Let $X\sim Bin(n,p)$. The problem is to show that a UMVUE can exist for $g(p)$ only if $g(p)$ is a polynomial in $p$ of degree at most $n$.

For the case when $g(p) = \frac{1}{p}$ we can show that it is impossible by taking $p \to 0$ and showing no such $\delta(X)$ that is unbiased for $\frac{1}{p}$ can exist.

But I am stuck on showing this is the case for any function that is not a polynomial in $p$ of degree at most $n$.

I guess the two approaches would be:

  1. Show $\sum_{x=1}^n \delta(x) \,{n \choose x}\,p^x (1-p)^{n-x}= g(p)$ only has a solution $\delta (x)$ if $g(p)$ is such a polynomial.

  2. Show that $\delta(X)$ can only be uncorrelated to every unbiased estimator of $0$ if $\Bbb E[\delta(X)]$ is a polynomial in $p$.

But I am unsure how to do this.

Is it enough to simply observe that

$$\sum_{x=1}^n \delta(x) \,{n \choose x} \, p^x (1-p)^{n-x}$$

is a polynomial in $p$ of degree at most $n$?

  • 1
    $\begingroup$ Yes the last expression is quite clear. The expectation is [as a function of $p$] a polynomial of degree $n$ or less. $\endgroup$ – Xi'an May 25 at 12:35

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