Does inconsistency imply biasedness?

Question

If an estimator $\theta$ is inconsistent, can I always conclude that $\theta$ is also biased?

Answer 1

To explain the relation between the bias and inconsistence, let's take a look at their mathematical definitions:

The definition of the bias is:

$Bias(\theta)=E(\hat{\theta})-\theta$

whis is the expected value of the estimator $\hat{\theta}$ minus the true parameter $\theta$.

Furthermore, one calls an estimator inconsistent in mean squared error, if

$lim_{ (N \to \infty )}MSE(\hat{\theta})=Bias(\hat{\theta})^2+Var(\hat{\theta}) \neq 0$

holds. ($N$ denotes the sample size)

Recall, that often the convergence in probability is used to check consistency, which is given by:

$plim_{N \to \infty} \hat{\theta}=\theta$

Both versions refer to the asymptotic behaviour of $\hat{\theta}$ and expresses that, as data accumulates, $\hat{\theta}$ gets closer and closer to the true value of $\theta$. This argumetation is outligned in: http://www.stats.ox.ac.uk/~steffen/teaching/bs2siMT04/si2c.pdf

Now to answer the question "Is an estimator allways biased if he is inconsistent?" just look at the formula: If the estimator is inconsistent, this might be due to a zero Bias(=unbiased) but a non-zero variance. Therefore, an estimator might be inconsistent but unbiased.