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I am not sure where I should even start with this problems. I know that the sum of negative binomial random variables is itself a negative binomial random variable. I am sure that I can show that the sum is also a sufficient and complete statistic. However, beyond this point I am not sure where to go with finding the "best" unbiased estimator. Any suggestions would be greatly appretiated

Michael Hardy
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mathestats
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1 Answers1

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Cite the Lehmann–Scheffé theorem. The conditional expected value of an unbiased estimator, given a complete sufficient statistic, is the best unbiased estimator.

As an unbiased estimator of the probability that $X_1\le 3,$ you can use the indicator function of that event: $$ Y = \begin{cases} 1 & \text{if } X_1\le 3, \\ 0 & \text{otherwise.} \end{cases} $$ You then need the conditional expected value of $Y$ given your complete sufficient statistic.

Michael Hardy
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