Repeated median regression

Method

The repeated median method estimates the slope of the regression line ${\displaystyle y=A+Bx}$ for a set of points ${\displaystyle (X_i,Y_i)}$ as

${\displaystyle \hat{B}=\underset{i}{\mathrm{median}}\underset{j\ne i}{\mathrm{median}}\mathrm{slope}(i,j)}$

where ${\displaystyle \mathrm{slope}(i,j)}$ is defined as ${\displaystyle (Y_j-Y_i)/(X_j-X_i)}$.

The estimated Y-axis intercept is defined as

${\displaystyle \hat{A}=\underset{i}{\mathrm{median}}\underset{j\ne i}{\mathrm{median}}\mathrm{intercept}(i,j)}$

where ${\displaystyle \mathrm{intercept}(i,j)}$ is defined as ${\displaystyle (X_j{Y}_i-X_i{Y}_j)/(X_j-X_i)}$.

A simpler and faster alternative to estimate the intercept ${\displaystyle \hat{A}}$ is to use the value ${\displaystyle \hat{B}}$ just estimated, thus:

${\displaystyle \hat{A}=\underset{i}{\mathrm{median}}(y_i-\hat{B}{x}_i)}$

Note: The direct and hierarchical methods of estimating ${\displaystyle \hat{A}}$ give slightly different values, with the hierarchical method normally being the best estimate. This latter hierarchical approach is identical to the method of estimating ${\displaystyle \hat{A}}$ in Theil–Sen estimator regression.