I would like to know whether there exists a closed-form solution for the $\lambda$-parameter that maximizes the log-likelihood function of Yeo-Johnson transformed random variables that (before the transformation) are distributed according to the normal-inverse Gaussian distribution.
I.e., given the parameters $\mu$, $\delta$, $\beta$, and $\alpha$, which specify a normal-inverse Gaussian distribution, my goal is to find the Yeo-Johnson transformation parameter $\lambda$ that maximizes the log-likelihood function (given for instance here and here) in closed form (instead of sampling from the distribution, transforming those sampled values, and repeating that transformation for various $\lambda$ to maximize the log-likelihood function).
I suspect that such a closed-form solution does not exist, but I am not sure. Any insights or suggestions are greatly appreciated!
