xerror: Create Table of X-Errors for Numerical Inversion Method
In rvgtest: Tools for Analyzing Non-Uniform Pseudo-Random Variate Generators

Description Usage Arguments Details Value Note Author(s) See Also Examples

Function for creating a table of x-errors of a numerical inversion method (i.e., it uses an approximate quantile function of the target distribution). Thus the domain of the inverse distribution function is partitioned into intervals for which maxima, minima and some other quantiles of the x-errors are computed.

Currently the function only works for generators for continuous univariate distribution.

1 2	xerror(n, aqdist, qdist, ..., trunc=NULL, udomain=c(0,1), res=1000, kind=c("abs","rel"),tails=FALSE, plot=FALSE)

`n`	sample size for one repetition.
`aqdist`	approximate inverse distribution function (quantile function) for a continuous univariate distribution.
`qdist`	(Exact) quatile function of distribution.
`...`	parameters to be passed to `qdist`.
`trunc`	boundaries of truncated domain. (optional)
`udomain`	domain of investigation for (approximate) quantile function `aqdist`.
`res`	resolution (number of intervals).
`kind`	kind of x-error.
`tails`	logical. If `TRUE`, then the tail regions are treated more accurately. However, this doubles the given sample size.
`plot`	logical. If `TRUE`, the (range of the) x-errors is plotted.

The absolute x-error of an approximate inverse distribution function (quantile function) G^[-1] for some u in (0,1) is given by

e_x(u) = |F^[-1](u) - G^[-1](u)|

where F^[-1] denotes the (exact) quantile function of the distribution. The relative x-error is then defined as

e_x(u) / |F^[-1](u)|

Computing, plotting and analyzing of such x-errors can be quite time consuming.

e_x(u)

is a very volatile function and requires the computation at a lot of points. For plotting we can condense the information by partitioning (0,1) into intervals of equal length. In each of these the x-error is computed at equidistributed points and some quantiles (see below) are estimated and stored. Thus we save memory and it is much faster to plot and compare x-errors for different methods or distributions.

If trunc is given, then function qdist is rescaled to this given domain. Notice, however, that this has some influence on the accuracy of the results of the “exact” quantile function qdist.

Using argument udomain it is possible to restrict the domain of the given (approximate) quantile function aqdist, i.e., of its argument u.

When tails=TRUE we use additional n points for the first and last interval (which correspond to the tail regions of the distribution).

An object of class "rvgt.ierror", see uerror for details.

It should be noted that xerror computes the difference between the approximate inversion function aqdist(u) and the given ‘exact’ quantile function qdist. Thus one needs a quantile function qdist that is numerically (much) more accurate than aqdist.

The random variate generator rdist can alternatively be a generator object form the Runuran package.

Josef Leydold josef.leydold@wu.ac.at

See plot.rvgt.ierror for the syntax of the plotting method. See uerror for computing u-errors.

## Create a table of absolute x-errors for spline interpolation of
## the inverse CDF of the standard normal distribution.
aq <- splinefun(x=pnorm((-100:100)*0.05), y=(-100:100)*0.05,
                method="monoH.FC")
## Use a sample of size of 10^5 random variates.
xerr <- xerror(n=1e5, aqdist=aq, qdist=qnorm, kind="abs")

## Plot x-errors
plot(xerr)


## Same for the relative error.
## But this time we use a resolution of 500, and
## we immediately plot the error.
xerr <- xerror(n=1e5, aqdist=aq, qdist=qnorm,
               res=500, kind="rel", plot=TRUE)


## An inverse CDF for a truncated normal distribution
aqtn <- splinefun(x=(pnorm((0:100)*0.015) - pnorm(0))/(pnorm(1.5)-pnorm(0)),
                  y=(0:100)*0.015, method="monoH.FC")
xerrtn <- xerror(n=1e5, aqdist=aqtn, qdist=qnorm, trunc=c(0,1.5),
                 plot=TRUE)