Using your Head is Permitted

Let X₁,...,X_n be independent random variables, all distributed uniformly in [a, b] for some unknown a and b.

This month's question: find (with proof) the minimum variance unbiased estimator of the expectation of the n random variables. (Or prove it does not exist.)

Some explanations regarding the terminology:

1. A function from the values of random variables X₁,...,X_n that are drawn from a distribution characterised by (among others) some parameter, θ, is called an unbiased estimator of θ if for any setting of the distribution parameters its expectation over all X₁,...,X_n equals θ. In our case, we are looking for a function, f, such that the expectation of f(X₁,...,X_n) equals (a+b)/2.
2. An unbiased estimator of θ is called a minimum variance unbiased estimator if for every setting of the distribution parameters the value of the estimator (which, in itself is a random variable) has the lowest variance, as compared to any other unbiased estimator. Note that if there is no single estimator that achieves this for the entire range of all possible parameters, then the minimum variance unbiased estimator does not exist. There are many cases where such estimators do not exist.

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