What is the relationship of bias and consistency of an estimator? Let $(\mathfrak{X}_n, (P_\vartheta)_{\vartheta \in \Theta})$ be a statistical model with $n$ samples. Then $\hat{\vartheta}_n: \mathfrak{X}_n \rightarrow \tilde{\Theta}$ is called an estimator for $\vartheta$.


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*$\hat{\vartheta}_n$ is called unbiased, if $\mathbb{E}(\hat{\vartheta}_n) = \vartheta$.

*$\hat{\vartheta}_n$ is called concistent, if $\lim_{n \rightarrow \infty} P(|\hat{\vartheta}_n - \vartheta| > \varepsilon) = 0$ for $\varepsilon > 0$.


There are estimators which are both unbiased and consistent:
Let $X_1, \dots, X_n \stackrel{iid}{\sim} Bin(1, \vartheta)$ with $\vartheta \in (0, 1)$. Then $\hat{\vartheta}_n = \frac{1}{n} \sum_{i=1}^n x_i$ is unbiased and consistent.
There are estimators which are neither unbiased nor consistent. The estimator $\hat{\vartheta} = 0.5$ for the setting from before (if $\vartheta \neq 0.5$).
But are there unbiased estimators which are not consistent? Are there consistent estimators which are not unbiased?

Yes to both. For the sequence of random variables you give:
$\hat{\vartheta}^\prime_n =x_n$ is unbiased but clearly not consistent.
 $\hat{\vartheta}^{\prime\prime}_n =\frac1n\sum_{i=1}^n x_i + \frac1n$ is biased but consistent.
