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I'd like to know if there is a multivariate regression model with uniform errors. More specifically, I'd like to find the Maximum Likelihood Estimators for $\beta^a$ and $\beta^b$. The model is given by

\begin{equation} \left( \begin{array}{c} Y^a \\ Y^b \end{array} \right) = % \left( \begin{array}{c} \mu_i^a \\ \mu_i^b \end{array} \right) + \Sigma^{1/2} % \left( \begin{array}{c} \varepsilon_i^a \\ \varepsilon_i^b \end{array} \right) \qquad i = 1, \ldots, n \end{equation},

$Y^a$ and $Y^b$ are random variables jointly uniformly distributed and response variable, $\left( \begin{array}{c} \varepsilon_i^a \\ \varepsilon_i^b \end{array} \right)$ is uniformly distributed, $\mu_i^a = (X^a)^T\beta^a$, and $\mu_i^b = (X^b)^T\beta^b$. $X^a = (X_1^a, \ldots, X_p^a)$ and $X^b = (X_1^b, \ldots, X_p^b)$ are $p$ non-stochastic variables, called regressors.

Wagner Jorge
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    You can use regression with Subbotin errors: https://stats.stackexchange.com/questions/203629/is-there-a-plateau-shaped-distribution/203634#203634 – Tim Apr 12 '18 at 07:34
  • @Tim That's a good answer--the post you reference is not a mere duplicate. – whuber Apr 12 '18 at 14:04
  • @whuber I didn't suggest it is, I thought it may be helpful for OP – Tim Apr 12 '18 at 14:07
  • @Tim That's a good answer and start point, but I'd like to solve the problem for bi-dimensional uniform errors. – Wagner Jorge Apr 12 '18 at 15:22

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