Maximum Likelihood Estimation for data with non normal distribution I have a set of data (results) that does not follow Normal distribution. In this case, wow can I get Maximum Likelihood Estimation?
Thank you very much
 A: Maximum Likelihood Estimation is about finding the parameters that maximize the likelihood of a probability model that you assume the data is generated from.
So if your model was
$$ y = \alpha + \beta'x + \epsilon$$
and you assumed $\epsilon \sim \mathcal{N}(0,\sigma)$, then you can express this as a probability model $Y|X,\theta \sim \mathcal{N}(\alpha+\beta'x, \sigma)$ where $\theta$ are the parameters. This implies the probability of observing the data given parameters is
$$ f(D; \theta) = \prod_{i=1}^{N} \frac{1}{\sigma\sqrt{2\pi}} e^{ (\frac{(y - (\alpha - \beta'x))}{\sigma})^2 }$$
where we view $f$ as a "likelihood" function of $\theta$.
To estimate non-normal models, you can simply switch out $f(D;\theta)$ for whatever probability model you have (for instance, a binomial pdf for a binary choice model). Finding the maximum likelihood estimates is usually done iteratively (for instance, gradient descent, Newton-Raphson, etc). The simplest way is to simply generate a grid over $\theta$, evaluate $f(D;\theta)$ at each grid point, and find the points that maximize the function.
