# Chi-square approximation in homogeneity

I am interested in testing homogeneity in mixture of Gaussians (testing no mixture vs. 2 populations) given that we know the weights of the two distributions. We can first use MLE to estimate the mean and variance of each distribution and try to use approximation in order to test the homogeneity.

Goffinet et al. (1992) suggested a solution for the case of known component weights, but I have a problem understanding it: according to Theorem 1 in the paper the limiting distribution is a “chi-square distribution with one degree of freedom if the variance is unknown and p is not equal to 0.5”. The p I’m using is far from 0.5, however my intuition led me to think that I should use chi-square with two degrees of freedom as I look for both mean and variance (I do not claim my intuition is always right though). I must note that using chi with two degrees of freedom in my algorithm is highly consistent with using permutation tests as distribution, which only added to my confusion.

I would very much appreciate if someone can help me with this issue..

: Bruno Goffinet , Patrice Loisel and Beatrice Laurent (1992),
"Testing in Normal Mixture Models when the Proportions are Known,"
Biometrika, 79:(4) (Dec.), pp. 842-846

• When you say "try to use approximation" ... what approximation are you referring to? Can you provide a full reference for the paper you refer to? – Glen_b Aug 20 '15 at 0:23
• Hi, thanks for your help :) I meant chi-square approximation. The title of the paper is: "Testing in normal mixture models when the proportions are known". The pdf can be found here: jstor.org/stable/2337241?seq=1#page_scan_tab_contents – Roni Aug 20 '15 at 7:54
• What chi-square approximation do you mean? Are you referring to a Wald test, or something else? – Glen_b Aug 20 '15 at 11:38
• Likelihood ratio test – Roni Aug 21 '15 at 15:07