# ecmb: Exposure Curve for the MBBEFD Distribution In MBBEFDLite: Statistical Functions for the Maxwell-Boltzmann-Bose-Einstein-Fermi-Dirac (MBBEFD) Family of Distributions

 ecmb R Documentation

## Exposure Curve for the MBBEFD Distribution

### Description

Returns the limited average severity at x of a random variable with an MBBEFD distribution with parameters g and b.

### Usage

ecmb(x, g, b, c = NULL, lower.tail = TRUE)


### Arguments

 x numeric; vector of quantiles. g numeric; (vector of) the g parameter, which is also the reciprocal of the probability of a maximum loss. b numeric; (vector of) the b parameter. c 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}. lower.tail 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.

### Details

Given random variable X with an MBBEFD distribution with parameters g and b, the exposure curve 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:

LAS(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.

### Value

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.

### References

Bernegger, S. (1997) The Swiss Re Exposure Curves and the MBBEFD Distribution Class. ASTIN Bulletin 27(1), 99–111. \Sexpr[results=rd]{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))