laeken (version 0.3.2)

thetaISE: Integrated squared error (ISE) estimator

Description

The integrated squared error (ISE) estimator estimates the shape parameter of a Pareto distribution based on the relative excesses of observations above a certain threshold.

Usage

thetaISE(x, k = NULL, x0 = NULL, w = NULL, ...)

Arguments

x

a numeric vector.

k

the number of observations in the upper tail to which the Pareto distribution is fitted.

x0

the threshold (scale parameter) above which the Pareto distribution is fitted.

w

an optional numeric vector giving sample weights.

...

additional arguments to be passed to optimize (see Details).

Value

• The estimated shape parameter.

Details

The arguments k and x0 of course correspond with each other. If k is supplied, the threshold x0 is estimated with the $n - k$ largest value in x, where $n$ is the number of observations. On the other hand, if the threshold x0 is supplied, k is given by the number of observations in x larger than x0. Therefore, either k or x0 needs to be supplied. If both are supplied, only k is used (mainly for back compatibility). The ISE estimator minimizes the integrated squared error (ISE) criterion with a complete density model. The minimization is carried out using optimize.

References

Vandewalle, B., Beirlant, J., Christmann, A., and Hubert, M. (2007) A robust estimator for the tail index of Pareto-type distributions. Computational Statistics & Data Analysis, 51(12), 6252--6268.

See Also

paretoTail, fitPareto, thetaPDC, thetaHill

Examples

data(eusilc)
# equivalized disposable income is equal for each household
# member, therefore only one household member is taken
eusilc <- eusilc[!duplicated(eusilc$db030),]

# estimate threshold
ts <- paretoScale(eusilc$eqIncome, w = eusilc$db090)

# using number of observations in tail
thetaISE(eusilc$eqIncome, k = ts$k, w = eusilc$db090)

# using threshold
thetaISE(eusilc$eqIncome, x0 = ts$x0, w = eusilc$db090)