 Solution

For an estimator with finite second moment, its bias and mean squared error areWrite $\hat{\theta }-\theta =(\hat{\theta }-{\mathbb{E}}_{\theta }\hat{\theta })+b_{\theta }$. On squaring and taking expectations, the cross term vanishes. This proves the bias-variance decomposition of mean squared error:For the sample mean, independence gives $\mathbb{E}\bar{X}=\theta$, $\mathrm{Var}(\bar{X})={\theta }^2/n$. Thus $k\bar{X}$ has bias $(k-1)\theta$. Unbiasedness throughout the parameter family requiresAt the single parameter value $\theta =0$, all these estimators are zero almost surely; that special value does not change the requirement of unbiasedness for all $\theta$.