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

Description

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.

Usage

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

Arguments

 `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.

Details

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).

Value

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`).

Note

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.

Author(s)

Josef Leydold [email protected]

References

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.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37``` ```## 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) ```