paretoQPlot  R Documentation 
The Pareto quantile plot is a graphical method for inspecting the parameters of a Pareto distribution.
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",
...
)
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 ( 
pch , cex , col , bg 
graphical parameters for the plot symbol of each data
point (see 
... 
additional arguments to be passed to

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.
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). 
The functionality to account for sample weights and to select the threshold (scale parameter) interactively was introduced in version 0.2. Also starting with version 0.2, a logarithmic yaxis is now used to display the axis labels in the scale of the original values.
Andreas Alfons and Josef Holzer
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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v054.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.
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.
paretoScale
, paretoTail
,
minAMSE
, meanExcessPlot
,
identify
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)
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