laeken (version 0.4.3)

paretoQPlot: Pareto quantile plot

Description

The Pareto quantile plot is a graphical method for inspecting the parameters of a Pareto distribution.

Usage

paretoQPlot(x, w = NULL, xlab = NULL, ylab = NULL,
    interactive = TRUE, x0 = NULL, theta = NULL,
    pch = par("pch"), cex = par("cex"), col = par("col"),
    bg = "transparent", ...)

Arguments

x

a numeric vector.

w

an optional numeric vector giving sample weights.

xlab,ylab

axis labels.

interactive

a logical indicating whether the threshold (scale parameter) can be selected interactively by clicking on points. Information on the selected threshold is then printed on the console.

x0,theta

optional; if estimates of the threshold (scale parameter) and the shape parameter have already been obtained, they can be passed through the corresponding argument (x0 for the threshold, theta for the shape parameter). I

pch,cex,col,bg

graphical parameters for the plot symbol of each data point (see points).

...

additional arguments to be passed to plot.default.

Value

• If interactive is TRUE, the last selection for the threshold is returned invisibly as an object of class "paretoScale", which consists of the following components:
• x0
  the selected threshold (scale parameter).
• k
  the number of observations in the tail (i.e., larger than the threshold).

Details

If the Pareto model holds, there exists a linear relationship between the lograrithms of the observed values and the quantiles of the standard exponential distribution, since the logarithm of a Pareto distributed random variable follows an exponential distribution. Hence the logarithms of the observed values are plotted against the corresponding theoretical quantiles. If the tail of the data follows a Pareto distribution, these observations form almost a straight line. The leftmost point of a fitted line can thus be used as an estimate of the threshold (scale parameter). The slope of the fitted line is in turn an estimate of $\frac{1}{\theta}$, the reciprocal of the shape parameter.

The interactive selection of the threshold (scale parameter) is implemented using identify. For the usual X11 device, the selection process is thus terminated by pressing any mouse button other than the first. For the quartz device (on Mac OS X systems), the process is terminated either by a secondary click (usually second mouse button or Ctrl-click) or by pressing the ESC key.

References

Beirlant, J., Vynckier, P. and Teugels, J.L. (1996) Tail index estimation, Pareto quantile plots, and regression diagnostics. Journal of the American Statistical Association, 91(436), 1659--1667.

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, accepted for publication.

See Also

paretoScale, paretoTail, minAMSE, meanExcessPlot, identify

Examples

data(eusilc)
# equivalized disposable income is equal for each household
# member, therefore only one household member is taken
eusilc <- eusilc[!duplicated(eusilc$db030),]

# with sample weights
paretoQPlot(eusilc$eqIncome, w = eusilc$db090)

# without sample weights
paretoQPlot(eusilc$eqIncome)