A criterion for estimating the largest linear homoscedastic zone in Gaussian data

A criterion is constructed to identify the largest homoscedastic region in a Gaussian dataset. This can be reduced to a one-sided non-parametric break detection, knowing that up to a certain index the output is governed by a linear homoscedastic model, while after this index it is different (e.g. a different model, different variables, different volatility, ….). We show the convergence of the estimator of this index, with asymptotic concentration inequalities that can be exponential. A criterion and convergence results are derived when the linear homoscedastic zone is bounded by two breaks on both sides. Additionally, a criterion for choosing between zero, one, or two breaks is proposed. Monte Carlo experiments will also confirm its very good numerical performance.