Find a sufficient statistic for $σ^2$ with $μ$ known, where $X_i$ is a random sample from $N(μ,σ^2)$

Question

Find a sufficient statistic for $σ^2$ with $μ$ known, where $X_i$ is a random sample from $N(μ,σ^2)$

I was able to find a sufficient statistic for $μ$ with $σ^2$ known, but I'm stuck on finding one for $σ^2$ when $μ$ is known. Can anyone give me some help?

I was using the factorization method before, is this the best way?

Answer 1

Hint:

$$f(x_1,\dots,x_n|\sigma^2)=\left(\frac{1}{\sqrt{2\pi\sigma^2}}\right)^ne^{-\frac{1}{2\sigma^2}\sum_{i=1}^n(x_i-\mu)^2}$$

So if you let

$$T(X_1,\dots,X_n)=\sum_{i=1}^n(x_i-\mu)^2$$

could you use the factorization theorem to conclude $T$ is sufficient?

Answer 2

The conditional density of the sample is $$\displaystyle\prod_i \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x_i - \mu)^2}{2 \sigma^2}} =\left(\frac{1}{\sigma\sqrt{2\pi}}\right)^n \exp\left({-\dfrac{\displaystyle\sum_i (x_i - \mu)^2}{2 \sigma^2}} \right)$$

so you might expect sufficient statistics to be something like $\displaystyle\sum_i (x_i - \mu)^2 $ or $\displaystyle\left(\sum_i x_i^2,\sum_i x_i\right)$.

The former has the merit of being a single value and being minimal sufficient, while the latter has the merit of also being a sufficient statistic for $\mu$ and $\sigma^2$ together even if $\mu$ is also unknown