fCantelli {tailloss}

Cantelli Bound.

Description

Function to bound the total losses via the Cantelli inequality.

Usage

fCantelli(ELT, s, t = 1, theta = 0, cap = Inf)

Arguments

ELT

Data frame containing two numeric columns. The column Loss contains the expected losses from each single occurrence of event. The column Rate contains the arrival rates of a single occurrence of event.

s

Scalar or numeric vector containing the total losses of interest.

t

Scalar representing the time period of interest. The default value is t = 1.

theta

Scalar containing information about the variance of the Gamma distribution: sd[X] = x * theta. The default value is theta = 0: the loss associated to an event is considered as a constant.

cap

Scalar representing the level of truncation of the Gamma distribution, i.e. the maximum possible loss caused by a single event. The default value is cap = Inf.

Details

Cantelli's inequality states:

\Pr( S \geq s) \leq \frac{\sigma^2}{\sigma^2 + (s - \mu)^2} \quad \mbox{for } s \geq \mu,

where \mu = E[S] and \sigma^2 = Var[S] < \infty are the mean and the variance of the distribution of S.

Value

A numeric matrix, containing the pre-specified losses s in the first column and the upper bound for the exceedance probabilities in the second column.

Examples

data(UShurricane)

# Compress the table to millions of dollars

USh.m <- compressELT(ELT(UShurricane), digits = -6)
EPC.Cantelli  <- fCantelli(USh.m, s = 1:40)
plot(EPC.Cantelli, type = "l", ylim = c(0, 1))
# Assuming the losses follow a Gamma with E[X] = x, and Var[X] = 2 * x
EPC.Cantelli.Gamma  <- fCantelli(USh.m, s = 1:40, theta = 2, cap = 25)
EPC.Cantelli.Gamma
plot(EPC.Cantelli.Gamma, type = "l")
# Compare the two results:
plot(EPC.Cantelli, type = "l", main = "Exceedance Probability Curve", ylim = c(0, 1))
lines(EPC.Cantelli.Gamma, col = 2, lty = 2)
legend("topright", c("Dirac Delta", expression(paste("Gamma(",
alpha[i] == 1 / theta^2, ", ", beta[i] ==1 / (x[i] * theta^2), ")", " cap =", 5))),
lwd = 2, lty = 1:2, col = 1:2)