## another Bernoulli factory

**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.)

May 18, 2020 at 8:20 am

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May 18, 2020 at 8:11 am

Regarding the very last sentence: I have read of this disturbing scenario, which did materialize, where open problems have been assigned to students as “exercises” (students had no idea that these problems were unsolved), so that the professor could benefit from their solutions and publish in academic journals (as a single author), in the unlucky scenario where a correct solution would emerge

May 18, 2020 at 8:53 am

This is very disturbing..! And rather unrealistic to expect undergraduates with no background to solve such a conjecture. But one (extreme value) is enough for wasting the 99.99% remainders’ effort, from such unscrupulous individuals…