# How to draw a complete statistic from an incomplete statistical model?

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).