Is there an unbiased estimator of PMF of a random variable $Y=\sum_{i=1}^{n} X_n $ where $X_i$ are independent Bernoulli trials with probability $p$, that is, the estimator of: \begin{equation}\tag{1} f(k,n)=P(Y=k|n)={n\choose k}p^k(1-p)^{n-k} \end{equation} for an arbitrary $n \geq 0$ and $n\geq k \geq 0$ when based on a sample of $s $ observations from that said Bernoulli RV; $\lbrace x_i\rbrace_{i=1}^{s}$?

More info and progress so far:
An obvious candidate for the estimator of PMF would be: \begin{equation}\tag{2} \hat{f}(k,n)={n\choose k}\hat{p}^k(1-\hat{p})^{n-k} \end{equation} where $\hat{p}= \frac{\sum_{i=1}^{s}x_i}{s}$. That is consistent but clearly not unbiased as apparent from Jensen's inequality. So far I found that the estimator based on hypergeometric distribution: \begin{equation}\tag{3} \hat{f}(k,n)=\dfrac{{\hat{p}*s\choose k} {s-\hat{p}*s\choose n-k}}{{n\choose k}} \end{equation} with $\hat{p}$ defined as above seems to be unbiased (at least in my simulations, although I wasn't able to prove it). This function $\hat{f}(k,n)$ is however usefull only for $n\leq s$. Does there exist some unbiased estimator of $f(k,n)$ which would work also for $n>s$? (Possibly some correction of estimator (2) or some generalization of estimator (3). Or something completely different.)


Since, for a Binomial $\text{B}(n,p)$ variable $X$, and $k\le n$, the factorial moment is given by $$\mathbb{E}_p[X(X-1)\cdots(X-k+1)] = n(n-1)\cdots(n-k+1)p^k,$$ the $s$ Bernoulli rvs $\lbrace X_i\rbrace_{i=1}^{s}$ can easily return independent unbiased estimates of both $p^k$ and $(1-p)^{n-k}$ if $k+(n-k)\le s$, that is, if $n\le s$. And hence of $\mathbb{P}(X=k)\propto p^k(1-p)^{n-k}$.

It sounds likely that an unbiased estimator of the above does not exist when $n>s$, because, for instance, developing $(X_1+\ldots+X_s)^k$ shows that the maximum number of terms in a product is $s$, with expectation $p^s$. No higher power of $p$ or $(1-p)$ can appear for this reason. Actually, the proof is straightforward: consider there exists such an unbiased estimator, denoted by $G(X_1+\ldots+X_s)$ since by sufficiency there exists an unbiased estimator based on the sum. Then it satisfies $$\mathbb{E}_p[G(X_1+\ldots+X_s)]=\sum_{j=1}^s \underbrace{G(j){s \choose j}}_\text{independent from $p$}p^j(1-p)^{s-j}=p^k(1-p)^{n-k}$$ or $$\sum_{j=1}^s \overbrace{G(j){s \choose j}}^{\text{non-negative}}p^{j-k}(1-p)^{s-j-n+k}=1$$ Letting $p$ tend to $0$ or $1$ leads to explosive terms when $j-k<0$ and when $s-j-n+k<0$, unless the coefficient $$G(j){s \choose j}p^{j-k}$$ is equal to zero. If $n>s$, then, for all $0\le j\le s$ and all $0\le k\le n$, either $j<k$ or $s-j<n-k$, which leads to an impossibility since $G(m)=0$ for all $0\le m\le s$. Therefore there is no unbiased estimator of $p^k(1-p)^{n-k}$ when $n>s$.

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    $\begingroup$ Thank you very much, professor Xi'an. That is a very neat counterexample. Now I know that my search for such estimator is futile and can allocate my time accordingly. Thanks again. $\endgroup$ – Nicolle Jan 6 '19 at 19:07
  • $\begingroup$ Thank you, this is a very neat case about the rarity of functions that allow for an unbiased estimator, a rarity that is often forgotten by textbooks. $\endgroup$ – Xi'an Jan 6 '19 at 19:50

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