Calculating the Breakdown Point of Sparse Linear Models

In robust statistics, the concept of breakdown point was introduced to quantify the robustness of an estimator in a linear regression model. Computing the breakdown point is useful in tuning some robust regression estimators (e.g., the least trimmed squares estimator). Computing the breakdown point for a structured linear model (i.e., one with dependencies among some p rows of the n×p design matrix X) can be very demanding. This article presents an algorithm for calculating the maximum breakdown point for sparse linear models, which are a special type of structured linear model whose design matrix has many zero entries. The algorithm decomposes a sparse design matrix into smaller submatrixes on which the computation is performed, thereby leading to substantial savings in computation. An assembly process, along with a few numerical examples, illustrate the application of the algorithm and demonstrate its computational benefits.