R: Partial density component (PDC) estimator

thetaPDC {laeken} | R Documentation

## Partial density component (PDC) estimator

### Description

The partial density component (PDC) estimator estimates the shape parameter of a Pareto distribution based on the relative excesses of observations above a certain threshold.

### Usage

    thetaPDC(x, k = NULL, x0 = NULL, w = 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.
`w` | an optional numeric vector giving sample weights.
`...` | additional arguments to be passed to `optimize` (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 PDC estimator minimizes the integrated squared error (ISE) criterion with an incomplete density mixture model. The minimization is carried out using `nlm`. By default, the starting value is obtained with the Hill estimator (see `thetaHill`). `optimize`.

### Value

The estimated shape parameter.

### Note

The arguments `x0` for the threshold (scale parameter) of the Pareto distribution and `w` for sample weights were introduced in version 0.2.

### Author(s)

Andreas Alfons and Josef Holzer

### References

A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators from Complex Surveys: The R Package laeken. Journal of Statistical Software, 54(15), 1–25. URL http://www.jstatsoft.org/v54/i15/

A. Alfons, M. Templ, P. Filzmoser (2013) Robust estimation of economic indicators from survey samples based on Pareto tail modeling. Journal of the Royal Statistical Society, Series C, 62(2), 271–286.

Vandewalle, B., Beirlant, J., Christmann, A., and Hubert, M. (2007) A robust estimator for the tail index of Pareto-type distributions. Computational Statistics & Data Analysis, 51(12), 6252–6268.

### See Also

`paretoTail`, `fitPareto`, `thetaISE`, `thetaHill`

### 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
    thetaPDC(eusilc$eqIncome, k = ts$k, w = eusilc$db090)

    # using threshold
    thetaPDC(eusilc$eqIncome, x0 = ts$x0, w = eusilc$db090)

[Package laeken version 0.4.6 Index]