elev: Empirical Limited Expected Value

In actuar: Actuarial Functions and Heavy Tailed Distributions

Empirical Limited Expected Value

Description

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

Usage

elev(x, ...)

## Default S3 method:
elev(x, ...)

## S3 method for class 'grouped.data'
elev(x, ...)

## S3 method for class 'elev'
print(x, digits = getOption("digits") - 2, ...)

## S3 method for class 'elev'
summary(object, ...)

## S3 method for class 'elev'
knots(Fn, ...)

## S3 method for class 'elev'
plot(x, ..., main = NULL, xlab = "x", ylab = "Empirical LEV")

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.

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.

Value

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

Author(s)

Vincent Goulet [e-mail] and Mathieu Pigeon

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)

actuar documentation built on Nov. 8, 2023, 9:06 a.m.