1. ## asymptotic unbiased esitimators

This a past paper question I would just like to know what would be the best way to answer to get full marks. Just as it says 'briefly' you never know how little or much to write!

a) Suppose we can estimate a parameter theta using an esitmate theta^ based on a random sample of data X1,...,Xn, and suppose that for n={1,2,...}, theta^ is unbiased. Briefly explain why theta^ must be asympototically unbiased as an estimator for theta.

b) Briefly explain why an asymptotically unbiased estimator theta^ for theta is not necessarily unbiased for any finite sample size n, and give an example of an estimator for a parameter which is asymptotically unbiased, while not being unbiased for any fixed n.

2. I'll take a crack at this, but I'm a student so this is partly for my benefit too.

1) If we think of the sequence $\{E \hat {\theta}_n \}_{n = 1} ^ \infty$, what it would mean for the estimator to be asymptotically unbiased would be for the limit of this sequence to converge to $\theta$. So, just going from what it means for a sequence of reals to converge, we would let $\epsilon >0$ be given and we have to find an N such that if n > N, then $|E\hat{\theta}_n - \theta| < \epsilon$. This is easy, since that difference will be 0 for all n since the estimator is unbiased.

2) If the bias of the estimator is strictly positive, but tends to zero as the sample size tends to infinity, then this will be the case. The standard example would be $S^2 = \frac{\sum (X_i - \bar{X}) ^ 2}{n}$ which has a multiplicative bias of $(n-1)/n$. Clearly as n goes to infinity this term goes to 1 which eliminates the bias.

3. I think you've approached it in a more sort of pure sense rather than statistical. I've had a look through my notes and this is what I came up with.

a)
If the bias is zero for all n, then in the limit as n tends to infinity this bias must also be zero, which is to be asymptotic unbiased.

(It also follows directly from theta^ unbiased for an n that theta^ is asymptotically unbiased.)

b)
It is possible that the bias is not equal to zero for any n, but instead tends to zero as n tends to infinity.
An example could be the normal distribution when the mean is known. The mle for the variance, is asymptotically unbiased, but not unbiased for any n.