3 The exponential family 3.3 Examples of linear EF distributions 3.5 Properties of

Y\sim\mathrm{EF}

3.4 Transformation of the random variable

Assume that pdf/pmf for the random variable $Y$ belongs to the exponential family. Then will the random variable $X$ under some one-to-one transformation $Y=T(X)$ also belong to the exponential family? To assess this, recall that when transforming a random variable, the rate of change between $X$ and $Y$ needs to be accounted for via the Jacobian when deriving the pdf/pmf for $X$ :

f_{X}(x|\theta)=f_{Y}(y=T(x)|\theta)\left|\frac{dy}{dx}\right|

Substituting the exponential family pdf/pmf for $f_{Y}(y|\theta)$ and the Jacobian, $|T^{\prime}(x)|$ , gives:

\displaystyle f_{X}(x|\theta)=q(y=T(x))|T^{\prime}(x)|~{}~{}\exp\{\theta T(x)-% \kappa(\theta)\}=h(x)\exp\{\theta T(x)-\kappa(\theta)\}.

Here, $h(x)$ does not depend on $\theta$ and is in its own right a pdf/pmf as it is a transformation of the pdf/pmf $q(y)$ . Furthermore, since the transform $T(x)$ also does not depend on $\theta$ , the pdf/pmf $f_{X}(x|\theta)$ also belongs to the exponential family. The function $T(x)$ is called the sufficient statistic for the parameter $\theta$ .

Definition 3.4.1.

A statistic $t=T(X)$ is a sufficient statistic for the parameter $\theta$ if the conditional distribution of the data given the statistic does not depend on the parameter $\theta$ , i.e.:

\mathbb{P}(x|t,\theta)=\mathbb{P}(x|t)

Exercise 3.25
The pdf for the random variable $X\sim\mathrm{Gamma}(\alpha,1)$ is:

f_{X}(x|\alpha)=\frac{1}{\Gamma(\alpha)}x^{\alpha-1}\exp\{-x\}\quad\mathrm{for% }~{}~{}x>0\quad\mathrm{and}\quad\alpha>0.

Show that this belongs to the exponential family. What is the sufficient statistic for the canonical parameter?