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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?

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    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),...2012-11-07
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    Now the sequence theta(a,...,a), theta(a,...,a),.... will not have theta (the actual parameter) as a cluster point.2012-11-07
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    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.2012-11-07

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