An analysis of the least median of squares regression problem

The optimization problem that arises out of the least median of squared residuals method in linear regression is analyzed. To simplify the analysis, the problem is replaced by an equivalent one of minimizing the median of absolute residuals. A useful representation of the last problem is given to examine properties of the objective function and estimate the number of its local minima. It is shown that the exact number of local minima is equal to $ {p+\lfloor (n-1)/2 \rfloor \choose{p}} $, where $ p $ is the dimension of the regression model and $ n $ is the number of observations. As applications of the results, three algorithms are also outlined.