Let \(\widehat{\sigma}\) be the U.M.V.Uestimate of \(\sigma\) when \(\mu\) is unknown

show that:

a) \(\sqrt{n}{(\widehat{\sigma}-\sigma})\rightarrow\,{N(0,\sigma^2/2)}\)

b) \(\displaystyle nVar_\theta\rightarrow{\sigma^2/2},\)\(\sqrt{n}b(\sigma,\widehat{\sigma})\rightarrow{0}\)

hints:

a)\(\sqrt{n}{(\widehat{\sigma^2}-\sigma^2})/\sigma^2\approx {1/\sqrt{n} \displaystyle\sum^n_i=1(Z_i^2-1)}\) where the \(\displaystyle Z_i^2\) are independent \(\chi^2_1\)

b) Use the stirling's approximation, \(\lim_{x\rightarrow \infty} \Gamma(x + 1)/(\sqrt{2\pi}e^{-x}x^{x + \frac{1}{2}})=1\)

note: the estimator \(\widehat{\sigma}\) is defined by:

\(\widehat{\sigma}=\sqrt{\frac{1}{n}\displaystyle\sum^n_{i=1}(X_i-\bar{X})}\) where \(X_i\) is a sample form a \(\mathbb N(\mu,\sigma^2)\)