# obtain (minimal) sufficient statistic for $\gamma$ knowing the canonical statistic $\theta(\gamma)$

A statistical model for a data set y is an exponential family , with canonical parameter vector $$\theta= (\theta_1,\theta_2,..\theta_k)$$ and canonical statistic $$t(y) = (t_1(y),t_2(y),..t_k(y))$$ if f has structure

$$f(y;\theta) = a(\theta)h(y) exp(\theta^T t(y))$$

For the birnbaum saunders distribution, $$\beta$$ being constant, it is easy to see that the canonical parameter is $$\theta = \gamma^{-2}$$ and canonical statistic $$t(y) = \frac{-(\sqrt{y/\beta} - \sqrt{\beta/y})^2 }{2}$$.

$$f(y) = \dfrac{\sqrt{y/\beta} + \sqrt{\beta/y}}{2\gamma y \sqrt{2 \pi}}exp \Big(- \frac{(\sqrt{y/\beta} - \sqrt{\beta/y})^2 }{2\gamma^2} \Big)$$

However I was asked to find a (minimal) sufficient statistic for $$\gamma$$ is it possible given that I have a (minimal)sufficient statistic $$t(y)$$ for $$\theta$$ ? to find a (minimal)statistic for $$\gamma$$ ?

• The notion of sufficiency is independent from the parameterisation of the model. – Xi'an Mar 22 at 12:55