3 Not Exponential Family

Distributions that are not members of the Exponential Family of Distributions:

• Uniform distribution

\[f(x)=\left\{\begin{array}{cr} \frac{1}{b-a},& a\leq x \leq b\\0,& \textrm{otherwise}\end{array}\right. \]

• Cauchy distribution

  \[f(x)=\frac{1}{\pi}\left[\frac{\beta}{\left(x-\alpha\right)^{2}+\beta^{2}}\right]\ \beta>0 \ \textrm{(scale)}, \alpha\ \textrm{(location)}\]

• Laplace family of distributions with non-zero mean

  \[f(x)=\frac{1}{2\beta}\exp\left(-\frac{|x-\mu|}{\beta}\right)\ \beta>0 \ \textrm{(scale)}, \alpha\ \textrm{(location)}\]

• Weibull distribution with unknown shape parameter

  \[f(x)=\left\{\begin{array}{cr}\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}\exp\left(-\left(\frac{x}{\lambda} \right)^{k}\right),& x\geq 0\\ 0, & x<0 \end{array}\right.\]

Can you demonstrate why these are not in the exponential family?