Assume a scalar random variable $X$ belongs to a vector-parameter exponential family with p.d.f.
$$ f_X(x|\boldsymbol \theta) = h(x) \exp\left(\sum_{i=1}^s \eta_i({\boldsymbol \theta}) T_i(x) – A({\boldsymbol \theta}) \right) $$
where ${\boldsymbol \theta} = \left(\theta_1, \theta_2, \cdots, \theta_s \right )^T$ is the parameter vector and $\mathbf{T}(x)= \left(T_1(x), T_2(x), \cdots,T_s(x) \right)^T$ is the joint sufficient statistic.
It can be show that the mean and the variance for each $T_i(x)$ exist. However, do the mean and the variance for $X$ (i.e. $E(X)$ and $Var(X)$) always exist as well? If not, is there an example of an exponential family distribution of this form whose mean and variable do not exist?
Thank you.
Best Answer
Taking $s=1$, $h(x)=1$, $\eta_1(\theta)=\theta$, and $T_1(x)=\log(|x|+1)$ gives $A(\theta)=\log\left(-2/(1+\theta)\right)$ provided $\theta \lt -1$, producing
$$f_X(x|\theta) = \exp\left(\theta\log(|x|+1) - \log\left(\frac{-2}{1+\theta}\right)\right) = -\frac{1+\theta}{2}(1+|x|)^\theta. $$
Graphs of $f_X(\ |\theta)$ are shown for $\theta=-3/2, -2, -3$ (in blue, red, and gold, respectively).
Clearly the absolute moments of weights $\alpha=-1-\theta$ or greater do not exist, because the integrand $|x|^\alpha f_X(x|\theta)$, which is asymptotically proportional to $|x|^{\alpha+\theta}$, will produce a convergent integral at the limits $\pm\infty$ if and only if $\alpha+\theta\lt -1$. In particular, when $-2 \le \theta \lt -1,$ this distribution does not even have a mean (and certainly not a variance).