Page 86 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes As an instance of the rejection method, consider the logistic distribution as a hurdle for the
standard normal. From discrete choice methods, it is identified that binary probit and binary
logit are very comparable , so the logistic may be a fine candidate that can be reversed logically.
2 2
The logistic distribution with scale zero and variance   /3 has cdf:
G(x) = [1 + exp(–x/)] .
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–1
–1
The matching thickness is g(x) =  G(x)[1 – G(x)] and the quantiles are G (u) = – log[(1 – u)/u].
Execution needs a pick for the scaling parameter c in (1) and the discrepancy of the logistic
distribution during , such that the rejection probability is (approximately) diminished. Devroye
demonstrates the principle of how these parameters can be calculated. Following this, we
maximize f(x)/g (x), or more expediently, d(x) = log f(x) – log g (x) to locate one solution at zero
 
for   0.71. For lesser ’s, zero is a local minimum. Solving numerically, it is found that min

max d(x) is at x = 0.98226 when  = 0.65. The value of c is then f(x )/g (x ) = 1.081. This shows
x m m m m m
a rejection algorithm:
repeat
w = log(u ), w = log(1 – u )
1 1 2 1
v = 0.65 _ (w – w );
1 2
until log(u ) + w + w + log(1.081/0.65) < –v /2 “ log(2)/2 return v. (5)
2
2 1 2
It will be slower than (4), as it needs at least two uniforms and three logarithms for each standard
normal.
There is a large journalism on the generation of non-uniform random numbers.

Efficient Methods for the Standard Normal Distribution

The abovementioned polar technique is simple to execute, but not chiefly fast. Due to the
significance of the normal distribution, many more well-organized methods have been
anticipated. The method of Wallace (1996) is attractive, as it generates normal random numbers
directly, in addition to being very well-organized. Though, due to this it is also rather harder to
understand the properties of this technique.
Perhaps the fastest obtainable rejection method at the moment is the so called ‘ziggurat method,’
commenced by Marsaglia and Tsang (1984) and consequently refined (Marsaglia and Tsang,
2000). This technique has some scarcities, which were corrected by Doornik (2005b) at the cost of
some of its speed.
The ziggurat partitions the standard normal thickness into horizontal blocks of equal area. The
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standardization can be mislaid, using f(x) = exp(–x /2) as an alternative. All blocks are rectangular
boxes, apart from the bottom one, which includes a box joined with the remainder of the
density.
This is demonstrated in Figure 6.1, using four boxes, labeled from bottom to top as B ,B ,B ,B .
0 1 2 3
Equal areas of size V entails:


x
x [f(x ) — f(x )] = x [f(x ) — f(x )] = x [f(x ) — f(x )] = x f(x ) +  f ( )dx V .
3 4 3 2 3 2 1 2 1 1 1
1 x
Note that, working backward along the X-axis, earlier values of x can be easily obtained. For
Instance, if x is known:
1
x = f (f(x ) + V/x ) .
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2 1 1

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