# How to show a UMVUE exists only if $g(p)$ is a polynomial of degree at most $n$?

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 $$n$$?

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