Isotonic regression for variance estimation and its role in mean estimation and model validation

We study isotonic regression which is a non-parametric rank-preserving regression technique. Under the assumption that the variance function of a response is monotone in its mean functional, we investigate a novel application of isotonic regression as an estimator of this variance function. Our proposal of variance estimation with isotonic regression is used in multiple classical regression problems focused on mean estimation and model validation. In a series of numerical examples, we (1) explore the power variance parameter of the variance function within Tweedie's family of distributions, (2) derive a semi-parametric bootstrap under heteroskedasticity, (3) provide a test for auto-calibration, (4) explore a quasi-likelihood approach to benefit from best-asymptotic estimation, (5) deal with several difficulties under lognormal assumptions. In all these problems we verify that the variance estimation with isotonic regression is essential for proper mean estimation and beneficial compared to traditional statistical techniques based on local polynomial smoothers.