1. Biasedness & Consistency

Q: Is the estimator (1/n) ∑ (X_i - X bar)^2] consistent or inconsistent for σ^2 ?

I showed that the above estimator is biased, does this automatically imply that it is not consistent?

Can a biased esimtator ever be consistent?

Thanks!

2. You have an example of a bias estimator that's consistent right here.
Constistency usually refers to convergence in probability (MOO)
Some people talk about strong (almost sure) consistency
and weak constistency (convergence in probability).
But most just say an estimator is consistent if it converges in probability to the parameter it's estimating.
That's consistent with Wackerly (pun intented).

3. Originally Posted by matheagle
You have an example of a bias estimator that's consistent right here.
[snip]
Indeed.

Another 'counter-example':

Let $\displaystyle X_1, \, X_2, \, .... X_n$ be a sample taken from the uniform distribution on $\displaystyle (0, \theta]$ for some positive $\displaystyle \theta$ and consider the maximum likelihood estimator of $\displaystyle \theta$: $\displaystyle \hat{\theta}_n = X_{(n)}$.

$\displaystyle E\left(\hat{\theta}_n\right) = \frac{n \theta}{n + 1}$ and is therefore biased for any finite sample size $\displaystyle n$.

It's not difficult to prove that the sequence of random variables $\displaystyle \hat{\theta}_1, \, \hat{\theta}_2, \, ....$ converges to $\displaystyle \theta$ in probability and so is consistent.

Note: I'm not going to be inconsistent and make more work for myself by typing the proof. The interested reader can work through it him/herself or do some research.

4. Originally Posted by matheagle
You have an example of a bias estimator that's consistent right here.
Constistency usually refers to convergence in probability (MOO)
Some people talk about strong (almost sure) consistency
and weak constistency (convergence in probability).
But most just say an estimator is consistent if it converges in probability to the parameter it's estimating.
That's consistent with Wackerly (pun intented).
But how can (1/n) ∑ (X_i - X bar)^2] and σ^2 stay "close" as n->infinity when it is a BIASED estimator?

5. Originally Posted by kingwinner
But how can (1/n) ∑ (X_i - X bar)^2] and σ^2 stay "close" as n->infinity when it is a BIASED estimator?