# Moment estimator

### Description

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

### Usage

1 | ```
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

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$db090)
# using number of observations in tail
thetaMoment(eusilc$eqIncome, k = ts$k)
# using threshold
thetaMoment(eusilc$eqIncome, x0 = ts$x0)
``` |