Let $X_1$ and $X_2$ be a random sample from the geometric distribution with $Pr(X_i=j)=p(1-p)^{j-1}$ for $i = 1, 2$, $j = 1, 2, \ldots$ and $0<p<1$.

Which statistics $T(X)$ could be an unbiased estimator for $\frac{p}{(1+p)}$?

My attempt: Let $T(X)$ be the required unbiased estimator. Then, $E(T(X))=\frac{p}{(1+p)}$ and $$ \sum T(j)p(1-p)^{j-1}=p(1+p)^{-1}=p(1-p+p^{2}-p^{3}+...) \\ \sum T(j)(1-p)^{j-1}=(1-p+p^{2}-p^{3}+...) $$

Can we find the estimator by comparing the two sides of above equation? If not, how can we find the required unbiased estimator?

As it is said that $X$ is a random sample of size $2$, should we consider the joint pmf while finding the unbiased estimator?

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    $\begingroup$ That summation should be actually $\sum_{j=1}^{\infty} T(j)(1-p)^{j-1}$ instead of $\sum_{j=1}^{\infty} T(X)(1-p)^{j-1}$. $\endgroup$ – Marcelo Ventura May 30 '16 at 18:07
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    $\begingroup$ Recall the definition of expected value (to correct your formulas) and use $q=1-p$. $\endgroup$ – Viktor May 30 '16 at 18:12
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    $\begingroup$ Since the pgf of this geometric pmf is $E(s^X)=\frac{ps}{1-(1-p)s}$, setting $\frac{ps}{1-(1-p)s}=\frac{p}{1+p}$ gives $s=\frac{1}{2}$. Hence an unbiased estimator based on two observations is simply $\frac{2^{-X_1}+2^{-X_2}}{2}$. $\endgroup$ – StubbornAtom Jun 1 '19 at 14:55
  • $\begingroup$ @StubbornAtom , you should transform your comment into an answer. $\endgroup$ – Marcelo Ventura May 21 at 5:48

As is shown here in a similar question, one can simply work with the PGF or MGF to find an unbiased estimator based on a single observation.

For a geometric variable $X$ with given pmf, we have

\begin{align} E\left[t^X\right]&=\sum_{j=1}^\infty t^jp(1-p)^{j-1} \\&=pt\sum_{j=1}^\infty ((1-p)t)^{j-1} \\&=\frac{pt}{1-(1-p)t}\qquad\quad,\,\,\small|t|<\frac1{1-p} \end{align}

Solving $\frac{pt}{1-(1-p)t}=\frac{p}{1+p}$ for $t$ gives $t=\frac12$.

That is, for every $p\in(0,1)$,


So an unbiased estimator of $\frac{p}{1+p}$ based on two observations $X_1$ and $X_2$ is simply $$T(X_1,X_2)=\frac{2^{-X_1}+2^{-X_2}}2$$

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