Adjustment Coefficient

Description

Returns the adjustment coefficient of a risk process with Gaussian diffusion.

Usage

adjcoef(process)

Arguments

Details

The moment-generating function of the individual claim amounts is required to be contained within the "claiminfo" element of process (see claiminfo). The only exception to this is the case of hypo-exponentially distributed claim amounts.

Value

The adjustment coefficient of the risk process represented by process. If the adjustment coefficient does not exist or cannot be computed for any reason, NULL is returned instead.

See Also

riskproc and claiminfo for more details on how to provide the information necessary to compute the adjustment coefficient.

Examples

## Setting up a risk process with hypo-exponentially distributed claims
myprocess <- riskproc(
    claims   = claiminfo(hypoexp = list(rates = c(1, 10))),
    premium  = 2,
    freq     = 1,
    variance = 0.4
)

## Return the adjustment coefficient
adjcoef(myprocess)

Distribution Information about Individual Claim Amounts

Description

Creates or tests for claim information objects.

Usage

claiminfo(...)
is.claiminfo(x)
is.hypoexp(x)

Arguments

Details

Typicall usages are:

claiminfo(mgf, mgf.d1, mgf.d2, pdf, cdf, mean)

To be completed.

Value

claiminfo returns an object of the class "claiminfo" (see details section).

is.claiminfo returns TRUE if x is a "claiminfo" object, and FALSE otherwise.

is.hypoexp returns TRUE if x is a "claiminfo" object describing hypo-exponentially distributed individual claim amounts (see dhypoexp), and FALSE otherwise. If x is a riskproc object, the function is applied to the "claiminfo" object contained within it.

Note

To be completed.

See Also

riskproc

Examples

## For hypo-exponentially distribution claim amounts
claiminfo(hypoexp = list(rates = c(1, 10)))

## A more complicated example
## Not run: claiminfo()

Hypo-Exponential Distribution

Description

Density, distribution function, quantile function, random generation and moment-generating function (and its first two derivatives) for the hypo-exponential distribution with rates rate.

Usage

dhypoexp(x, rate = 1, log = FALSE)
phypoexp(q, rate = 1, lower.tail = TRUE, log.p = FALSE, tailarea = FALSE)
qhypoexp(p, rate, interval = c(0.0, 1.0e+10))
rhypoexp(n = 1, rate = 1)
mgfhypoexp(x, rate = 1, difforder = 0)

Arguments

Details

The sum of n independent exponentially distributed random variables X_{i} with rate parameters \lambda_{i} has a hypo-exponential distribution with rate vector (\lambda_{1}, \dots, \lambda_{n}).

The hypo-exponential distribution is a generalization of the Erlang distribution (a Gamma distribution with an integer-valued shape parameter) and a special case of the phase-type distribution (see References section).

The quantile function is computed by numeric inversion (using uniroot).

Value

dhypoexp gives the density, phypoexp gives the distribution function (or the integrated tail area distribution function), qhypoexp gives the quantile function, rhypoexp generates random deviates and mgfhypoexp gives the moment-generating function (or its derivative up to the second order).

Note

If length(rate) == 1, dhypoexp, phypoexp and rhypoexp are equivalent to dexp, pexp and rexp with rate parameter rate and should, in fact, be replaced by the latter ones for computation speed.

References

Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, reprinted and corrected.

See Also

dexp, dgamma

Examples

## Random generation
rhypoexp(10, c(3, 5))

## Mean
mu <- mgfhypoexp(0, c(3, 5), difforder = 1)

## Variance
mgfhypoexp(0, c(3, 5), difforder = 2) - mu^2

## Quantile
qhypoexp(0.5, c(3, 5))

Compound Poisson Risk Process with Diffusion

Description

Creates an R object representing a compound Poisson risk process with Gaussian diffusion, which contains some or all information necessary for further processing.

Usage

riskproc(claims, premium, freq, variance)
is.riskproc(x)

Arguments

Details

Given the arguments, most prominently claims, various auxialiary parameters and functions associated with the risk process to be represented are calculated.

Value

riskproc returns an object of the class "riskproc". Internally, this is a list containing various elements (depending on the information provided in the arguments).

is.riskproc returns TRUE if x is a "riskproc" object, and FALSE otherwise.

See Also

claiminfo for more details about passing on information about the distribution of the individual claim amounts.

Examples

## A risk process with hypo-exponentially distributed individual claim amounts
riskproc(
    claims   = claiminfo(hypoexp = list(rates = c(1, 10))),
    premium  = 2,
    freq     = 1,
    variance = 0.4
)

Calculation or Approximation of the Probability of Ruin

Description

This functions provide various approximation methods for the (total) probability of ruin, the probability of ruin due to oscillation and the probability of ruin due to a claim. Exact calculations are possible in the case of hypo-exponentially distrubuted claim amounts.

Usage

ruinprob(process, method = c("saddlepoint", "fft", "bounds", "hypoexp", "lundberg"), ...)
boundsRuinprob(process, interval, maxreserve, richardson = TRUE, use.splines = FALSE)
fftRuinprob(process, interval, maxreserve, n, use.splines = FALSE)
hypoexpRuinprob(process)
saddlepointRuinprob(process, jensen = FALSE, normalize = TRUE)

Arguments

Details

ruinprob is a wrapper function for the other ones given here.

Value

References

Daniels, H. E. (1954) Saddlepoint Approximations in Statistics. Annals of Mathematical Statistics 25(4), pp. 631–650.

Gatto, R. and Mosimann, M. (2012) Four Approaches to Compute the Probability of Ruin in the Compound Poisson Risk Process with Diffusion. Mathematical and Computer Modelling 55(3–4), pp. 1169–1185

Jensen, J. L. (1992) The Modified Signed Likelihood Statistic and Saddlepoint Approximations. Biometrika 79(4), pp. 693–703.

Lugannani, R. and Rice, S. (1980) Saddle Point Approximation for the Distribution of the Sum of Independent Random Variables. Advances in Applied Probability 12(2), pp. 475–490.

See Also

riskproc, claiminfo

Sensitivity of the Value and Tail Value at Ruin

Description

The sensitivities of both the value and the tail value at ruin are defined as their respective derivatives with respect to the probability level.

Usage

sensitivity(process, method = c("saddlepoint", "hypoexp"), ...)
hypoexpSensitivity(process)
saddlepointSensitivity(process, ...)

Arguments

Value

Tail Value at Ruin

Description

The tail value at ruin for a given probability level \varepsilon is defined as the conditional expectation of the maximal aggregate loss given that it is above the value at ruin of level \varepsilon.

Usage

tvaru(process, method = c("saddlepoint", "hypoexp"), ...)
hypoexpTvaru(process)
saddlepointTvaru(process, type = c("tail", "density"), ...)

Arguments

Details

tvaru is a wrapper function for hypoexpTvaru and saddlepointTvaru.

hypoexpTvaru calculates the tail value at ruin in the case of hypo-exponentially distributed claim amounts by numerical integration of the probability of ruin, which can be computed exactly.

saddlepointTvaru uses saddlepoint techniques for the approximation of the tail value at ruin. More precisely, the saddlepoint approximation to the probability of is numerically integrated in the frequency domain, and implicitly also the saddlepoint approximation to the value at ruin (see varu) is used. If type = "tail" the integrand is the probability of ruin (as function in the frequency domain), otherwise (type = "density") it is essentially a re-scaled version of the probability of ruin due to claims. The former requires fewer calculations and seems to produce slightly more accurate results.

Value

A function returning the tail value at ruin of a given probability level is returned.

If method = "saddlepoint" or if saddlepointTvaru is used, the returned function has an additional second argument giving the number of iterations used for the approximation of the value at ruin (i. e., the lower integration limit), see varu.

See Also

varu

Value at Ruin

Description

The value at ruin at a given probability level \varepsilon is defined as the minimal capital that is required in order to have a ruin probability of at most \varepsilon. This is equivalent to the (1-\varepsilon)-quantile of the maximal aggregate loss.

Usage

varu(process, method = c("saddlepoint", "hypoexp"), ...)
hypoexpVaru(process)
saddlepointVaru(process, type = 2)

Arguments

Details

varu is a wrapper function for hypoexpVaru and saddlepointVaru.

hypoexpVaru calculates the value at ruin in the case of hypo-exponentially distributed claim amounts by numerical inversion of the probability of ruin, which can be computed exactly.

saddlepointVaru uses saddlepoint techniques for the approximation of the value at ruin, more specifically, the inversion algorithms provided by Wang (1995). The first one (type = 1) is only given for completeness (or comparison purposes), because, due to repeatedly switching back and forth between the monetary domain the frequency (saddlepoint) domain, it is much slower than the second one (type = 2), which is performed entirely in the frequency domain. Refer to the references given below for more details.

Value

A function returning the value at ruin of a given probability level is returned.

If method = "saddlepoint" or if saddlepointVaru is used, the returned function has an additional second argument giving the number of iterations.

References

Wang, Suojin (1995) One-Step Saddlepoint Approximations for Quantiles. Computational Statistics and Data Analysis 20(1), pp. 65–74.