unuran.verify.hat: Verify hat and squeezes in a "unuran" generator object

In Runuran: R Interface to the 'UNU.RAN' Random Variate Generators

Verify hat and squeezes in a "unuran" generator object

Description

Verify hat function and squeezes in a given unuran generator that implements a rejection method.

[Advanced] – Verify rejection method.

Usage

unuran.verify.hat(unr, n=1e5, show=TRUE)

Arguments

unr	an unuran object.
n	sample size. (integer)
show	whether the result is printed on the console. (boolean)

Details

UNU.RAN is a library of so called black-box algorithms. For algorithms based on the rejection method this means that hat and squeezes are created automatically during the setup. Obviously not all algorithms work for all distribution. Then usually the setup fails (which is good, since then one does not silently obtain a random sample from a distribution other then the requested.)

Although we have tested these algorithms with a lot of distributions (including those with extreme properties) there is still some (minor) chance that hat and squeezes are computed without any warnings, but are incorrect, i.e., the inequalities

squeeze(x) ≤ density(x) ≤ hat(x)

are not satisfied for all x. This might happen due to serious round-off errors for densities with extreme properties (e.g., sharp and narrow peaks). But it also might be caused by some incorrect additional information about the distribution given by the user which has not been detected by various checks during the setup. If one is unsure about his or her chosen generation method one can check these inequalities.

Routine unuran.verify.hat allows to run generator unr and check whether the two inequalities are violated. This is done for every point x that is sampled from the hat distribution. This includes also those points that are rejected. The function counts the occurrences of such evaluations and returns the ratio of this number and the sample size n. (It is thus a little bit too high since the total number of generated but rejected points is not known.) Yet, it does not provide any information about the magnitude of violation of the inequality.

If show is TRUE then this routine prints this ratio and some diagnostcs to the console.

Routine unuran.verify.hat does not work for algorithms that do not implement a rejection method.

Value

Ratio of number occurrences where the hat and squeezes violate the inequality and the sample size.

Note

Due to round-off errors there might exist a few points where the ratio density(x) / hat(x) is slightly larger than 1. In our experiments we observed a few cases where this ratio was as large as 1+1.e-8 for some points although we could proof (using real numbers instead of floating point numbers) that hat and squeeze are computed correctly.

On the other hand, there are cases where, due to the limitation of floating point arithmetic, it is not possible to sample from the target distribution at all. The Gamma distribution with extremely small shape parameter, say 0.0001, is such an example. Then the continuous Gamma distribution degenerates to a point distribution with only a few points with significant mass.

Author(s)

Josef Leydold and Wolfgang H\"ormann [e-mail].

See Also

unuran.

Examples

## Create a generator object that implements a rejection method
unr <- tdrd.new(udnorm())

## Verify hat and squeeze
unuran.verify.hat(unr)

Runuran documentation built on Jan. 17, 2023, 5:17 p.m.