In the coin tossing setup, we saw that the fraction of observed heads in trails approaches as . The function is called an estimator that takes on a value in . It’s a special kind of estimator called an unbiased estimator because it satisfies
It says that on average, this estimator deduces the correct answer. Consider a different estimator which essentially starts with an assumed ‘tails’. Then
which, on average, slightly underestimates the success probability.
A problem asks the following. Let it be known a priori that has a value in the set . Construct an unbiased estimator for , taking values only in .
Consider the case where for . Then is an unbiased estimator because
which is unbiased because can only be . Now, I can’t seem to proceed further than this. For example, what is the estimator when ? I’ll have to return with an answer another day.