actuar (version 3.3-4)

elev: Empirical Limited Expected Value

Description

Compute the empirical limited expected value for individual or grouped data.

Usage

elev(x, ...)
# S3 method for default
elev(x, ...)
# S3 method for grouped.data
elev(x, ...)
# S3 method for elev
print(x, digits = getOption("digits") - 2, ...)
# S3 method for elev
summary(object, ...)
# S3 method for elev
knots(Fn, ...)
# S3 method for elev
plot(x, ..., main = NULL, xlab = "x", ylab = "Empirical LEV")

Value

For elev, a function of class "elev", inheriting from the

"function" class.

Arguments

x

a vector or an object of class "grouped.data" (in which case only the first column of frequencies is used); for the methods, an object of class "elev", typically.

digits

number of significant digits to use, see print.

Fn, object

an R object inheriting from "ogive".

main

main title.

xlab, ylab

labels of x and y axis.

...

arguments to be passed to subsequent methods.

Author

Vincent Goulet and Mathieu Pigeon

Details

The limited expected value (LEV) at \(u\) of a random variable \(X\) is \(E[X \wedge u] = E[\min(X, u)]\). For individual data \(x_1, \dots, x_n\), the empirical LEV \(E_n[X \wedge u]\) is thus $$E_n[X \wedge u] = \frac{1}{n} \left( \sum_{x_j < u} x_j + \sum_{x_j \geq u} u \right).$$

Methods of elev exist for individual data or for grouped data created with grouped.data. The formula in this case is too long to show here. See the reference for details.

References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (1998), Loss Models, From Data to Decisions, Wiley.

See Also

grouped.data to create grouped data objects; stepfun for related documentation (even though the empirical LEV is not a step function).

Examples

data(gdental)
lev <- elev(gdental)
lev
summary(lev)
knots(lev)            # the group boundaries

lev(knots(lev))       # empirical lev at boundaries
lev(c(80, 200, 2000)) # and at other limits

plot(lev, type = "o", pch = 16)