Simple Jacobian Adjustment question

Modeling
1

Hi, I have a simple Jacobian adjustment question:

Assume I have data: (\vec{x}_i, y_i),\vec{x}_i \in \mathbb{R}^K, i=1\ldots,N; and I know that the \vec{x}_i's are noise free, but the y_i's are Beta distributed (with known a_i and b_i \; Beta parameters, so y_i \sim Beta(a_i,b_i). I would like to find a best fit exponential (for the sake of example) through this data. So \exp\left(\beta^T \vec{x}\right) \sim Beta(\vec{a},\vec{b}).

First of all, is this the correct way to do this?

Secondly, what if there were error in both \vec{x}_i AND y_i?. Say, we knew the \vec{x}_i were normally distributed with mean \mu and covariance \Sigma.

Do we have to do implicit derivatives to adjust the Jacobian?

Can someone help me out here? (maybe I need more coffee to see it…)

Thanks

2

Known data do not necessitate Jacobians. If you believe \mathbf{x} is multivariate normal that is fine. The more common way to do regressions with outcomes that are proportions would be to set the conditional mean to some function of a linear predictor, usually a CDF of the standard logistic, normal, etc. Then there is another parameter, \psi > 0, that may or may not be a function of predictors such that a = \mu \psi and b = \left(1 - \mu\right) \psi.