**A** question that came out on X validated is asking for help in figuring out the UMVUE (uniformly minimal variance unbiased estimator) of (1-θ)^{½} when observing iid Bernoulli B(θ). As it happens, there is no unbiased estimator of this quantity and hence not UMVUE. But there exists a Bernoulli factory producing a coin with probability (1-θ)^{½} from draws of a coin with probability θ, hence a mean to produce unbiased estimators of this quantity. Although of course UMVUE does not make sense in this sequential framework. While Nacu & Peres (2005) were uncertain there was a Bernoulli factory for θ^{½}, witness their Question #1, Mendo (2018) and Thomas and Blanchet (2018) showed that there does exist a Bernoulli factory solution for θ^{a}, 0≤a≤1, with constructive arguments that only require the series expansion of θ^{½}. In my answer to that question, using a straightforward R code, I tested the proposed algorithm, which indeed produces an unbiased estimate of θ^{½}… (Most surprisingly, the question got closed as a “self-study” question, which sounds absurd since it could not occur as an exercise or an exam question, unless the instructor is particularly clueless.)

## Archive for existence of unbiased estimators

## another Bernoulli factory

Posted in Books, Kids, pictures, R, Statistics with tags Bernoulli factory, Cockatoo Island, cross validated, existence of unbiased estimators, industrial ruins, Sydney Harbour, UMVUE on May 18, 2020 by xi'an## unbiased estimators that do not exist

Posted in Statistics with tags binomial distribution, counterexample, cross validated, exam, existence of unbiased estimators, monomial, teaching, unbiasedness on January 21, 2019 by xi'an**W**hen looking at questions on X validated, I came across this seemingly obvious request for an unbiased estimator of **P**(X=k), when X~**B**(n,p). Except that X is not observed but only Y~**B**(s,p) with s<n. Since **P**(X=k) is a polynomial in p, I was expecting such an unbiased estimator to exist. But it does not, for the reasons that Y only takes s+1 values and that any function of Y, including the MLE of **P**(X=k), has an expectation involving monomials in p of power s at most. It is actually straightforward to establish properly that the unbiased estimator does not exist. But this remains an interesting additional example of the rarity of the existence of unbiased estimators, to be saved until a future mathematical statistics exam!