Page 107 - Computational Statistics Handbook with MATLAB

94 Computational Statistics Handbook with MATLAB

 k 
X = – 2log  ∏ U i .  (4.15)
 
i = 1
ν
When is odd, say 2k + 1 , we can use the fact that the chi-square distribu-
ν
ν
tion with degrees of freedom is the sum of squared independent stan-
dard normals [Ross, 1997]. We obtain the required random variable by first
simulating a chi-square with 2k degrees of freedom and adding a squared
standard normal variate Z, as follows
k
 
2
X = Z – 2log  ∏ U i .  (4.16)
 
i = 1
Example 4.8
In this example, we provide a function that will generate chi-square random
variables.
% function X = cschirnd(n,nu)
% This function will return n chi-square
% random variables with degrees of freedom nu.

function X = cschirnd(n,nu)
% Generate the uniforms needed.
rm = rem(nu,2);
k = floor(nu/2);
if rm == 0 % then even degrees of freedom
U = rand(k,n);
if k ~= 1
X = -2*log(prod(U));
else
X = -2*log(U);
end
else % odd degrees of freedom
U = rand(k,n);
Z = randn(1,n);
if k ~= 1
X = Z.^2-2*log(prod(U));
else
X = Z.^2-2*log(U);
end
end
The use of this function to generate random variables is left as an exercise.

© 2002 by Chapman & Hall/CRC