risks (e.g. fire, explosion, etc.). We exclude natural catastrophe events, and started by focusing on claims that made the data provider incur a loss amount of at least EUR 1 million. We then extended our dataset to include claims leading to loss amounts smaller that EUR 1 million. Given time constraints, we only partially extended loss data by obtaining FGU losses larger than EUR 140k. One should note that any selection bias arising from the data collection exercise is driven by both data quality and reliability. Based on our experience, the latter two attributes are homogeneous across developed and developing countries APAC claims.

For further details, see the technical report: Benedetti, Biffis and Milidonis (2015a).

Usage

data(asiacomrisk)

Format

asiacomrisk contains 7 columns:

Period

A character string for the period: "2000-2003", "2004-2008", "2009-2010", "2011-2013".

FGU

From the Ground Up Loss (USD).

TIV

Total Insurable Value (TIV) replaced with Total Sum Insured (TSI) when the TIV is not available (USD).

CountryStatus

A character string for the country status: "Developped", "Emerging".

Usage

A character string for the type of exposure hit by the loss: "Commercial", "Energy", "Manufacturing", "Misc.", "Residential".

SubUsage

A character string for a precise type of exposure hit by the loss: "Commercial", "Energy", "General industry", "Metals/Mines/Chemicals", "Misc.", "Residential", "Utility".

DR

A numeric for the destruction rate (FGU divided TIV capped to 1).

References

Benedetti, D., Biffis, E., and Milidonis, A. (2015a). Large Commercial Risks (LCR) in Insurance: Focus on Asia-Pacific, Insurance Risk and Finance Research Centre Technical report.

Benedetti, D., Biffis, E., and Milidonis, A. (2015b). Large Commercial Exposures and Tail Risk: Evidence from the Asia-Pacific Property and Casualty Insurance Market, Working paper.

Chavez-Demoulin, V., Embrechts, P., and Hofert, M. (2015). An extreme value approach for modeling operational risk losses depending on covariates, The Journal of Risk and Insurance.

Examples

# (1) load of data
#
data(asiacomrisk)
dim(asiacomrisk)

# (2) basic boxplots
#

asiacomrisk
boxplot(DR ~ Usage, data=asiacomrisk)
boxplot(DR ~ SubUsage, data=asiacomrisk)
boxplot(DR ~ Period, data=asiacomrisk)
boxplot(DR ~ CountryStatus, data=asiacomrisk)

---

AON Re Belgian dataset

Description

The dataset was collected by the reinsurance broker AON Re Belgium and comprise 1,823 fire losses for which the building type and the sum insured are available.

Usage

data(beaonre)

Format

beaonre contains three columns and 1823 rows:

BuildType

The building type either A, B, C, D, E or F.

ClaimCost

The loss amount in thousand of Danish Krone (DKK).

SumInsured

The sum insured in thousand of Danish Krone (DKK).

References

Dataset used in Beirlant, Dierckx, Goegebeur and Matthys (1999), Tail index estimation and an exponential regression model, Extremes 2, 177-200, doi:10.1023/A:1009975020370.

Examples

# (1) load of data
#
data(beaonre)

# (2) plot and description of data
#

boxplot(ClaimCost ~ BuildType, data=beaonre, log="y", 
xlab="Building type", ylab="Claim size", main="AON Re Belgium data")

---

Bootstrap simulation of destruction rate models

Description

Uses parametric or nonparametric bootstrap resampling in order to simulate uncertainty in the parameters of the distribution fitted to destruction rate data. Generic methods are print, plot, summary.

Usage

bootDR(f, bootmethod="param", niter=1001, silent=TRUE,
  parallel = c("no", "snow", "multicore"), ncpus)

Arguments

f	An object of class "fitDR", output of the fitDR function.
bootmethod	A character string coding for the type of resampling : "param" for a parametric resampling and "nonparam" for a nonparametric resampling of data.
niter	The number of samples drawn by bootstrap.
silent	A logical to remove or show warnings and errors when bootstraping.
parallel	The type of parallel operation to be used, "snow" or "multicore" (the second one not being available on Windows), or "no" if no parallel operation.
ncpus	Number of processes to be used in parallel operation : typically one would fix it to the number of available CPUs.

Details

Samples are drawn by parametric bootstrap (resampling from the distribution fitted by fitDR) or nonparametric bootstrap (resampling with replacement from the data set). On each bootstrap sample the estimation process is used to estimate bootstrapped values of parameters. When that function fails to converge, NA values are returned. Medians and 2.5 and 97.5 percentiles are computed by removing NA values.

This method returns an object of class "bootDR" inheriting from the "bootdist" class. Therefore the following generic methods are defined: print, plot, summary.

Value

bootDR returns an object of class "bootDR" inheriting from the "bootdist" class. That is a list with 6 components,

estim	a data frame containing the bootstrapped values of parameters.
converg	a vector containing the codes for convergence obtained if an iterative method is used to estimate parameters on each bootstraped data set (and 0 if a closed formula is used).
method	A character string coding for the type of resampling : "param" for a parametric resampling and "nonparam" for a nonparametric resampling.
nbboot	The number of samples drawn by bootstrap.
CI	bootstrap medians and 95 percent confidence percentile intervals of parameters.
fitpart	The object of class "fitDR" on which the bootstrap procedure was applied.

Generic functions:

print

The print of a "bootDR" object shows the bootstrap parameter estimates. If inferior to the whole number of bootstrap iterations, the number of iterations for which the estimation converges is also printed.

summary

The summary provides the median and 2.5 and 97.5 percentiles of each parameter. If inferior to the whole number of bootstrap iterations, the number of iterations for which the estimation converges is also printed in the summary.

plot

The plot shows the bootstrap estimates with stripchart function for univariate parameters and plot function for multivariate parameters.

density

density computes the empirical density of "bootDR" objects using the density function (with Gaussian kernel by default). It returns an object of class density.bootdist for which print and plot methods are provided.

Author(s)

Christophe Dutang

References

Cullen AC and Frey HC (1999), Probabilistic techniques in exposure assessment. Plenum Press, USA, pp. 181-241.

Delignette-Muller ML and Dutang C (2015), fitdistrplus: An R Package for Fitting Distributions. Journal of Statistical Software, 64(4), 1-34, doi:10.18637/jss.v064.i04.

See Also

See mledist for details on parameter estimation. See bootdist for details on generic function. See fitDR for estimation procedures.

Examples

## Not run: 
n <- 1e3
set.seed(12345)
x <- roigbeta(n, 3, 2, 5/2, 1/6)

f1 <- fitDR(x, "oigbeta", method="mle") 
b1 <- bootDR(f1, niter=100, silent=TRUE)
summary(b1)
plot(b1)

## End(Not run)

---

Empirical Exposure Curve Function

Description

Compute an empirical exposure curve function, with several methods for plotting, printing, computing with such an object.

Usage

eecf(x)


## S3 method for class 'eecf'
plot(x, ..., ylab="Gn(x)", do.points=TRUE, 
     col.01line = "gray70", pch = 19, main=NULL, ylim=NULL, 
     add=FALSE)

## S3 method for class 'eecf'
lines(x, ...)

## S3 method for class 'eecf'
print(x, digits= getOption("digits") - 2, ...)

## S3 method for class 'eecf'
summary(object, ...)

Arguments

x, object	numeric vector of the observations for eecf; for the methods, an object of class "eecf".
...	arguments to be passed to subsequent methods, e.g., to the plot method.
ylab	label for the y-axis.
do.points	logical; if TRUE, also draw points at the (xlim restricted) knot locations.
col.01line	numeric or character specifying the color of the horizontal lines at y = 0 and 1, see colors.
pch	plotting character.
main	main title.
ylim	the y limits of the plot.
add	logical; if TRUE add to an already existing plot.
digits	number of significant digits to use, see print.

Details

Compute a continuous empirical exposure curve and returns an object of class "eecf" similar to what an object returned by ecdf.

Value

For eecf, a function of class "eecf", inheriting from the "function" class.

For the summary method, a summary of the knots of object with a "header" attribute.

Author(s)

Dutang Christophe

See Also

exposureCurve, ecdf.

Examples

x <- c(0.4756816, 0.1594636, 0.1913558, 0.2387725, 0.1135414, 0.7775612,
  0.6858736, 0.4340655, 0.3181558, 0.1134244)

#print
eecf(x)

#summary
summary(eecf(x))

#plot
plot(eecf(x))

#lines
lines(eecf(x[1:4]), col="red")

---

Empirical total loss

Description

Compute the empirical total loss.

Usage

etl(x, na.rm=FALSE)

Arguments

x	numeric vector of the observations.
na.rm	a logical value indicating whether NA values should be stripped before the computation proceeds.

Details

Compute the empirical total loss defined as the proportion of full destruction rates, that is observations that equal 1.

Value

A numeric value or a vector.

Author(s)

Dutang Christophe

Examples

x <- c(1, 0.000495134903027804, 0.787229130724068, 0.71154311082138, 
0.0669802789251427, 0.310872967333683, 1, 1, 1, 1, 0.162030982251957, 
1, 1, 0.322530106394859, 1, 1, 1, 0.60805410798081, 0.660941675188664, 1)

#empirical total loss (true value is 1/2)
etl(x)

---

Exposure curves for the beta and the uniform distributions.

Description

An exposure curve is defined between x between 0 and 1 and represents the ratio of the limited expected value to unlimited expected value.

Usage

ecbeta(x, shape1, shape2)
ecunif(x, min = 0, max =1)

Arguments

x	x value, percentage of damage to total loss
shape1, shape2	parameters for the beta distribution.
min, max	parameters for the uniform distribution.

Details

ecbeta, ecunif is the theoretical exposure curve function for beta and uniform distribution.

Value

A numeric value

Author(s)

Giorgio Spedicato, Christophe Dutang

References

BERNEGGER, STEFAN (1997). The Swiss Re Exposure Curves And The MBBEFD Distribution Class, ASTIN Bulletin, 27(1), pp99-111, doi:10.2143/AST.27.1.563208.

See Also

ecmbbefd and ecMBBEFD are implemented in mbbefd-distr. See also Uniform, Beta, swissRe.

Examples

x <- 0.2
ecbeta(x, 2, 3)
ecunif(x)

---

Fit of destruction rate models

Description

Fit of univariate distributions to destruction rate data by maximum likelihood (mle), moment matching (mme), quantile matching (qme) or maximizing goodness-of-fit estimation (mge). The latter is also known as minimizing distance estimation. Generic methods are print, plot, summary, quantile, logLik, vcov and coef. For bootstrap, use bootDR.

Usage

fitDR(x, dist, method="mle", start=NULL, optim.method="default", 
  control = list(trace = 0), ...)

Arguments

x	A numeric vector.
dist	A character string "name" naming a distribution among "oiunif", "oistpareto", "oibeta", "oigbeta", "mbbefd", "MBBEFD".
method	A character string coding for the fitting method: "mle" for 'maximum likelihood estimation', "tlmme" for 'total-loss-moment matching estimation'.
start	A named list giving the initial values of parameters of the named distribution or a function of data computing initial values and returning a named list. This argument may be omitted (default) for some distributions for which reasonable starting values are computed (see the 'details' section of mledist).
optim.method	"default" or an optimization method to pass to optim or constrOptim.nl.
control	a control argument passed to optim and/or constrOptim.nl both control parameters can be set. The trace level control$trace is decreased by one when passed to those optimization functions, i.e., control=list(trace=1) displays high-level traces of fitDR() but not of optimization functions; control=list(trace=2) or higher add traces from optimization functions.
...	Further arguments to be passed to "fitdist" when method != "tlmme". See fitdist for details on parameter estimation.

Details

The fitted distribution (dist) has its d, p, q, r functions defined in the man page: oiunif, oistpareto, oibeta, oigbeta, mbbefd, MBBEFD.

The two possible fitting methods are described below:

When method="mle"

Maximum likelihood estimation consists in maximizing the log-likelihood. A numerical optimization is carried out in mledist via optim to find the best values (see mledist for details). For one-inflated distributions, the probability parameter is estimated by a closed-form formula and other parameters use a two-optimization procedures.

When method="tlmme"

Total loss and moment matching estimation consists in equalizing theoretical and empirical total loss as well as theoretical and empirical moments. The theoretical and the empirical moments are matched numerically, by minimization of the sum of squared differences between observed and theoretical quantities via constrOptim.nl as in mmedist.

For one-inflated distributions, by default, direct optimization of the log-likelihood (or other criteria depending of the chosen method) is performed using optim, with the "L-BFGS-B" method for distributions characterized by more than one parameter and the "Brent" method for distributions characterized by only one parameter. Note that when errors are raised by optim, it's a good idea to start by adding traces during the optimization process by adding control=list(trace=2).

A pre-fitting process is carried out for the following distributions "mbbefd", "MBBEFD" and "oigbeta" before the main optimization.

The estimation process is carried out via fitdist from the fitdistrplus package and the output object will inherit from the "fitdist" class. Therefore, the following generic methods are available print, plot, summary, quantile, logLik, vcov and coef.

Value

fitDR returns an object of class "fitDR" inheriting from the "fitdist" class. That is a list with the following components:

estimate	the parameter estimates.
method	the character string coding for the fitting method : "mle" for 'maximum likelihood estimation', "tlmme" for 'matching total loss moment estimation'.
sd	the estimated standard errors, NA if numerically not computable or NULL if not available.
cor	the estimated correlation matrix, NA if numerically not computable or NULL if not available.
vcov	the estimated variance-covariance matrix, NULL if not available.
loglik	the log-likelihood.
aic	the Akaike information criterion.
bic	the the so-called BIC or SBC (Schwarz Bayesian criterion).
n	the length of the data set.
data	the data set.
distname	the name of the distribution.
fix.arg	the named list giving the values of parameters of the named distribution that must be kept fixed rather than estimated by maximum likelihood or NULL if there are no such parameters.
fix.arg.fun	the function used to set the value of fix.arg or NULL.
discrete	the input argument or the automatic definition by the function to be passed to functions gofstat, plotdist and cdfcomp.
dots	further arguments used in fitdist or NULL otherwise.
convergence	the convergence code: 0 indicates successful completion, for positive values see optim or constrOptim.nl.
discrete	always FALSE.
weights	the vector of weigths used in the estimation process or NULL.
optim.method	the optimization method used in fitDR and bootdist.
control	the control parameters used in fitDR and bootdist.

Generic functions:

print

The print of a "fitDR" object shows few traces about the fitting method and the fitted distribution.

summary

The summary provides the parameter estimates of the fitted distribution, the log-likelihood, AIC and BIC statistics and when the maximum likelihood is used, the standard errors of the parameter estimates and the correlation matrix between parameter estimates.

plot

The plot of an object of class "fitDR"returned by fitdist uses the function plotdist.

*comp

An object of class "fitdist" or a list of objects of class "fitDR" corresponding to various fits using the same data set may also be plotted using a cdf plot (function cdfcomp), a density plot(function denscomp), a density Q-Q plot (function qqcomp), or a P-P plot (function ppcomp). There is also empirical exposure curve plot (eecf) available via (function eccomp)

logLik

Extracts the estimated log-likelihood from the "fitDR" object.

vcov

Extracts the estimated var-covariance matrix from the "fitDR" object (only available when method = "mle").

coef

Extracts the fitted coefficients from the "fitDR" object.

bootDR

Computes bootstrap procedures dedicated to "fitDR" object as in bootdist.

Author(s)

Christophe Dutang.

References

Cullen AC and Frey HC (1999), Probabilistic techniques in exposure assessment. Plenum Press, USA, pp. 81-155.

Venables WN and Ripley BD (2002), Modern applied statistics with S. Springer, New York, pp. 435-446, doi:10.1007/978-0-387-21706-2.

Vose D (2000), Risk analysis, a quantitative guide. John Wiley & Sons Ltd, Chischester, England, pp. 99-143.

Delignette-Muller ML and Dutang C (2015), fitdistrplus: An R Package for Fitting Distributions. Journal of Statistical Software, 64(4), 1-34, doi:10.18637/jss.v064.i04.

See Also

See mledist, mmedist, for details on parameter estimation. See gofstat for goodness-of-fit statistics. See plotdist, graphcomp for graphs. See bootDR for bootstrap procedures See optim for base R optimization procedures. See quantile.fitdist, another generic function, which calculates quantiles from the fitted distribution. See quantile for base R quantile computation.

Examples

# (1) fit of a one-inflated beta distribution by maximum likelihood estimation
#
n <- 500
set.seed(12345)
x <- roibeta(n, 3, 2, 1/6)

f1 <- fitDR(x, "oibeta", method="mle")
summary(f1)
coef(f1)
cdfcomp(f1)
eccomp(f1)

plot(bootdist(f1, niter=11), enhance=TRUE, trueval=c(3, 2, 1/6))

---

Get a parameter known g and b

Description

g2a returns the a parameter known g and b

Usage

g2a(g, b)

Arguments

g	the g parameter
b	the b parameter

Value

a real value

See Also

mbbefd-distr.

Examples

g2a(10,2)

---

The generalized Beta of the first kind Distribution

Description

Density, distribution function, quantile function and random generation for the GB1 distribution with parameters shape0, shape1 and shape2.

Usage

dgbeta(x, shape0, shape1, shape2, log = FALSE)
pgbeta(q, shape0, shape1, shape2, lower.tail = TRUE, log.p = FALSE)
qgbeta(p, shape0, shape1, shape2, lower.tail = TRUE, log.p = FALSE)
rgbeta(n, shape0, shape1, shape2)
ecgbeta(x, shape0, shape1, shape2)
mgbeta(order, shape0, shape1, shape2)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
shape0, shape1, shape2	positive parameters of the GB1 distribution.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
order	order of the raw moment.

Details

The GB1 distribution with parameters shape0 $= g$, shape1 $= a$ and shape2 $= b$ has density

$f(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}{x}^{a/g-1} {(1-x^{1/g})}^{b-1}/g$

for $a,b,g > 0$ and $0 \le x \le 1$ where the boundary values at $x=0$ or $x=1$ are defined as by continuity (as limits).

Value

dgbeta gives the density, pgbeta the distribution function, qgbeta the quantile function, and rgbeta generates random deviates.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language, Wadsworth & Brooks/Cole, doi:10.1201/9781351074988.

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. Chapter 6: Gamma and Related Functions.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, especially Chapter 25. Wiley, New York, doi:10.1080/00224065.1996.11979675.

See Also

Distributions for other standard distributions.

Examples

#density
curve(dgbeta(x, 3, 2, 3))

#cdf
curve(pgbeta(x, 3, 2, 3))

---

Graphical comparison of multiple fitted distributions

Description

eccomp plots the empirical exposure curve distribution against fitted exposure curve functions.

Usage

eccomp(ft, xlim, ylim, main, xlab, ylab, do.points=TRUE,
                    datapch, datacol, fitlty, fitcol, addlegend = TRUE, 
                   legendtext, xlegend = "bottomright", 
                    ylegend = NULL, ...)

Arguments

ft	One "DR" object or a list of objects of class "DR".
xlim	The $x$-limits of the plot.
ylim	The $y$-limits of the plot.
main	A main title for the plot, see also title.
xlab	A label for the $x$-axis, defaults to a description of x.
ylab	A label for the $y$-axis, defaults to a description of y.
datapch	An integer specifying a symbol to be used in plotting data points, see also points.
datacol	A specification of the color to be used in plotting data points.
fitcol	A (vector of) color(s) to plot fitted distributions. If there are fewer colors than fits they are recycled in the standard fashion.
fitlty	A (vector of) line type(s) to plot fitted distributions/densities. If there are fewer colors than fits they are recycled in the standard fashion. See also par.
addlegend	If TRUE, a legend is added to the plot.
legendtext	A character or expression vector of length $\geq 1$ to appear in the legend, see also legend.
xlegend, ylegend	The $x$ and $y$ co-ordinates to be used to position the legend. They can be specified by keyword or in any way which is accepted by 'xy.coords': see legend for details.
do.points	logical; if TRUE, also draw points at the x-locations. Default is true. For large dataset (n > 1e4), do.points is ignored and no point is drawn.
...	Further graphical arguments passed to graphical functions used in cdfcomp, denscomp, ppcomp and qqcomp.

Details

eccomp provides a exposure curve plot of each fitted distribution along with the eecf.

By default a legend is added to these plots. Many graphical arguments are optional, dedicated to personalize the plots, and fixed to default values if omitted.

Author(s)

Christophe Dutang.

See Also

See plot, legend, eecf.

Examples

# (1)

---

Italian grade scores

Description

This dataset contains scores of an university admission test. The total score is subdivided into four areas (Italian, English, abstract reasoning, science). Each subitem can have a point of pass at the end.

Usage

data(itagradescore)

Format

itagradescore contains 10 columns:

Number

a numeric for the record number.

ID

a factor for the identification code.

Correct

A score of correct answers.

Wrong

A score of wrong answers.

Null

A score of null answers.

ItalianLanguage

A score for the Italian language test.

EnglishLanguage

A score for the English language test.

LogicalReasoning

A score for the logic test.

Science

A score for the science test.

TotalScore

The sum of the four scores (i.e. four previous columns).

Source

Internal

Examples

# (1) load of data
#
data(itagradescore)
dim(itagradescore)

---

General Liability Claims

Description

The lossalae is a data frame of 1500 rows and 4 columns containing 1,500 general liability claims randomly chosen from late settlement lags and were provided by Insurance Services Office, Inc. Each claim consists of an indemnity payment (the loss, X1) and an allocated loss adjustment expense (ALAE). ALAE are types of insurance company expenses that are specifically attributable to the settlement of individual claims such as lawyers' fees and claims investigation expenses. The third column is the underwriting limit of the policy and and the fourth column indicates a censored observation.

Usage

data(lossalaefull)

Format

lossalaefull contains four columns:

Loss

A numeric vector containing the indemnity payments (USD).

ALAE

A numeric vector containing the allocated loss adjustment expenses (USD).

Limit

A numeric vector containing the policy limit (USD).

Censored

A binary indicating that the payments are capped to their policy limit (USD).

Source

Frees, E. W. and Valdez, E. A. (1998) Understanding relationships using copulas. North American Actuarial Journal, 2, 1–15, doi:10.1080/10920277.1998.10595749.

References

Klugman, S. A. and Parsa, R. (1999) Fitting bivariate loss distributions with copulas. Insurance: Mathematics and Economics, 24, 139–148, doi:10.1016/S0167-6687(98)00039-0.

Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. L. (2004) Statistics of Extremes: Theory and Applications., Chichester, England: John Wiley and Sons, doi:10.1002/0470012382.

Cebrian, A.C., Denuit, M. and Lambert, P. (2003). Analysis of bivariate tail dependence using extreme value copulas: An application to the SOA medical large claims database, Belgian Actuarial Bulletin, Vol. 3, No. 1, https://dial.uclouvain.be/pr/boreal/object/boreal:17222.

Examples

# (1) load of data
#
data(lossalaefull)

---

The MBBEFD distribution (two parametrizations)

Description

These functions perform probabilistic analysis as well as random sampling on the MBBEFD distribution: the 1st parametrization MBBEFD(a,b) is implemented in <d,p,q,r>mbbefd, the 2nd parametrization MBBEFD(g,b) is implemented in <d,p,q,r>MBBEFD. We also provide raw moments, exposure curve function and total loss.

Usage

dmbbefd(x, a, b, log=FALSE)
pmbbefd(q, a, b, lower.tail = TRUE, log.p = FALSE)
qmbbefd(p, a, b, lower.tail = TRUE, log.p = FALSE)
rmbbefd(n, a, b)
ecmbbefd(x, a, b)
mmbbefd(order, a, b)
tlmbbefd(a, b)

dMBBEFD(x, g, b, log=FALSE)
pMBBEFD(q, g, b, lower.tail = TRUE, log.p = FALSE)
qMBBEFD(p, g, b, lower.tail = TRUE, log.p = FALSE)
rMBBEFD(n, g, b)
ecMBBEFD(x, g, b)
mMBBEFD(order, g, b)
tlMBBEFD(g, b)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is take to be the number required.
a, b, g	shape parameters. For .mbbefd functions, g is computed from a.
order	order of the raw moment.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X <= x]$, otherwise, $P[X> x]$.

Details

it shall be remebered that $g=\frac{1}{p_1}=\frac{a+b}{\left(a+1\right)*b}$.

Value

A numeric value or a vector.

Author(s)

Giorgio Spedicato, Dutang Christophe

References

BERNEGGER, STEFAN (1997). The Swiss Re Exposure Curves And The MBBEFD Distribution Class, ASTIN Bulletin, 27(1), pp99-111, doi:10.2143/AST.27.1.563208.

See Also

swissRe, exposureCurve.

Examples

#1st parametrization
#
aPar=0.2
bPar=0.04
rmbbefd(n=10,a=aPar,b=bPar) #for random generation
qmbbefd(p=0.7,a=aPar,b=bPar) #for quantiles
dmbbefd(x=0.5,a=aPar,b=bPar) #for density
pmbbefd(q=0.5,a=aPar,b=bPar) #for distribution function

#2nd parametrization
#
gPar=2
bPar=0.04
rMBBEFD(n=10,g=gPar,b=bPar) #for random generation
qMBBEFD(p=0.7,g=gPar,b=bPar) #for quantiles
dMBBEFD(x=0.5,g=gPar,b=bPar) #for density
pMBBEFD(q=0.5,g=gPar,b=bPar) #for distribution function

---

One-inflated beta distribution

Description

These functions perform probabilistic analysis as well as random sampling on one-inflated beta distribution.

Usage

doibeta(x, shape1, shape2, p1, ncp=0, log=FALSE)
poibeta(q, shape1, shape2, p1, ncp=0, lower.tail = TRUE, log.p = FALSE)
qoibeta(p, shape1, shape2, p1, ncp=0, lower.tail = TRUE, log.p = FALSE)
roibeta(n, shape1, shape2, p1, ncp=0)
ecoibeta(x, shape1, shape2, p1, ncp=0)
moibeta(order, shape1, shape2, p1, ncp=0)
tloibeta(shape1, shape2, p1, ncp=0)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is take to be the number required.
p1, shape1, shape2, ncp	parameters.
order	order of the raw moment.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X <= x]$, otherwise, $P[X> x]$.

Details

d,p,q,ec,m,tl-oibeta functions computes the density function, the distribution function, the quantile function, the exposure curve function, raw moments and total loss of the one-inflated beta distribution. roibeta generates random variates of this distribution.

Value

A numeric value or a vector.

Author(s)

Dutang Christophe

See Also

mbbefd-distr and oidistribution.

Examples

#density
curve(doibeta(x, 3, 2, 1/3), n=200)

#cdf
curve(poibeta(x, 3, 2, 1/3), n=200)

---

One-inflated distributions

Description

These functions perform probabilistic analysis as well as random sampling on one-inflated distributions.

Usage

doifun(x, dfun, p1, log=FALSE, ...)
poifun(q, pfun, p1, lower.tail = TRUE, log.p = FALSE, ...)
qoifun(p, qfun, p1, lower.tail = TRUE, log.p = FALSE, ...)
roifun(n, rfun, p1, ...)
ecoifun(x, ecfun, mfun, p1, ...)
moifun(order, mfun, p1, ...)
tloifun(p1, ...)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is take to be the number required.
dfun, pfun, qfun, rfun	d, p, q, r functions of the original distribution.
p1	parameter for the probability at x=1.
ecfun, mfun	exposure curve and moment functions which should have arguments x, ... and order, ... respectively.
order	order of the raw moment.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X <= x]$, otherwise, $P[X> x]$.
...	further arguments to pass to dfun, pfun, qfun, rfun, ecfun, mfun.

Details

d,p,q,ec,m,tl functions of oifun computes the density function, the distribution function, the quantile function, the exposure curve function, raw moments and total loss of an one-inflated distribution of an original distribution specified by d,p,q,ec,m-fun. roifun generates random variates of the resulting distribution.

Value

A numeric value or a vector.

Author(s)

Dutang Christophe

See Also

oibeta, oiunif, oistpareto and oidistribution.

---

One-inflated generalized beta of the first kind (GB1)) distribution

Description

These functions perform probabilistic analysis as well as random sampling on one-inflated GB1 distribution.

Usage

doigbeta(x, shape0, shape1, shape2, p1, log=FALSE)
poigbeta(q, shape0, shape1, shape2, p1, lower.tail = TRUE, log.p = FALSE)
qoigbeta(p, shape0, shape1, shape2, p1, lower.tail = TRUE, log.p = FALSE)
roigbeta(n, shape0, shape1, shape2, p1)
ecoigbeta(x, shape0, shape1, shape2, p1)
moigbeta(order, shape0, shape1, shape2, p1)
tloigbeta(shape0, shape1, shape2, p1)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is take to be the number required.
p1, shape0, shape1, shape2	shape parameters.
order	order of the raw moment.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X <= x]$, otherwise, $P[X> x]$.

Details

d,p,q,ec,m,tl-oigbeta functions computes the density function, the distribution function, the quantile function, the exposure curve function, raw moments and total loss of the one-inflated GB1 distribution. roigbeta generates random variates of this distribution.

Value

A numeric value or a vector.

Author(s)

Dutang Christophe

See Also

mbbefd-distr and oidistribution.

Examples

#density
curve(doigbeta(x, 3, 2, 3, 1/3), n=200)

#cdf
curve(poigbeta(x, 3, 2, 3, 1/3), n=200)

---

One-inflated shifted truncated pareto distribution

Description

These functions perform probabilistic analysis as well as random sampling on one-inflated shifted truncated pareto distribution.

Usage

doistpareto(x, a, p1, log=FALSE)
poistpareto(q, a, p1, lower.tail = TRUE, log.p = FALSE)
qoistpareto(p, a, p1, lower.tail = TRUE, log.p = FALSE)
roistpareto(n, a, p1)
ecoistpareto(x, a, p1)
moistpareto(order, a, p1)
tloistpareto(a, p1)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is take to be the number required.
a, p1	parameters.
order	order of the raw moment.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X <= x]$, otherwise, $P[X> x]$.

Details

d,p,q,ec,m,tl-oistpareto functions computes the density function, the distribution function, the quantile function, the exposure curve function, raw moments and total loss of the one-inflated shifted truncated pareto distribution. roistpareto generates random variates of this distribution.

Value

A numeric value or a vector.

Author(s)

Dutang Christophe

See Also

mbbefd-distr and oidistribution.

Examples

#density
curve(doistpareto(x, 2, 1/3), n=200)

#cdf
curve(poistpareto(x, 2, 1/3), n=200)

---

One-inflated uniform distribution

Description

These functions perform probabilistic analysis as well as random sampling on one-inflated uniform distribution.

Usage

doiunif(x, p1, log=FALSE)
poiunif(q, p1, lower.tail = TRUE, log.p = FALSE)
qoiunif(p, p1, lower.tail = TRUE, log.p = FALSE)
roiunif(n, p1)
ecoiunif(x, p1)
moiunif(order, p1)
tloiunif(p1)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is take to be the number required.
p1	parameter.
order	order of the raw moment.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X <= x]$, otherwise, $P[X> x]$.

Details

d,p,q,ec,m,tl-oiunif functions computes the density function, the distribution function, the quantile function, the exposure curve function, raw moments and total loss of the one-inflated uniform distribution. roiunif generates random variates of this distribution.

Value

A numeric value or a vector.

Author(s)

Dutang Christophe

See Also

mbbefd-distr and oidistribution.

Examples

#density
curve(doiunif(x, 1/3), n=200, ylim=0:1)

#cdf
curve(poiunif(x, 1/3), n=200)

---

The shifted truncated Pareto distribution

Description

These functions perform probabilistic analysis as well as random sampling on the shifted truncated Pareto distribution.

Usage

dstpareto(x, a, log=FALSE)
pstpareto(q, a, lower.tail = TRUE, log.p = FALSE)
qstpareto(p, a, lower.tail = TRUE, log.p = FALSE)
rstpareto(n, a)
mstpareto(order, a)
ecstpareto(x, a)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is take to be the number required.
order	order of the raw moment.
a	shape parameter.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X <= x]$, otherwise, $P[X> x]$.

Details

The distribution is based on the Pareto 2 truncated at 1. The distribution function is given by $P(X<=x) = (1-(x+1)^(-a))/(1-2^(-a)).$

Value

A numeric value or a vector.

Author(s)

Dutang Christophe

See Also

mbbefd-distr, exposureCurve

Examples

#density
curve(dstpareto(x, 3))

#cdf
curve(pstpareto(x, 3))

---

Swiss Re exposure curve generation function

Description

This function turns out the MBBEFD b and g parameters for the famous Swiss Re (SR) exposure curves.

Usage

swissRe(c)

Arguments

c	A numeric value

Details

The four Swiss Re Y1-Y4 are defined for c=1.5, 2, 3, 4. In addition c=5 coincides with a curve used by Lloyds for industrial risks exposure rating.

Value

A named two dimensional vector

Author(s)

Giorgio Spedicato

References

BERNEGGER, STEFAN (1997). The Swiss Re Exposure Curves And The MBBEFD Distribution Class, ASTIN Bulletin, 27(1), pp99-111, doi:10.2143/AST.27.1.563208.

See Also

mbbefd-distr.

Examples

pars <- swissRe(4)
losses <- rMBBEFD(n=1000,b=pars[1],g=pars[2])
mean(losses)

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