Estimate the shape parameter of a Pareto distribution using a trimmed mean approach.

1 |

`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. |

`beta` |
A numeric vector of length two giving the trimming proportions for the lower and upper end of the tail, respectively. If a single numeric value is supplied, it is recycled. |

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).

The estimated shape parameter.

The argument `x0`

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

Andreas Alfons and Josef Holzer

Brazauskas, V. and Serfling, R. (2000) Robust estimation of tail
parameters for two-parameter Pareto and exponential models via generalized
quantile statistics. *Extremes*, **3**(3), 231–249.

Brazauskas, V. and Serfling, R. (2000) Robust and efficient estimation of the
tail index of a single-parameter Pareto distribution. *North American
Actuarial Journal*, **4**(4), 12–27.

`paretoTail`

, `fitPareto`

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

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