Hill estimator
The Hill estimator uses the maximum likelihood principle to estimate the shape parameter of a Pareto distribution.
thetaHill(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. |
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 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
Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution. The Annals of Statistics, 3(5), 1163–1174.
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)
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