Here is the problem:
Suppose we have $L$ groups, and from each of these $L$ groups we observe $n$ random variables $X_{j,i}$, with $j = 1,\cdots,L$ and $i = 1,\cdots, n$. We also assume that $X_{j,i}$ is normally distributed with mean $\mu$ and standard deviation $\sigma$ for all $j = 1,\cdots,L$ and $i = 1,\cdots, n$.
Now suppose that from these observations, we want to construct an upper confidence bound for the variance. I found this upper bound in an article, and it is given by $\sigma^2_u$ in the following:
$$ \sigma^2_u = \left[ \frac{1}{n}MS_L + \left( 1- \frac{1}{n} \right)MS_E \right] + \left\{ \left[ \left( \frac{L-1}{\chi^2_{L-1,1-\alpha}}-1 \right) MS_L \right]^2 + \left[ \left( \frac{L(n-1)}{\chi^2_{L(n-1), 1-\alpha}} -1 \right)MS_E \right]^2 \right\}^\frac{1}{2}, $$ where $\chi^2_{N, 1-\alpha}$ is the $1-\alpha$ quantile of the chi square distribution with $N$ degrees of freedom. $MS_L$ is the mean squares between location and $MS_E$ is the mean squares within location.
In this same article, it is also said that the relation between the empirical variance $\hat{\sigma}^2$ and the empirical variance between location $\hat{\sigma}^2_L$ and the empirical variance within location $\hat{\sigma}^2_E$, is:
$$ \hat{\sigma}^2 = \hat{\sigma}^2_L + \hat{\sigma}^2_E $$
My problem is that I don't know which definition is true for $MS_E$ and $MS_L$ and how to relate them to $\hat{\sigma}^2_L$ and $\hat{\sigma}^2_E$ since it is not specified in the article. Here are the relationships and definitions I used:
Empirical mean: $\bar{X}:= \frac{1}{nL} \sum_{j = 1}^L \sum_{i = 1}^n X_{ij}$
Within location variance $\hat{\sigma}^2_E = \frac{1}{L(n-1)} \sum_{j = 1}^L \sum_{i = 1}^n \left(X_{ij} - \bar{X}_j \right)^2 = MS_E$
Between-location variance $\hat{\sigma}^2_L = \frac{1}{L-1} \sum_{j = 1}^L \left( \bar{X}_j - \bar{X} \right)^2 - \frac{1}{n}\hat{\sigma}^2_E = \frac{1}{n}(MS_L - MS_E)$
$MS_L = \frac{n}{L-1} \sum_{j = 1}^L \left( \bar{X}_j - \bar{X} \right)^2$
$\bar{X}_j = \frac{1}{n} \sum_{i = 1}^n X_{ij}$
But I'm really not sure about these definitions and I was hoping that someone who is really familiar with ANOVA could help me. I would be very grateful.
After some research, the definitions presented above for the within and between locations variance are correct.
We can find these definitions in the book: Anand M. Joglekar. Statistical Methods for Six Sigma. John Wiley and Sons Inc. 2003 on pages 191-193.