Description Usage Arguments Details Value Examples
Function to bound the total losses via the Chernoff inequality.
1 2 |
ELT |
Data frame containing two numeric columns. The column |
s |
Scalar or numeric vector containing the total losses of interest. |
t |
Scalar representing the time period of interest. The default value is |
theta |
Scalar containing information about the variance of the Gamma distribution: sd[X] = x * |
cap |
Scalar representing the financial cap on losses for a single event, i.e. the maximum possible loss caused by a single event. The default value is |
nk |
Number of optimisation points. |
verbose |
Logical. If |
Chernoff's inequality states:
Pr(S ≥ s) ≤ inf_{k > 0} e^{-k s} M_S(k)
where M_S(k) is the Moment Generating Function (MGF) of the total loss S.
The fChernoff
function optimises the bound over a fixed set of nk
discrete values.
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.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | data(UShurricane)
# Compress the table to millions of dollars
USh.m <- compressELT(ELT(UShurricane), digits = -6)
EPC.Chernoff <- fChernoff(USh.m, s = 1:40)
EPC.Chernoff
plot(EPC.Chernoff, type = "l", ylim = c(0, 1))
# Assuming the losses follow a Gamma with E[X] = x, and Var[X] = 2 * x
EPC.Chernoff.Gamma <- fChernoff(USh.m, s = 1:40, theta = 2, cap = 5)
EPC.Chernoff.Gamma
plot(EPC.Chernoff.Gamma, type = "l", ylim = c(0, 1))
# Compare the two results:
plot(EPC.Chernoff, type = "l", main = "Exceedance Probability Curve", ylim = c(0, 1))
lines(EPC.Chernoff.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)
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