A pseudo-algorithmic separation of lines from pseudo-lines

The least-trimmed squares estimation (LTS) is a robust solution for regression problems. On the one hand, it can achieve any given breakdown value by setting a proper trimming fraction. On the other hand, it has $\text{n}$-consistency and asymptotic normality under some conditions. In addition, the LTS estimator is regression, scale, and affine equivariant. In practical regression problems, we often need to impose constraints on slopes. In this paper, we describe a stable algorithm to compute the exact LTS solution for simple linear regression with constraints on the slope parameter. Without constraints, the overall complexity of the algorithm is $\text{O}\text{(n}{}^2\text{log}\text{n)}$ in time and O(n²) in storage. According to our numerical tests, constraints can reduce computing load substantially. In order to achieve stability, we design the algorithm in such a way that we can take advantage of well-developed sorting algorithms and softwares. We illustrate the algorithm by some examples.