# thetaHill: Hill estimator In laeken: Estimation of indicators on social exclusion and poverty

## Description

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

## Usage

 `1` ```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.

`paretoTail`, `fitPareto`, `thetaPDC`, `thetaWML`, `thetaISE`, `minAMSE`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```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) ```