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Suppose that $ X_1,\ldots,X_n$ are iid poisson($b$); $c = b^2$ and $S_n = \sum X_i$

To Show that $S_n$ is a complete sufficient statistic for $c$. I can prove using exponential family that $S_n$ is sufficient for $b$, but how may I prove it for $b^2$? Will anybody help me?

Michael Hardy
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Enas
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1 Answers1

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If the conditional probability distribution of $X_1,\ldots,X_n$ given $S_n$ depends on $b^2$, then it depends on $b$. Since it does not depend on $b$, it therefore does not depend on $b^2$.

Michael Hardy
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  • Are there any theorem that proves this? in order to mention it in the question solution – Enas Apr 01 '14 at 16:55
  • You can write an argument as follows: The conditional distribution of $X_1,\ldots,X_n$ given $S_n$ when $c=b_1^2$ is the same as the conditional distribution of $X_1,\ldots,X_n$ given $S_n$ when $b=b_1$. That last is the same as the conditional distribution of $X_1,\ldots,X_n$ given $S_n$ when $b=\text{a different number }b_2$, since you've already proved that $S_n$ is sufficient, and finally that distribution_ is the same as the conditional distribution of $X_1,\ldots,X_n$ given $S_n$ when $c=b_2^2$. ${}\qquad{}$ – Michael Hardy Apr 01 '14 at 17:14