The partial density component (PDC) estimator estimates the shape parameter of a Pareto distribution based on the relative excesses of observations above a certain threshold.
a numeric vector.
the number of observations in the upper tail to which the Pareto distribution is fitted.
the threshold (scale parameter) above which the Pareto distribution is fitted.
an optional numeric vector giving sample weights.
additional arguments to be passed to
x0 of course correspond with each other.
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 needs to be supplied. If both are supplied,
k is used (mainly for back compatibility).
The PDC estimator minimizes the integrated squared error (ISE) criterion with
an incomplete density mixture model. The minimization is carried out using
nlm. By default, the starting value is obtained with
the Hill estimator (see
The estimated shape parameter.
x0 for the threshold (scale parameter) of the
Pareto distribution and
w for sample weights were introduced in
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. URL http://www.jstatsoft.org/v54/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.
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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$db090) # using number of observations in tail thetaPDC(eusilc$eqIncome, k = ts$k, w = eusilc$db090) # using threshold thetaPDC(eusilc$eqIncome, x0 = ts$x0, w = eusilc$db090)