Moment estimator

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Description

Estimate the shape parameter of a Pareto distribution based on moments.

Usage

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thetaMoment(x, k = NULL, x0 = 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.

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 argument x0 for the threshold (scale parameter) of the Pareto distribution was introduced in version 0.2.

Author(s)

Andreas Alfons and Josef Holzer

References

Dekkers, A.L.M., Einmahl, J.H.J. and de Haan, L. (1989) A moment estimator for the index of an extreme-value distribution. The Annals of Statistics, 17(4), 1833–1855.

See Also

paretoTail, fitPareto

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
thetaMoment(eusilc$eqIncome, k = ts$k)

# using threshold
thetaMoment(eusilc$eqIncome, x0 = ts$x0)

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