With Least Absolute Deviations (LAD) regression coefficients are estimated through minimization of the sum of the absolute values of the residuals.
Quantile regression aims at estimating either the conditional median or other quantiles of the response variable.
Is nonlinear quantile regression which aims to estimate the conditional median equivalent to nonlinear LAD regression?
Assume we have the following regression model:
$\mathbf{y} = f(\mathbf{x},\mathbf{\beta}) + \mathbf{\epsilon}$
The $\beta$ estimate of LAD regression is given by:
$ \hat{\beta}_{LAD} = \text{argmin}_{ b} \sum_{i=1}^n |y_i - f(\mathbf{b},x_i)|$
The $\beta$ estimate of Quantile regression is given by:
$ \hat{\beta}_{Quantile} = \text{argmin}_{ b} \sum_{i:y_{i} \geq f(\mathbf{b},x_i) }^n q|y_i -f(\mathbf{b},x_i)| + \sum_{i:y_{i} < f(\mathbf{b},x_i) }^n (1-q)|y_i -f(\mathbf{b},x_i)| $
If $q = 0.5$ (which is the case if we want to estimate the conditional median), this simplifies to:
$ \hat{\beta}_{Quantile, q =0.5} = \text{argmin}_{ b} \sum_{i=1}^n 0.5|y_i - f(\mathbf{b},x_i)|$, which is equivalent to $\hat{\beta}_{LAD}$
