Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $\vec{X}=(X_1,X_2,\ldots,X_n)$ be a random sample of some random variable $X$ whose distribution $F$ depends on some real-valued parameter $\theta_F$. Let $\hat\theta(\vec{X})$ be an unbiased estimator of $\theta_F$. Let $(\vec{x}_n)$ be a sequence of observations of $\vec{X}$. Is it true that the sequence $(\hat\theta(\vec{x}_n))$ will have $\theta_F$ as a cluster point?

My guess is that it should be true, but I am not sure how to prove this from the fact that the estimator is unbiased. Any tips?

share|improve this question
    
What if the sequence of observations is a constant sequence? For example let theta(a,a,...,a) be different from the actual parameter. Now consider the sequence of observations (a,a,...,a),(a,a,...,a),(a,a,...,a),... –  Amr Nov 7 '12 at 20:07
    
Now the sequence theta(a,...,a), theta(a,...,a),.... will not have theta (the actual parameter) as a cluster point. –  Amr Nov 7 '12 at 20:08
    
You are right about that. Hmm...I am trying to get a feel for what unbiased means in terms of actual data. I guess I will have to think harder. –  echoone Nov 7 '12 at 20:43
add comment

1 Answer

No. In particular, if the possible range of $\theta_F$ is compact, then you can create an unbiased estimator $\tilde\theta$ which only takes values on the boundary, and so no interior point will be a cluster point of any sequence of estimates even if you put additional constraints (e.g. independent samples) to deal with Amr's concern. If $\theta_F$ takes values on the entire real line you can build an unbiased estimator which only takes integer values by considering nested compact sets.

In addition, a biased estimator can have the true value be a cluster point. For example, if $X_i$ are normal with known variance $\sigma^2$ and we are trying to estimate the mean, we can use the biased estimator $\tilde\theta =\frac1n \sum x_i + 1$, which is clearly biased but has every point on the real line as a cluster point if we have an infinite sequence of independent random samples, since the sampling distribution of $\tilde \theta$ has support on the entire real line.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.