Question

Consider taking one sample from the distribution N(0,02). Is |x| a sufficient statistic?

Answer 1

The probability density function of one sample from $N(0, \sigma^2)$ is,

$f(x; \sigma) = \frac{1}{\sqrt{2\pi} \sigma} e^{-x^2/2\sigma^2}$

$= \frac{1}{\sqrt{2\pi} \sigma} e^{- \left | x \right |^2/2\sigma^2}$

$= \phi (u(\left | x \right |); \sigma) \times h(x)$

where

$\phi (u(\left | x\right |); \sigma) =\frac{1}{\sqrt{2\pi} \sigma} e^{-\left | x \right |^2/2\sigma^2}$

and

$h(x) = 1$

Thus, by factorization theorem and $f(x;\sigma) = \phi (u(\left | x \right |); \sigma) \times h(x)$ for all x and $\sigma$.

|x| is a sufficient statistic for $\sigma$.