Returns the limited average severity at x of a random variable with an
MBBEFD distribution with parameters g and b.
ecmb(x, g, b, c = NULL, lower.tail = TRUE)A numeric vector containing the values of the exposure curve for the passed
x, b, and g vectors. If lower.tail is FALSE,
the return value is the complement of the exposure curve.
numeric; vector of quantiles.
numeric; (vector of) the g parameter, which is also
the reciprocal of the probability of a maximum loss.
numeric; (vector of) the b parameter.
numeric; (vector of) the optional c parameter. Should
be NULL if g and b are passed. Otherwise,
\(g = e^{(0.78 + 0.12c)c}\) and
\(b = e^{3.1 - 0.15(1+c)c}\).
logical; if TRUE (default), percentages are of the
total loss being less than or equal to x. Otherwise they are the
percentage of total loss greater than x.
Avraham Adler Avraham.Adler@gmail.com
Given random variable \(X\) with an MBBEFD distribution with parameters \(g\) and \(b\), the exposure curve (EC) is defined as the ratio of the limited average severity (LAS) of the variable at \(x\) divided by the unlimited expected severity of the distribution: $$EC(x) = \frac{LAS(x)}{E(X)} = \frac{E(X\wedge x)}{E(X)} = \frac{\int_0^x t f(t) dt + x \int_x^\infty f(t) dt }{\int_0^\infty t f(t) dt}$$
If one considers \(x\) as a policy limit, then the value of
ecmb(x, g, b) is the percentage of the total expected loss which will be
contained within the (reinsurance) policy limit. If lower.tail is
FALSE, the return value is the percentage of total loss which will exceed
the limit.
Bernegger, S. (1997) The Swiss Re Exposure Curves and the MBBEFD Distribution Class. ASTIN Bulletin 27(1), 99--111. tools:::Rd_expr_doi("10.2143/AST.27.1.563208")
dmb and pmb for the density and distribution.
all.equal(ecmb(c(0, 1), 20, 5), c(0, 1))
ecmb(0.25, 100, 34)
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