A sufficient statistic for a normal variance

Question

You have independent observations X1, \dots, Xn from a normal distribution with known mean \mu but unknown variance \sigma^2. Using the factorization idea, find a sufficient statistic for \sigma^2.

Accepted Answer

The plan. Write the joint density as a \sigma^2-piece (touching the data only through some summary) times a piece free of \sigma^2. The summary in the first piece is sufficient. Look at the density. The joint density is f(x; \sigma^2) = (2\pi\sigma^2)^{-n/2} \exp\!\Big(-\frac{1}{2\sigma^2}\sumi (xi - \mu)^2\Big). Because \mu is known, the quantity \sumi (xi - \mu)^2 is a number you can compute straight from the data. Everything involving \sigma^2 depends on the data only through that single sum. So set g(T, \sigma^2) = (2\pi\sigma^2)^{-n/2} e^{-T/(2\sigma^2)}, \quad T = \sumi (xi - \mu)^2, \q…

A sufficient statistic for a normal variance

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