The Factorization Criterion


Let $U$ be a statistic based on the random sample $Y_1, Y_2,...Y_n$. Then $U$ is a sufficient statistics for the estimation of a parameter $\theta$ if and only if the likelihood $L(\theta)$ can be factored into two non-negative functions, $$L(\theta)=g(u,\theta)h(y_1,y_2,...,y_n)$$

where $g(u,\theta)$ is a function only of $u$ and $\theta$ and $h(y_1,y_2,...,y_n)$ is not a function of $\theta$.

What is the proof for the factorization criterion?

  • $\begingroup$ Any standard textbook on mathematical statistics would include a proof. $\endgroup$ – StubbornAtom Aug 17 at 15:16
  • $\begingroup$ I'm using Mathematical Statistics with Applications 7e. There's no proof for it in that textbook. $\endgroup$ – Andrew Aug 18 at 16:30
  • $\begingroup$ See Statistical Inference by Casella/Berger and Introduction to Mathematical Statistics by Hogg/Craig for example. A more rigorous proof can be found in Lehmann/Romano's Testing Statistical Hypotheses. $\endgroup$ – StubbornAtom Aug 18 at 16:42
  • $\begingroup$ Awesome. Thanks $\endgroup$ – Andrew Aug 18 at 16:59

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