Several Unbiased Estimators

Question

If I have some data set $ D={X_1,...X_N} $ and have an esitmator be "pick the first point" $X_1$, how can I show that this estimator is unbiased? I also have to show why its highly undesirable, and I would assume it uses some variance/convergance argument. I would like some help proving that the estimator is indeed unbasied and in finding its variance.

Answer 1

If every $X_i$ has mean $\mu$ and variance $\sigma^2$, then the estimator $Y=X_1$ is such that $\mathbb E(Y)=\mu$ and $\text{var}(Y)=\sigma^2$.

Proof: $Y=X_1$. End of proof.

The fact that $\mathbb E(Y)=\mu$ may be seen as desirable if $Y$ is meant to estimate $\mu$, the fact that $\text{var}(Y)=\sigma^2$ less so if another estimator $\widetilde Y$ is such that $\mathbb E(\widetilde Y)=\mu$ and $\text{var}(\widetilde Y)\lt\sigma^2$, even less so if $\text{var}(\widetilde Y)\ll\sigma^2$.

You probably have an idea for some of these other, more efficient, estimators $\widetilde Y$.