Confidence Intervals on Variance in Normal Distribution

Let X₁, X₂, cdots, Xn be a random sample from a normal distribution with mean zero and variance σ². And let {bar X} = sum_{i=1}^n X_i /n and S² = sum from { i=1 } to { n } (X_i - bar {X} )² /n. In normal distribution case, we usually use the confidence intervals of σ² obtained by using the fact that nS² / σ² has a chi square distribution with parameter (n-1). In this paper we find three other confidence intervals of σ² by using the central limit theorem and variance stabilization method. And we compare them through simulation method. The simulation result shows that the confidence interval obtained by variance stabilization is better than that obtained by the exact distribution of nS² / σ² in view of the means of lengths of each confidence intervals and the number of the case, where the confidence intervals of variance contains variance σ².