TITLE: Problem with unbiased but not consistent estimator
QUESTION [6 upvotes]: I have some troubles with understanding of this explanation taken from wikipedia:
"An estimator can be unbiased but not consistent. For example, for an iid sample $\{x
_1,..., x_n\}$ one can use $T(X) = x_1$ as the estimator of the mean $E[x]$. This estimator is obviously unbiased, and obviously inconsistent."
Why unbiased, and why inconsistent ...
Can someone explain it in more details?

REPLY [9 votes]: Suppose your sample was drawn from a distribution with mean $\mu$ and variance $\sigma^2$. Your estimator $\tilde{x}=x_1$ is unbiased as $\mathbb{E}(\tilde{x})=\mathbb{E}(x_1)=\mu$ implies the expected value of the estimator equals the population mean. Your estimator is on the other hand inconsistent, since $\tilde{x}$ is fixed at $x_1$ and will not change with the changing sample size, i.e. will not converge in probability to $\mu$. 
Perhaps an easier example would be the following. Let $\beta_n$ be an estimator of the parameter $\beta$. Suppose $\beta_n$ is both unbiased and consistent. Now let $\mu$ be distributed uniformly in $[-10,10]$. Consider the estimator $\alpha_n=\beta_n+\mu$. This estimator will be unbiased since $\mathbb{E}(\mu)=0$ but inconsistent since $\alpha_n\rightarrow^{\mathbb{P}} \beta + \mu$ and $\mu$ is a RV.