In the previous post I needed to know the tail percentile points for a chi squared distribution with a huge number of degrees of freedom. When the number of degrees of freedom ν is large, a chi squared random variable has approximately a normal distribution with the same mean and variance, namely mean ν and variance 2ν.
In that post, ν was 9999 and we needed to find the 2.5 and 97.5 percentiles. Here are the percentiles for χ²(9999):
>>> chi2(9999).ppf([0.025, 0.975])
array([ 9723.73223701, 10278.05632026])
And here are the percentiles for N(9999, √19998):
>>> norm(9999, (2*9999)**0.5).ppf([0.025, 0.975])
array([ 9721.83309451, 10276.16690549])
So the results on the left end agree to three significant figures and the results on the right agree to four.
Fewer degrees of freedom
When ν is more moderate, say ν = 30, the normal approximation is not so hot. (We’re stressing the approximation by looking fairly far out in the tails. Closer to the middle the fit is better.)
Here are the results for χ²(30):
>>> chi2(30).ppf([0.025, 0.975])
array([16.79077227, 46.97924224])
And here are the results for N(30, √60):
>>> norm(30, (60)**0.5).ppf([0.025, 0.975])
array([14.81818426, 45.18181574])
The normal distribution is symmetric and the chi squared distribution is not, though it becomes more symmetric as ν → ∞. Transformations of the chi squared distribution that make it more symmetric may also improve the approximation accuracy. That wasn’t important when we had ν = 9999, but it is more important when ν = 30.
Fisher transformation
If X ~ χ²(ν), Fisher suggested the approximation √(2X) ~ N(√(2ν − 1), 1).
Let Y be a N(√(2ν − 1), 1) random variable and Z a standard normal random variable, N(0, 1). Then we can estimate χ² probabilities from normal probabilities.
So if we want to find the percentage points for X, we can solve for corresponding percentage points for Z.
If z is the point where P(Z ≤ z) = p, then
is the point where P(X ≤ x) = p.
If we use this to find the 2.5 and 97.5 percentiles for a χ²(30) random variable, we get 16.36 and 46.48, an order of magnitude more accurate than before.
When ν = 9999, the Fisher transformation gives us percentiles that are two orders of magnitude more accurate than before.
Wilson–Hilferty transformation
If X ~ χ²(ν), the Wilson–Hilferty transformation is (X/ν)1/3 is approximately normal with mean 1 − 2/9ν and variance 2/9ν.
This transformation is a little more complicated than the Fisher transform, but also more accurate. You could go through calculations similar to those above to approximate percentage points using the Wilson–Hilferty transformation.
The main use for approximations like this is now for analytical calculations; software packages can give accurate numerical results. For analytical calculation, the simplicity of the Fisher transformation may outweigh the improve accuracy of the Wilson-Hilferty transformation.