Solved – Asymptotic distribution of $sqrt{n}left(hat{sigma_{1}^{2}}-sigma^2right)$

No delta method is required to get the asymptotic distribution of $widehat{sigma}^2$.

Asymptotic Distribution of $widehat{sigma}^2 = frac{1}{n}sum_{i=1}^n (X_i-overline{X})^2$:

In the following we assume an iid sample $X_1, dots, X_n$ with $0<Var((X_i-mu)^2) = varsigma^2 <infty$, where $mathbb{E}(X_i)=mu$. We then have by the usual decomposition:

begin{align*} widehat{sigma}^2 & = frac{1}{n}sum_{i=1}^n (X_i-overline{X})^2\ & = frac{1}{n}sum_{i=1}^n ((X_i- mu) + (mu-overline{X}))^2\ & = frac{1}{n}sum_{i=1}^n (X_i- mu)^2 + 2 frac{1}{n} sum_{i=1}^n(X_i- mu)cdot(mu-overline{X}) + frac{1}{n}sum_{i=1}^n(overline{X}-mu)^2\ & = frac{1}{n}sum_{i=1}^n (X_i- mu)^2 + 2 frac{1}{n}sum_{i=1}^n (X_imu – X_ioverline{X}-mu^2+muoverline{X}) + (overline{X}^2-2overline{X}mu +mu^2)\ &=underbrace{frac{1}{n}sum_{i=1}^n (X_i- mu)^2}_{A} + underbrace{(overline{X}-mu)^2}_{B} end{align*} Part A is our main term. By assumption we have $0<Var((X_i-mu)^2) = varsigma^2 <infty$ and hence, by an application of the Lindeberg-Lévy central limit theorem with $mathbb{E}((X_i-mu)^2)=sigma^2$ we derive for Part A: $$sqrt{n}(widehat{sigma}^2-sigma^2) stackrel{d}{to}mathcal{N}(0,varsigma^2).$$

Part B on the other hand is asymptotical negligible: begin{align*} sqrt{n}B & = sqrt{n}(overline{X}-mu)^2\ & = (sqrt{n}(overline{X}-mu))(overline{X}-mu) \ & = (sqrt{n}(overline{X}-mu))(0 + o_p(1)) \ & = O_p(1)(0+o_p(1)) \ & = o_p(1). end{align*} In words: $(overline{X}-mu)$ converges in probablilty to the constant $0$. At the same time $(sqrt{n}(overline{X}-mu))$ converges in distribution to a normal distributed random variable. Hence $(sqrt{n}(overline{X}-mu))(overline{X}-mu)$ converges by the Slutsky theorem in distribution to $0$, and since $0$ is a constant, $(sqrt{n}(overline{X}-mu))(overline{X}-mu)$ converges additionally to $0$ in probability.

We can thus conclude by another application of the Slutsky theorem that: $$sqrt{n}(widehat{sigma}^2 – sigma^2) stackrel{d}{to} mathcal{N}(0,varsigma^2),$$ where $$varsigma^2= Var((X_i-mu)^2) = mathbb{E}((X_i-mu)^4) – (mathbb{E}(X_i-mu)^2)^2 = mathbb{E}((X_i-mu)^4) – sigma^4.$$

Extra: asymptotic $(1-alpha)$-confidence interval for $sigma^2$

Let $z_{alpha}$ be the $alpha$-quantil of a standard normal distribution. An asymptotical $1-alpha$-confidence interval for $sigma^2$ is then given by $$[widehat{sigma}^2- z_{1-alpha/2} frac{varsigma}{n},widehat{sigma}^2 + z_{1-alpha/2}frac{varsigma}{n}]$$ where we obviously have to replace the unknown $varsigma = sqrt{mathbb{E}((X_i-mu)^4) – sigma^4}$ by a consistent estimator for $varsigma$, i.e. using $$widehat{varsigma} = sqrt{frac{1}{n}sum_{i=1}^n (X_i – overline{X})^4 – widehat{sigma}^4}.$$

The confidence follows since: begin{align*} 1-alpha & = Pleft(-z_{1-alpha/2}leq sqrt{n}frac{widehat{sigma}^2-sigma^2}{varsigma}leq z_{1-alpha/2}right)\ & = Pleft(widehat{sigma}^2- z_{1-alpha/2} frac{varsigma}{n} leq sigma^2 leq widehat{sigma}^2 + z_{1-alpha/2}frac{varsigma}{n}right). end{align*}