2003-06.wh-ley
#
Smoothed Transformed Density Rejection

### Abstract

There are situations in the framework of quasi-Monte Carlo
integration where nonuniform low-discrepancy sequences are required.
Using the inversion method for this task usually results in the best
performance in terms of the integration errors. However,
this method requires a fast algorithm for evaluating the inverse of
the cumulative distribution function which is often not available.
Then a smoothed version of transformed density rejection is a good
alternative as it is a fast method and its speed hardly depends on
the distribution. It can easily be adjusted such that it is
almost as good as the inversion method.
For importance sampling it is even better to use the hat
distribution as importance distribution directly. Then the
resulting algorithm is as good as using the inversion method for the
original importance distribution but its generation time is much shorter.

**Mathematics Subject Classification:**
65C05 (Monte Carlo Methods)
65C10 (Random Number Generation)
65D30 (Numerical integration)

**CR Categories and Subject Descriptors:**
G.3 [Probability and Statistics]: Random number generation

**General Terms:**
Algorithms

**Key Words:**
Monte Carlo method,
quasi-Monte Carlo method,
nonuniform random variate generation,
transformed density rejection,
smoothed rejection,
inversion

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Josef.Leydold@statistik.wu-wien.ac.at