Let's consider the family of transformations given by $$g_a(Y)=\begin{cases} \frac{e^{aY}-1}{a} & \text{ for } a\neq 0 \\ Y & \text{ for } a=0 \end{cases}$$ for $Y\in\mathbb{R}$. Analogous to the estimation of the Box-Cox parameter $\lambda$, the parameter $a\in\mathbb{R}$ can be estimated using a profile likelihood approach. Let $Y_1,...,Y_n\in\mathbb{R}$ be independent responses together with corresponding predictors $\mathbf{x}_1,...,\mathbf{x}_n\in\mathbb{R}^n$. We assume for $a$ that there exist $\textbf{b}\in\mathbb{R}^p$ and $\sigma^2>0$ such that $g_a(Y_i)\sim N(\textbf{x}_i^T \textbf{b},\sigma^2)$ for $i=1,...,n$.
Here is my questions:
Can we derive such a density function $f_Y$ of the untransformed observations $Y_i$. If so, would it be $$f_{Y_i}(\textbf{x})=\frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{1}{2}\cdot \frac{(\textbf{x}-\textbf{x}_i^T\textbf{b})^2}{\sigma^2}}$$
How would we find the log-likelihood function $\ell(a,b,\sigma^2;y_1,...,y_n)$? I know that $\sum_{i=1}^n \log(f_{Y_i}(y_i))$.
Thanks in advance.
