Unbiasedness

We define the as

$$Bias({\hat{\theta }}_n):=E[{\hat{\theta }}_n-\theta ]=E({\hat{\theta }}_n)-\theta$$

Recall that for i.i.d. sample $X1,\cdot \cdot \cdot ,Xn$, we have $s_X^2=\frac{1}{n-1}{\sum }_{i=1}^n(X_i-\overline{X}{)}^2$. Show $E(s_X^2)={\sigma }_X^2.$

First,

$$\begin{array}{r}E(s_X^2)=E(\frac{1}{n-1}\sum _{i=1}^n(X_i-\overline{X}{)}^2)\\ =\frac{1}{n-1}\sum _{i=1}^{n}E[(X_i-\overline{X}{)}^2]\end{array}$$

Since $Y_i=X_i-\overline{X}$ , $E(Y_i)=E(X_i-\overline{X})=E(X_i)-{\mu }_X$

Since $Cov(X_i,\overline{X})=Cov(X_i,\frac{1}{n})=\frac{1}{n}{\sigma }_X^2$

E(X_i-\overline X)^2=Var(X_i-\overline X) \\=Var(X_i)-2Cov(X_i,\overline X)+Var(\overline X) \\ \end{aligned}