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

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

Function for creating a table of u-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 u-errors are computed.

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

1 2	uerror(n, aqdist, pdist, ..., trunc=NULL, udomain=c(0,1), res=1000, tails=FALSE, plot=FALSE)

`n`	sample size for one repetition.
`aqdist`	approximate inverse distribution function (quantile function) for a continuous univariate distribution.
`pdist`	cumulative distribution function for distribution.
`...`	parameters to be passed to `pdist`.
`trunc`	boundaries of truncated domain. (optional)
`udomain`	domain of investigation for (approximate) quantile function `aqdist`.
`res`	resolution (number of intervals).
`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) u-errors is plotted.

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

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

where F denotes the (exact) cumulative distribution. It is a convenient measure for approximation errors in non-uniform random variate generators bases on numerical inversion, see the reference below for our arguments.

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

e_u(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 u-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 u-errors for different methods or distributions.

If trunc is given, then function pdist is rescaled to this given domain. Notice, however, that this has some influence on the accuracy of the results of the distribution function pdist.

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" which is a list with components:

`n`	sample size.
`res`	resolution (number of intervals).
`kind`	kind of error (string).
`udomain`	domain for u.
`min`	an array of minimum u-errors (of length `res`).
`lqr`	an array of lower quartile of u-errors (of length `res`).
`med`	an array of median u-errors (of length `res`).
`uqr`	an array of upper quartile of u-errors (of length `res`).
`max`	an array of maximum u-errors (of length `res`).
`mad`	an array of mean absolute deviations (of length `res`).
`mse`	an array of mean squared errors (of length `res`).

It should be noted that uerror computes the numerical error of the composed function pdist(aqdist(u)). Thus one needs a distribution function pdist 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

G. Derflinger, W. H\"ormann, and J. Leydold (2009): ACM Trans. Model. Comput. Simul., to appear.

See plot.rvgt.ierror for the syntax of the plotting method. See xerror for computing x-errors.

## Create a table of u-errors for spline interpolation of
## the inverse CDF of the standard normal distribution.
aqn <- 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.
uerrn <- uerror(n=1e5, aqdist=aqn, pdist=pnorm)

## Plot u-errors
plot(uerrn)

## Investigate tails more accurately, and use
## a resolution of 1000 intervals.
uerrn <- uerror(n=1e5, aqdist=aqn, pdist=pnorm, res=1000, tails=TRUE)


## Same for a gamma distribution.
## But this time we immediately plot the error.
aqg <- splinefun(x=pgamma((0:500)*0.1,shape=5),
                 y=(0:500)*0.1, method="monoH.FC")
uerrg <- uerror(n=1e5, aqdist=aqg, pdist=pgamma, shape=5,
                plot=TRUE)


## Compute u-error for a subdomain of a beta distribution
aqb <- splinefun(x=pbeta((0:100)*0.01,shape1=2,shape2=5),
                 y=(0:100)*0.01, method="monoH.FC")
uerrb <- uerror(n=1e5, aqdist=aqb, pdist=pbeta, shape1=2, shape2=5,
                udomain=c(0.6,0.65), plot=TRUE)

## Show all u-errors in one plot
plot.rvgt.ierror(list(uerrn,uerrg,uerrb))

## 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")
uerrtn <- uerror(n=1e5, aqdist=aqtn, pdist=pnorm, trunc=c(0,1.5),
                 plot=TRUE)