A Bootstrap Test for the Probability of Ruin in the Compound Poisson Risk process

Description

This package provides a bootstrap test for the probability of ruin in the classical (compound Poisson) risk process and some procedures for comparing the performance of the bootstrap test and the test using the asymptotic normal approximation.

Details

See the reference for more information.

Author(s)

Benjamin Baumgartner benjamin@baumgrt.com,
Riccardo Gatto gatto@stat.unibe.ch

References

Baumgartner, B. and Gatto, R. (2010) A Bootstrap Test for the Probability of Ruin in the Compound Poisson Risk Process. ASTIN Bulletin, 40(1), pp. 241–255.

Density Estimation of Data in the Unit Interval

Description

This function computes density estimators for densities with the unit interval as support. One example of data with such a density are p-values. Currently, two methods are implemented that differ in the kernel function used for estimation.

Usage

pvaldens(x, bw, rho, method = c("jh", "chen"))

Arguments

Details

Depending on which method is selected, a different kernel function is used for the estimation. Since the support of the estimated function is bounded, those kernel functions are location-dependent.

If method = "jh", a Gaussian copula-based kernel function according to Jones and Henderson (2007) is used. In this case the bandwidth can either be specified directly or as correlation coefficient: if \rho>0 denotes the correlation coefficient and h>0 the bandwidth, then h^2=1-\rho. Note that rho and bw are mutually exclusive.

For method = "chen", the kernel function is based on a beta density, according to Chen (1999).

See the cited articles for more details.

Value

A function with a single vector-valued argument that returns the estimated density at any given point(s).

References

Jones, M. C. and Henderson, D. A. (2007) Kernel-Type Density Estimation on the Unit Interval. Biometrika, 94(4), pp. 977–984.

Chen, S. X. (1999) A Beta Kernel Estimation for Density Functions. Computational Statistics and Data Analysis, 31(2), pp. 131–145.

See Also

density

Examples

require(graphics)

x <- rbeta(100, 2, 5)
fhat <- pvaldens(x, rho = 0.9, method = "jh")

hist(x, freq = FALSE, xlim = c(0, 1))
curve(fhat(x), from = 0, to = 1, add = TRUE, col = 2)
box()

Distance Measures of Empirical Probability Functions

Description

This function provides a framework to evaluate various measures of distance between an empirical distribution (induced by the dataset provided) and a theoretical probability distribution.

Usage

pvaldistance(x, method = c("ks", "cvm"), dist.to = c("uniform"))

Arguments

Details

method = "ks" gives the Kolmogorov-Smirnov distance.

method = "cvm" yields the Cramér-von-Mises criterion (scaled with the sample size).

Value

A positive real number giving the distance measure.

Note

At the moment, dist.to = "uniform" (the uniform distribution on the unit interval) is the only valid option for the theoretical distribution, and hence the members of x have to lie in the unit interval.

See Also

See ks.test for the Kolmogorov-Smirnov test.

Examples

# A sample from the standard uniform distribution
x <- runif(100, 0, 1)

# Distance to uniformity should be small
pvaldistance(x, "ks")
pvaldistance(x, "cvm")

# A sample from the Beta(2, 7) distribution
y <- rbeta(100, 2, 7)

# Distance to uniformity should be much larger here
pvaldistance(y, "ks")
pvaldistance(y, "cvm")

Creating Bootstrap Replications from an Matrix of Observations

Description

This function provides a simple way to create bootstrap replications of a dataset. The replication is either non-parametrical or parametrical (for exponential or logarithmic normal data).

Usage

rpdataboot(x, b, method = c("nonp", "exp", "lnorm"))

Arguments

Details

The input matrix x is supposed to contain (independent) observations in each column. The bootstrap replication take this into account is done column-wise.

Depending on how the boostrap replications are further processed, the boostrap resampling should be done either non-parametrically (method = "nonp") or parametrically.

In the non-parametrical case, the bootstrap replications are samples drawn from the empirical distribution of the original observation, this is equivalent to drawing with replacement.

For the parametrical bootstrap replications there are currently two options: With method = "exp" each bootstrap replication is a vector simulated from an exponential distribution function whose parameter is estimated by the original observation. For method = "lnorm" the resampling is done by simulating from a logarithmic normal distribution whose log-mean and log-variance are estimated from the original observation.

Value

An array of dimension c(dim(x), b) containing column-wise bootstrap replications of x

Note

NA's are propagated consistently. More precisely, only the non-NA values undergo the resampling and thus, missing values remain unchanged in the bootstrap replications.

See Also

rpdataconv for creating a suitable data matrix from a list of observation vectors, and rpdatasim for creating such a matrix by simulation.

Examples

# Generate a data matrix of 5 samples with 10 observations each.
x <- matrix(rexp(50), nrow = 10, ncol = 5)

# Create (parametric) bootstrap replications
x.boot.par <- bootruin:::rpdataboot(x, b = 50, method = "exp")

# Create (non-parametric) bootstrap replications
x.boot.nonp <- bootruin:::rpdataboot(x, b = 50, method = "nonp")

Convert a List of Numerics to a Matrix

Description

This function converts a list whose members are numeric vectors (possibly of different length) to a matrix.

Usage

rpdataconv(x)

Arguments

Details

If the list entries do not have the same length, the shorter elements are not recycled, but filled with NA instead.

Value

A matrix whose columns contain the members of x.

Note

For consistency, x can also be a numeric vector or matrix. The return value is then the same as the one of as.matrix.

See Also

See rpdatasim for creating a similar data matrix with simulated data, and rpdataboot for creating boostrap replications of such a data matrix.

Examples

# Gemerate samples of different size from an
# exponential distribution with different parameters
x <- list(rexp(10, 0.2), rexp(7, 0.1), rexp(12, 0.5))

# Write x into a matrix that can be further processed
x.rp <- bootruin:::rpdataconv(x)

Simulating Data, Shaped into a Matrix

Description

This is a wrapper function for simulating data that have the correct structure for further processing with rpdataboot.

Usage

rpdatasim(n, replications, rdist, ...)

Arguments

Value

A numeric matrix with n rows and replications columns.

Note

Typical choices for rdist are rexp or rlnorm.

See Also

See rpdataconv for converting an existing dataset to a matrix, and rpdataboot for creating boostrap replications of such a data matrix.

Examples

# Generate 5 independent samples of size 10 from
# an exponential distribution with mean 10
x <- bootruin:::rpdatasim(n = 10, replications = 5, rexp, rate = 1/10)

A Jackknife Estimator of the Standard Error of the Probability of Ruin

Description

This function computes the jackknife estimator of the standard error of the estimator of the probability of ruin given observed claims.

Usage

rpjack(x, ...)

Arguments

Details

If x is a vector of observed claims, for each element of x the probability of ruin is estimated with said element of x left out. The resulting vector of ruin probabilities has the same length as x, and its standard error, properly rescaled, is used to approximate the standard error of the estimator of the probability of ruin of x.

This procedure is applied column-wise if x is a matrix.

Value

A numeric vector of length ncol(as.matrix(x)).

Note

The calculation of the jackknife standard error can be computationally intensive. In most cases the computation time can be drastically reduced at the price of a slightly lower accuracy, viz. a higher value for the interval argument of ruinprob.

References

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall.

Tukey, J. W. (1958) Bias and Confidence in Not Quite Large Samples. The Annals of Mathematical Statistics, 29(2), p. 614.

Quenouille, M. H. (1956) Note on Bias in Estimation. Biometrika, 43, pp. 353–360.

See Also

ruinprob for valid options that can be used for ....

P-values for the Test of the Probability of Ruin

Description

This function provides p-values for the test of the probability of ruin using one of two different methods.

Usage

rppvalue(x, method = c("bootstrap", "normal"), x.boot)

Arguments

Details

This function is not intended to be used by itself, but rather in combination with rpteststat. Hence, ideally, both x and x.boot stem from a call of the latter.

If method = "bootstrap", then bootstrap p-values are computed. The values of x are compared to those of x.boot. The number of rows of x.boot has to match the number of columns of x (or its length if it is a vector). For most applications, however, x will be a single number and x.boot will be the bootstrap replications of x

For method = "normal" the p-values are computed using the asymptotic normal approximation of the test statistic. x can be a vector, a matrix or an array of numerics.

The elements of x are interpreted as statistics of separate, independent tests, and adjusting the p-values for multiple comparison may be necessary.

Value

A numeric (vector, matrix or array) with the same dimension as x.

Note

If method = "normal", the argument x.boot is not used and a warning is issued if it is still provided.

References

Baumgartner, B. and Gatto, R. (2010) A Bootstrap Test for the Probability of Ruin in the Compound Poisson Risk Process. ASTIN Bulletin, 40(1), pp. 241–255.

See Also

See rpteststat for the computation of the test statistics, ruinprob for computating the probability of ruin, and rpjack for the computation of the standard errors.

p.adjust from the package stats provides methods to adjust p-values for multiple testing.

Computation of the Test Statistic for the Probability of Ruin

Description

This function computes the test statistic for the test of the probability of ruin

Usage

rpteststat(x, x.null, se)

Arguments

Details

This function studentizes (i.e., centers and rescales) the observed probabilities of ruin in a way that they can be further processed by rppvalue. This latter computes p-values for the test with the null hypothesis that the probability of ruin is equal to x.null versus the one-sided alternative that probability of ruin is smaller than x.null

There are two possible scenarios how to use this function:

The first one is to compute the test statistic for vector of estimated probabilities of ruin (where each element corresponds to an independent sample of observed claims) using a pre-determined value of x.null. In this case the output is a matrix with length(x) columns and length(x.null) rows.

The second use is to compute the bootstrap replications of the test statistic. In that case the estimated probabilities of ruin are used for x.null, and the numbers of rows of x has to match the length of x.null, i.e. there is one row of bootstrap replications of the probability of ruin for each sample of claim sizes.

Value

A numeric matrix that contains the test statistic. Its dimension depends on the input, see Details.

Note

The dimensions (or lengths) of x and se have to be the same.

See Also

See ruinprob on how to obtain the estimators of the probabilities of ruin, rpjack on how to get approximate standard errors of the former, and rppvalue on how to compute the p-values of the test described here.

The Probability of Ruin in the Classical Risk Process

Description

This function calculates or estimates the probability of ruin in the classical (compund Poisson) risk process using several different methods.

Usage

ruinprob(x, param.list, compmethod = c("dg", "exp"),
    flmethod = c("nonp", "exp", "lnorm", "custom"),
    reserve, loading, fl = NA, interval = 0.5,
    implementation = c("R", "C"), ...)

Arguments

Details

The classical risk process, also called Cramér-Lundberg risk process, is a stochastic model for an insurer's surplus over time and, for any t\ge0, it is given by

Y_{t} = r_{0} + ct - Z_{t},

where Z_{t} is a compund Poisson process, r_{0} \ge 0 is the initial surplus and c > 0 is the constant premium rate.

This function calculates, approximates or estimates (depending on what options are given) the probability of ruin in the infinite time horizon, i.e. the probability that Y_{t} ever falls below 0.

Currently there are two options for the compmethod argument. If compmethod = "exp", the claims are assumed to be from an exponential distribution. In that case, the probability of ruin is given by

\frac{1}{1 + \beta} \exp\left\{-\frac{\beta}{1+\beta} \frac{r_{0}}{\mu}\right\},

where \mu is the mean claim size (estimated from x) and \beta is the relative security loading.

For compmethod = "dg", the recursive algorithm due to Dufresne and Gerber (1989) is used. In this case, the parameter flmethod determines what cumulative distribution function is used for the discretization. The possible choices are either a non-parametric estimator, parametric estimators for exponential or log-normal claims, or a user-supplied function (in which the argument fl must be specified). See the reference for more details on how this algorithm works.

Value

The estimated or calculated probability of ruin. The shape and dimension of the output depends on the specifics of the claim data x. If x is a vector, the output is a single numeric value. In general, the dimension of the output is one less than that of x. More precisely, if x is an array, then the output value is an array of dimension dim(x)[-1], see the note below.

Note

If x is an array rather than a vector, the function acts as if it was called through apply with MARGIN = 2:length(dim(x))

If an option is given both explicitly and as part of the param.list argument, then the value given explicitly takes precedence. This way the parameter list, saved as a variable, can be reused, but modifications of one or more parameter values are still possible.

References

Dufresne, F. and Gerber, H.-U. (1989) Three Methods to Calculate the Probability of Ruin. ASTIN Bulletin, 19(1), pp. 71–90.

See Also

ruinprob.test

Examples

# Claims have an exponential distribution with mean 10
x <- rexp(10, 0.1)
print(x)

# The estimated probability of ruin
ruinprob(x, reserve = 100, loading = 0.2, interval = 0.25)

# The true probability of ruin of the risk process
ruinprob(
    10, reserve = 100, loading = 0.2,
    flmethod = "exp", compmethod = "exp"
)

A Bootstrap Test for the Probability of Ruin in the Classical Risk Process

Description

This function provides a testing framework for the probability of ruin in the classical, compound Poisson risk process. The test can be performed using the bootstrap method or using normal approximation.

Usage

ruinprob.test(x, prob.null, type = c("bootstrap", "normal"),
    nboot, bootmethod = c("nonp", "exp", "lnorm"), ...)

Arguments

Details

The null hypothesis is that the probability of ruin is equal to prob.null versus the one-sided alternative that probability of ruin is smaller than prob.null.

If type = "bootstrap", a bootstrap test is performed. The arguments nboot and bootmethod have to be specified. bootmethod determines the kind of bootstrap: "nonp" creates the usual nonparametric bootstrap replications, while "exp" and "lnorm" create parametric bootstrap replications, the former assuming exponentially distributed claims, the latter log-normally distributed ones.

type = "normal" makes use of an asymptotic normal approximation. The computations are a lot faster, but from a theoretical point of view the bootstrap method is more accurate, see References.

For details about the necessary and valid arguments that might have to be supplied for ..., see ruinprob.

Value

A list with class "htest" containing the following components:

Note

Using the bootstrap method is computationally intensive. Values for nboot should not be too large, usually numbers between 50 and 200 are reasonable choices.

References

Baumgartner, B. and Gatto, R. (2010) A Bootstrap Test for the Probability of Ruin in the Compound Poisson Risk Process. ASTIN Bulletin, 40(1), pp. 241–255.

See Also

ruinprob

Examples

# Generating a sample of 50 exponentially distributed claims with mean 10
x <- rexp(50, 0.1)

## Not run: 
# Given this sample, test whether the probability of ruin is smaller than
# 0.1 using a bootstrap test with 100 bootstrap replications.
ruinprob.test(
    x = x, prob.null = 0.10, type = "bootstrap",
    loading = 0.2, reserve = 100, interval = 1,
    bootmethod = "nonp", nboot = 100
)

## End(Not run)

# The same test using normal approximation. This is a lot faster.
ruinprob.test(
    x = x, prob.null = 0.15, type = "normal",
    loading = 0.2, reserve = 100, interval = 1
)