3

Consider $x_i \sim N(\mu,\sigma)$ I am interested in distribution of the the following statistic (arises from likelyhood ratio test): $$ \frac{ \sum_{n=1}^{n} (x_i - \overline{x})^2 }{ \sum_{n=1}^{n} (x_i - \mu)^2 } $$

where $\overline{x}$ is the sample mean

After observing distribution of this statistic with simulation for couple of N I noticed that it looks awfully like $Beta(\frac{1}{2}+\frac{n-2}{2},\frac{1}{2})$

Here are how empirical and beta CDF look like when plotted against each other for n=2,3 and 10:

n=2

n=3

n=10

Is my guess correct?

Bait Hoven
  • 31
  • 3
    This result is stated (in terms of Gamma distributions) in the Wikipedia article on Beta distributions at https://en.wikipedia.org/wiki/Beta_distribution#Derived_from_other_distributions. Note the typographic error in your question, though: Beta distributions have positive parameters, not negative ones. Closely related posts here on CV include https://stats.stackexchange.com/questions/520498 – whuber Jan 30 '24 at 20:45
  • 1
    Fixed the typo, thanks. Yes, the conclusion can be derived from the wiki page. If you post this as an answer I can accept it. – Bait Hoven Jan 30 '24 at 21:25

0 Answers0