Connfidence interval of a variance

What is-it?

A confidence interval of a variance is a statistical range that provides an estimate of where the true population variance is likely to fall. It is a way to quantify the uncertainty associated with estimating the population variance based on a sample.
When we want to estimate the variability or dispersion within a population, a confidence interval for the variance provides a range of values within which we can reasonably expect the true population variance to lie.

Data: \(x_{1:n}=\left(x_1,\cdots,x_n\right)\in\mathbb{R}^n\)
The observations \(x_{1:n}\) are supposed to be iid realizations of random variables \(X_{1:n}:=\left(X_1,\cdots,X_n\right)\), with \(X_i\overset{iid}{\sim}\mathcal{N}(\mu,\sigma^2)\)
If expectation is unknown: \(m=\overline{X}\)
If expectation is known: \(m=\mu:=\mathbb{E}\left[X_i\right]\)
Sum of squares: \(SS:=\sum_{i=1}^n\left(X_i-m\right)^2\)

A procedure of CI (or asymptotic CI) at a confidence level \(\beta\) for a variance \(\sigma^2\) is a random interval \[IC_{\beta}=\left[T_{1,n}, T_{2,n}\right],\] where \(T_{1,n}=T_{1,n}\left(X_{1:n}\right)\) and \(T_{2,n}=T_{2,n}\left(X_{1:n}\right)\) are statistics that satisfy \[\mathbb{P}\left(T_{1,n}\leq\sigma^2\leq T_{2,n}\right)\geq \beta=1-\alpha\] or \[\lim_{n\rightarrow +\infty}\mathbb{P}\left(T_{1,n}\leq\sigma^2\leq T_{2,n}\right)\geq \beta=1-\alpha.\]
A confidence interval is obtained by replacing the random variables \(X_i\) in the procedure with the observed data \(x_i\).
\(CI_{1-\alpha}\left(\sigma^2\right)=\left[\dfrac{SS}{q_1},\ \dfrac{SS}{q_2}\right]\)

\(\dfrac{SS}{\sigma^2}=\dfrac{\sum_i\left(X_i-\mu\right)^2}{\sigma^2}\sim\mathcal{X}_n^2\)
\(q_1:= q_{1-\alpha/2}\left(\mathcal{X}_n^2\right)\)
\(q_2:= q_{\alpha/2}\left(\mathcal{X}_n^2\right)\)

\(\dfrac{SS}{\sigma^2}=\dfrac{\sum_i\left(X_i-\overline{X}\right)^2}{\sigma^2}\rightarrow\mathcal{X}_{n-1}^2\)
\(q_1:= q_{1-\alpha/2}\left(\mathcal{X}_{n-1}^2\right)\)
\(q_2:= q_{\alpha/2}\left(\mathcal{X}_{n-1}^2\right)\)