The Hill estimator uses the maximum likelihood principle to estimate the shape parameter of a Pareto distribution.
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.
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 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
Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution. The Annals of Statistics, 3(5), 1163–1174.
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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$db090) # using number of observations in tail thetaHill(eusilc$eqIncome, k = ts$k, w = eusilc$db090) # using threshold thetaHill(eusilc$eqIncome, x0 = ts$x0, w = eusilc$db090)
Loading required package: boot Loading required package: MASS  3.965728  3.965728
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