thetaWML {laeken}

Weighted maximum likelihood estimator

Description

Estimate the shape parameter of a Pareto distribution using a weighted maximum likelihood approach.

Usage

thetaWML(
  x,
  k = NULL,
  x0 = NULL,
  weight = c("residuals", "probability"),
  const,
  bias = TRUE,
  ...
)

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.

weight

a character string specifying the weight function to be used. If "residuals" (the default), the weight function is based on standardized residuals. If "probability", probability based weighting is used. Partial string matching allows these names to be abbreviated.

const

Tuning constant(s) that control the robustness of the method. If weight="residuals", a single numeric value is required (the default is 2.5). If weight="probability", a numeric vector of length two must be supplied (a single numeric value is recycled; the default is 0.005 for both tuning parameters). See the references for more details.

bias

a logical indicating whether bias correction should be applied.

...

additional arguments to be passed to uniroot (see “Details”).

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).

The weighted maximum likelihood estimator belongs to the class of M-estimators. In order to obtain the estimate, the root of a certain function needs to be found, which is implemented using uniroot.

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

Dupuis, D.J. and Morgenthaler, S. (2002) Robust weighted likelihood estimators with an application to bivariate extreme value problems. The Canadian Journal of Statistics, 30(1), 17–36.

Dupuis, D.J. and Victoria-Feser, M.-P. (2006) A robust prediction error criterion for Pareto modelling of upper tails. The Canadian Journal of Statistics, 34(4), 639–658.

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

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