Let $f_\theta(x) = \exp(\theta T(x) - K(\theta))$ be a one-parameter exponential family of probability density functions with respect to the Lebesgue measure on $\mathbb{R}$, for $\theta$ in an open interval $\Theta$. Let $F_\theta(x)$ be the corresponding cumulative distribution function (CDF), and let $\Phi(z)$ be the CDF of the standard normal distribution $N(0,1)$.
I am interested in identifying conditions that $T$ and $K$ must satisfy in order for the normalizing transformation $g_\theta(x) := \Phi^{-1}(F_\theta(x))$ to be an elementary function of $x$ for all $\theta \in \Theta$. (Here, "elementary" is used in the sense of Liouville: functions built from constants, algebraic functions, exponentials, and logarithms through finite composition and arithmetic operations.)
Reformulation as a Differential Equation
The condition $\Phi(g_\theta(x)) = F_\theta(x)$ implies, by differentiation with respect to $x$, the following first-order nonlinear ordinary differential equation for $g_\theta(x)$: $$\phi(g_\theta(x)) \cdot g'_\theta(x) = f_\theta(x)$$ where $\phi(z) = (2\pi)^{-1/2} e^{-z^2/2}$ is the standard normal PDF. This can be rewritten as: $$g'_\theta(x) = \sqrt{2\pi} \exp\left(\theta T(x) - K(\theta) + \frac{1}{2} g_\theta(x)^2\right)$$ The problem is thereby equivalent to asking:
For which choices of the function $T(x)$ does this differential equation admit a solution $g_\theta(x)$ that is an elementary function of $x$ for every $\theta$?
The trivial case is the normal distribution itself, where $T(x)=x$ and $g_\theta(x)$ is linear in $x$. I am seeking non-trivial examples or, ideally, a complete characterization.
This question is motivated by an attempt to generalize the methods introduced in the paper "Bounds on Tail Probabilities in Exponential families."
