# Hill estimator

### Description

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

### Usage

1 |

### 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

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)
``` |