Let $X$ have the logistic distribution with the PDF
$$f(x) = \frac{\exp(-x-θ)}{(1+\exp(-x-θ))^{2}}$$
Does $f(x)$ belong to the exponential family?
My solution is
$\exp[(-2)\cdot \ln(1+\exp\{-x-θ\})-x-θ]$.
Since it does not have the form $Q(θ)T(x)$, it does not belongs to exponential family.
At the end of my book there is an answer, which is "Yes", meaning it belongs to the exponential family.
Is my solution right or wrong?
Best Answer
Summary: No, the logistic distribution is not an exponential family.
You are missing some parenthesis in your definition, so I repeat the logistic density is $$ f(x) = \frac{\exp(-(x-θ))}{(1+\exp(-(x-θ)))^{2}}. $$ One interesting way to see the result is to start with the centered logistic density $$ f_0(x)=\frac{e^{-x}}{(1+e^{-x})^2}, $$ construct an exponential family by exponential tilting, and then observe that it is not the logistic distribution we get. (The logistic distribution is a location-scale family.)
First we need the cgf (cumulant generating function, log of moment generating function) of $f_0$, which is (from Wikipedia) $$ k(t)=\log B(1-t,1+t),\qquad t\in (-1,1) $$ where $B$ is the Beta function. Then by exponential tilting define the density $$ f_\theta(x)=\frac{f_0(x)e^{\theta x}}{e^{k(\theta)}} $$ and this manifestly does not have the form of a logistic density.