Page 87 - DCAP601_SIMULATION_AND_MODELING

Unit 6: Discrete System Simulation (III)

Notes
Figure 6.1: Example of a Four-way Ziggurat Partitioning of the Standard Normal Density

Tasks Analyze the methods used for non-uniform distributed random numbers.

Testing Non-uniform Random Numbers

Testing the normal RNG is possibly more significant than the fundamental uniform, as it offers
the basis leading econometric simulation experiments. A simple technique is to convert the
~ ~
e
normal random numbers e back to consistency using the normal cdf : u  ( ) . The can
i i i i u
then be used in the obtainable testing software, in our case the crush test suite of L’Ecuyer and
~
1
(

Simard (2005). When inversion is used, testing is unnecessary: u i    ( )) u . Though, for
u
i
i
~
the other methods: u  u .
i i
Table 6.1 offers the failure count on the crush test for two techniques: the sum of 12 uniforms (2),
the polar method (4), in addition to the ziggurat method (but only for MWC8222). The first is
shown to be problematic for all uniform RNGs. The consequences for the polar method are
interesting, as it shows that the new uniform random numbers ˜u are significantly more uniform
i
than the underlying u for LCG31 and MWC60, but also to some extent for LFSR113 and WELL1024.
i
This is mainly the outcome of using the rejection method, whereas there is no added problem
from using consecutive numbers. So, though LCG31 and MWC60 were rejected as uniform
RNGs, they are more adequate for standard normal RNGs based on the polar method.
Lastly, consider the Student-t(n) distribution. This can be produced as
1
n
  2
n
   ,  N [0,1],x  x 2 ( )
x
 
2
The  distribution in turn can be obtained from a gamma distribution, for which algorithms
3.19 and 3.20 from Ripley (1987) can be used. Table 6.2 offers the failure count on the crush test for
n = 6 and n = 12 after mapping the Student-t numbers back to uniformity. Yet again, LCG31 and
MWC60 do not act as well as the others, but not devastatingly so.

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