Consistency implies that bias & variance vanish?

Question

One can easily show using Markov's inequality that when Bias and Variance vanish, then an estimator is consistent with respect to the parameter of interest. Yet, I wonder why standard text books do not claim the reverse: If an estimator is consistent, both bias and variance must vanish. This should be trivial or is my intuition wrong?

Answer 1

There exist consistent estimators whose bias and variance do not vanish as sample size goes to infinity.

Indeed, let $X\sim \mathcal N (\mu,\sigma^2)$ and consider the problem of estimating $\mu$ with $n$ i.i.d. copies of $X$ that we denote $X_1,\ldots,X_n$, and let $\bar X_n := \sum_{i=1}^n X_i/n$ be the sample mean.
Next, let $(Y_n)$ be any sequence of random variables such that for all $n$, $Y_n$ is independent of $\bar X_n$, $\mathbb E[Y_n] = 0$ and $\text{Var} [Y_n] \ge c^2$ for some constant $c>0$.

It is easy to see that $Z_n := \bar X_n + Y_n$ is a consistent estimator of $\mu$, however it's variance is lower bounded by $c^2$ and thus $\text{Var} [Z_n] \not\to 0$ as $n\to\infty$.

Similarly, this Wikipedia article gives an example of a sequence of consistent estimators whose bias stays uniformly bounded away from zero.

I suggest you give a read to this excellent thread on stats.SE where the relationship between consistency and biasedness is discussed with many illuminating examples.