Hill estimator

Description

The Hill estimator uses the maximum likelihood principle to estimate the shape parameter of a Pareto distribution.

Usage

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thetaHill(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.

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).

Value

The estimated shape parameter.

Note

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

Author(s)

Andreas Alfons and Josef Holzer

References

Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution. The Annals of Statistics, 3(5), 1163–1174.

See Also

paretoTail, fitPareto, thetaPDC, thetaWML, thetaISE, minAMSE

Examples

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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)

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