R: Weighted maximum likelihood estimator

thetaWML {laeken} | R Documentation

## 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)

[Package laeken version 0.4.6 Index]