thetaISE: Integrated squared error (ISE) estimator
In laeken: Estimation of Indicators on Social Exclusion and Poverty

Description Usage Arguments Details Value Note Author(s) References See Also Examples

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.

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

`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”).

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 nlm. By default, the starting value is obtained the Hill estimator (see thetaHill). optimize.

The estimated shape parameter.

The arguments x0 for the threshold (scale parameter) of the Pareto distribution and w for sample weights were introduced in version 0.2.

Andreas Alfons and Josef Holzer

A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators from Complex Surveys: The R Package laeken. Journal of Statistical Software, 54(15), 1–25. doi: 10.18637/jss.v054.i15

A. Alfons, M. Templ, P. Filzmoser (2013) Robust estimation of economic indicators from survey samples based on Pareto tail modeling. Journal of the Royal Statistical Society, Series C, 62(2), 271–286.

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.

paretoTail, fitPareto, thetaPDC, thetaHill

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)