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My textbook gives an example, that normal distribution family $\{N(0,\sigma^2):\sigma\in R^+\}$ is not complete, but a complete statistic, $T_n=\sum_{i=1}^n X_i^2$, can still be constructed from samples $(X_1,\cdots,X_n)$. So under what (sufficient / necessary) conditions can we draw complete statistics from an incomplete model? |
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Necessary conditions might be hard, at least for the question as wide-open as it currently stands. A pretty simple sufficient condition is for $X_{1},X_{2},\ldots,X_{n}$ to be a simple random sample from an exponential family. This covers your example above because the $N(0,\sigma)$ family can be written in the form $$ f(x|\sigma)=\frac{1}{\sqrt{2\pi}}\cdot \frac{1}{\sigma}\cdot \exp\left( \frac{-1}{\sigma^2}\cdot x^2 \right). $$ In one of Lehmann's books (I believe it is Testing Statistical Hypotheses) there is a proof that the canonical statistic $T$ when sampling from an exponential family is complete (plus it's minimal sufficient, ...the list goes on). Off the top of my head, with necessary conditions you're going on an adventure with image measures, transforms, functional analysis...? |
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