The answer to this question given by my professor was statistic T(x)= 1when X=0 and T(x) = 0 otherwise.
Can I consider E(x) = (1-p)/p and then cross multiply and take 1/(1+x) as an unbiased estimator of p?
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1$\begingroup$ What is T(x)? Since p is probability of success, with p=0 you'd observe no success and p=1 infinite number of successes so this doesn't make sense... $\endgroup$– TimCommented Nov 25, 2018 at 7:15
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1$\begingroup$ Do you have any doubts over the solution provided by your professor? Have you actually verified whether your proposed estimator is unbiased for $p$? $\endgroup$– StubbornAtomCommented Nov 25, 2018 at 7:57
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1$\begingroup$ Assuming distribution of $X$ is of the form $P(X=j)=p(1-p)^j\mathbf1_{j\in\{0,1,2,\ldots\}}$ and that $T(X)$ is a statistic based on the single observation $X$. $\endgroup$– StubbornAtomCommented Nov 25, 2018 at 8:21
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1 Answer
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Assuma as StubbornAtom in a comment that we have only one observation and that $P(X=j)=p(1-p)^j\mathbf1_{j\in\{0,1,2,\ldots\}}$ and $T(X) = \mathcal{I}_{\{X=0\}}$. Then just calculate the expectation of $T$, which in this case is just the probability that $X=0$, and you are done.
answered Oct 4, 2020 at 20:47