Question

Question:Suppose that a random variable X has the geometric distribution with an unknown parameter p (0<p<1). Show that the only unbiased estimator of p is the estimator \({\bf{\delta }}\left( {\bf{X}} \right)\) such that \({\bf{\delta }}\left( {\bf{0}} \right){\bf{ = 1}}\) and \({\bf{\delta }}\left( {\bf{X}} \right){\bf{ = 0}}\) forX>0.

Short answer

the only unbiased estimator of p is the estimator \(\delta \left( X \right)\) such that \(\delta \left( 0 \right) = 1\) and \(\delta \left( X \right) = 0\).

Step-by-step solution

Given information

Therefore, the pdf of X is \(P\left( {X = x} \right) = p{\left( {1 - p} \right)^x},x \ge 0\)

Define the unbiased estimator

An estimator \(\delta \left( X \right)\,\,of\,\,g\left( \theta \right)\) is unbiased if \(E\left( {\delta \left( X \right)} \right) = g\left( \theta \right)\) for all possible values of \(\theta \) .

Therefore,

\(\begin{aligned}{}E\left( {\delta \left( X \right)} \right) &= p\\ \Rightarrow \sum\limits_{x = 0}^n {p{{\left( {1 - p} \right)}^{x - 1}}} \delta \left( X \right) &= p\end{aligned}\)

Do the required calculation

It is given that \(\delta \left( 0 \right) = 1\) and \(\delta \left( X \right) = 0\), therefore, the above equation becomes,

\(\begin{aligned}{}p{\left( {1 - p} \right)^0}\delta \left( 0 \right) + \ldots + p{\left( {1 - p} \right)^n}\delta \left( n \right)\\ &= p \times 1 + 0 + \ldots 0\\ &= p\end{aligned}\)

Therefore, the only unbiased estimator of p is the estimator \(\delta \left( X \right)\) such that \(\delta \left( 0 \right) = 1\) and \(\delta \left( X \right) = 0\).