91Ó°ÊÓ

Let \({X_1},{\bf{ }}.{\bf{ }}.{\bf{ }}.{\bf{ }},{\bf{ }}{X_n}\)follow a Bernoulli distribution with parameter p, that is,\({X_1},{\bf{ }}.{\bf{ }}.{\bf{ }}.{\bf{ }},{\bf{ }}{X_n} \sim Ber\left( p \right)\). Here p lies from 0 to 1, that is, \(0 \le p \le 1\) .

Let g(p) be an estimator function of p such that there exists an unbiased estimator of g(p).

Let us define the unbiased estimator of g(p).

Let\(\delta \left( {{X_1},{\bf{ }}.{\bf{ }}.{\bf{ }}.{\bf{ }},{\bf{ }}{X_n}} \right)\)be the unbiased estimator of g(p).

By the definition, it follows,

\(E\left( {\delta \left( {{X_1},{\bf{ }}.{\bf{ }}.{\bf{ }}.{\bf{ }},{\bf{ }}{X_n}} \right)} \right) = g\left( p \right)\,\,\forall \,\,p \in \left( {0,1} \right) \ldots \left( 1 \right)\)

Solving equation (1),

\(\begin{aligned}{}E\left( {\delta \left( {{X_1},{\bf{ }}.{\bf{ }}.{\bf{ }}.{\bf{ }},{\bf{ }}{X_n}} \right)} \right) &= g\left( p \right)\\ \Rightarrow \sum\limits_{x = 0}^n {g\left( x \right)} \left( {{}^n{C_x}} \right){p^x}{\left( {1 - p} \right)^{n - x}} &= g\left( p \right)\\ \Rightarrow g\left( 0 \right)\left( {{}^n{C_0}} \right){p^0}{\left( {1 - p} \right)^{n - 0}} + \ldots + g\left( n \right)\left( {{}^n{C_n}} \right){p^n}{\left( {1 - p} \right)^{n - n}} &= g\left( p \right)\end{aligned}\)

We can check that the Left-Hand Side of the solved equation is a polynomial equation in p whose maximum degree is n.

For LHS to be equal to RHS, g(p) should also be a polynomial equation with its maximum degree being up to n.

Hence, it is shown that expectation of every function\({\bf{\delta }}\left( {{{\bf{X}}_{\bf{1}}}{\bf{, }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}} \right)\)is a polynomial in p whose degree does not exceed n.