Exact algorithms for computing the least median of squares estimate in multiple linear regression

Abstract

We propose two finite algorithms to compute the exact least median of squares (LMS) estimates of parameters of a linear regression model with p coefficients. The first algorithm is similar to Stromberg's (1993) exact algorithm. It is based on the exact fit to subsets of p cases and uses impossibility conditions to avoid unnecessary calculations. The second one is based on a branch and bound (BAB) technique. Empirical results suggest that the proposed algorithms are faster than the finite exact algorithms described earlier in the literature.

A Fast Algorithm for S-Regression Estimates

“…Exact algorithms to compute the LMS-estimator have been proposed by Steele and Steiger (1986), Stromberg (1993), and Agulló (1997a,b). Hawkins (1994) proposed an approximate algorithm for the LTS-estimator, while Agulló (2001) described an exact algorithm for the same estimator. Unfortunately, these algorithms are feasible only when n and p are small.…”

Analyzing the Number of Samples Required for an Approximate Monte-Carlo LMS Line Estimator

“…In Section 4 we consider specifically Gaussian and exponential pdf's. 1 In the full paper we also consider the uniform pdf [7].…”

“…Without loss of generality, we assume that y i , y j ∼ E (1), that is, y i , y j are independent, exponentially distributed with λ = 1. (This one-sided distribution is a special case of the gamma and Weibull distributions.)…”

“…His algorithm runs in O(n 4 log n) time for simple regression. A slightly improved algorithm which runs in O(n 4 ) time is due to Agulló [1]. The best algorithm known for finding the LMS strip (in the plane) is the topological plane-sweep algorithm due to Edelsbrunner and Souvaine [3].…”

QUANTILE APPROXIMATION FOR ROBUST STATISTICAL ESTIMATION AND k-ENCLOSING PROBLEMS

“…For our purposes, it suffices to use a simpler O(n d+2 ) time algorithm due to Agulló. 4 His algorithm considers all n d+1 elemental subsets and computes the number of points within each resulting hyperstrip. Among all hyperstrips containing the desired number of points, the strip of minimum width is returned.…”