Finding unbiased estimator - can it be a function of parameter

Question

The question is : Let $X_1,\ldots, X_n$ ($n\ge2$) be a random sample from Bin$(1, p)$. Find an unbiased estimator of $p^2$.

I got on solving that $T(0) = -p$; $T(1)=1.$
My doubt is...does the unbiased estimator qualify as being one if it is a function of the parameter it is trying to estimate? (like $p$ in this case)

Answer 1

does the unbiased estimator qualify as being one if it is a function of the parameter it is trying to estimate?

an estimator, biased or not, cannot be a function of the parameter....an estimator, by definition, is a function (alone) of the data.

First: the pmf of a $Bin(1;p)$ is the following

$$\mathbb{P}[X=x]=p^x(1-p)^{1-x}\mathbb{1}_{\{0;1\}}(x)$$

It is self evident that $\mathbb{E}[X^k]=p$ $\forall k$ and also

$\mathbb{E}[X_i\cdot X_j]=\mathbb{E}[X_i]\mathbb{E}[X_j]$ for $i \ne j$ for independence

Thus I choose

$T=X_1\cdot X_2$ as an estimator for $p^2$ because

$$\mathbb{E}[T]=p\cdot p= p^2$$

that is unbiased for $p^2$

Plase, try to use MathJax to write formulas