# thetaHill: Hill estimator In laeken: Estimation of Indicators on Social Exclusion and Poverty

 thetaHill R Documentation

## Hill estimator

### Description

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

### Usage

``````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`

### Examples

``````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)

``````

laeken documentation built on May 29, 2024, 4:42 a.m.