Date: May 19, 2013 9:10 AM
Author: Luis A. Afonso
Subject: Re: Chi-squared to normal approximation
A (perhaps futile) caution note
Back to Wilson and Hilferty formula to convert normal data into Chi-squared variable, namely:
W= [(Y/r)^(1/p) - (1- 2*p^2/r]/sqrt(2*p^2/r)________(1)
where y ~ Chi-squared, r degrees of freedom, W~ N(0,1), p=3.
Solving for y,
Y = r *[W*sqrt(2*p^2/r) + (1- 2*p^2/r]^p =
= r *[W*sqrt(2/(9*r)) + (1- 2/(9*r)]^3
Note that this equation allows, for example, to transform, whatever sum of squared deviations from W~N (mu, sigma): n, that follows rigorously a Chi-squared r-1 df, into normal Data.
Referring to (1), as Matthew Shultz states, W doesn´t transform Chi-squared variable into a Student one. The reason is clear (1) is a relationship (approximate) between random variables and so has nothing to do with sample mean values . . .
He says (wrongly):
___That is, if Z is a standard normal variable, P(Z<W(y)) approx. = P(Y<y). So W(Y) approximates a t statistic___END OF QUOTING.
Luis A. Afonso