thetaLS {laeken}

Least squares (LS) estimator

Description

Estimate the shape parameter of a Pareto distribution using a least squares (LS) approach.

Usage

thetaLS(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

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.

See Also

paretoTail, fitPareto

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
thetaLS(eusilc$eqIncome, k = ts$k)

# using threshold
thetaLS(eusilc$eqIncome, x0 = ts$x0)