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 $\gamma$ ?

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

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