Fitting Beta Distribution Parameters to Y conditional on X

Question

I have a bi-variate data set where Y is in [0,1]. X is some measure of intensity and in this example is (0,~200) though there is no hard upper bound. X has a strong positive skew but I am not interested in the distribution of X right now. I don't believe what I want is a bivariate distribution.

I'm looking for a way to fit Beta Parameters conditional on X. An additional constraint would be that the mean(Y|X) should be non-decreasing as X increases.

I've tried a rolling window over X and fitting Beta to each subset of Y. Finally running a Linear Model of Shape1 ~ X, and Shape2 ~ X.

I would like to know if I'm on the right path, or if there is a better way. My preferred platform is R, but any help is appreciated.

Answer 1

Beta regression as proposed by Ferrari & Cribari-Neto (2004, Journal of Applied Statistics) can be used to model the parameters of a beta regression by linear predictors. Specifically, the beta distribution is then not characterized by two shape parameters $p$ and $q$ but instead by mean $\mu$ and precision parameter $\phi$ where $\mu = p/(p + q)$ and $\phi = p + q$. These parameters are then linked to linear preditors by suitable link functions, e.g., \begin{eqnarray*} \mathrm{logit}(\mu_i) & = & x_i^\top \beta\\ \log (\phi_i) & = & x_i^\top \gamma \end{eqnarray*} This model is implemented in the R package betareg (Cribari-Neto & Zeileis, Journal of Statistical Software, doi:10.18637/jss.v034.i02). However, this does not implement any monotonicity restrictions on the regression coefficients.

Moreover, if you are uncertain about the specific form of the regression relationship, you might finde beta regression trees useful (Grün et al., Journal of Statistical Software, doi:10.18637/jss.v048.i11).