Quantile and CDF Regression Example

Quantile regression objective

$$ J(\tau) = E\left(\rho(\tau, Y - u(\tau, X)|X\right)$$

$J(\tau) = E\left(\rho(\tau, Y - u(\tau, X)|X\right)$

CDF regression objective

$$ J(y_c) = E\left(\mathbb{1}{Y < y_c} \log v(y_c, X) + (1 - \mathbb{1}{Y < yc}) \log(1 - v(y_x, X)) | X\right)$$

$J(y_c) = E\left(\mathbb{1}_{Y < y_c} \log v(y_c, X) (1 - \mathbb{1}_{Y < yc}) \log(1 - v(y_x, X)) | X\right)$

The functions $u$, $v$ must be monotonic in $\tau$ and $y_c$ respectively.

Unconditional distribution of $Y$

Quantile regression

CDF estimation via logistic regression with monotone network

Conditional distributional of $Y|X$

Quantile regression

CDF estimation via logistic regression with monotone network

TODO

Do more quantitative error plots etc.