# What is the proof for the factorization criterion?

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## The Factorization Criterion

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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?

• Any standard textbook on mathematical statistics would include a proof. – StubbornAtom Aug 17 at 15:16
• I'm using Mathematical Statistics with Applications 7e. There's no proof for it in that textbook. – Andrew Aug 18 at 16:30
• 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. – StubbornAtom Aug 18 at 16:42
• Awesome. Thanks – Andrew Aug 18 at 16:59