Calibration bands for mean estimates within the exponential dispersion family

A statistical model is said to be calibrated if the resulting mean estimates perfectly match the true means of the underlying responses. Aiming for calibration is often not achievable in practice as one has to deal with finite samples of noisy observations. A weaker notion of calibration is auto-calibration. An auto-calibrated model satisfies that the expected value of the responses being given the same mean estimate matches this estimate. Testing for auto-calibration has only been considered recently in the literature, and we propose a new approach based on calibration bands. Calibration bands denote a set of lower and upper bounds such that the probability that the true means lie simultaneously inside those bounds exceeds some given confidence level. Such bands were constructed by Yang and Barber ((2019). Contraction and uniform convergence of isotonic regression. Electronic Journal of Statistics, 13(1), 646-677. https://doi.org/10.1214/18-EJS1520) for sub-Gaussian distributions. Dimitriadis et al. ((2023). Honest calibration assessment for binary outcome predictions. Biometrika, 110(3), 663-680. https://doi.org/10.1093/biomet/asac068) then introduced narrower bands for the Bernoulli distribution. We use the same idea in order to extend the construction to the entire exponential dispersion family that contains, for example, the binomial, Poisson, negative binomial, gamma, and normal distributions. Moreover, we show that the obtained calibration bands allow us to construct various tests for calibration and auto-calibration, respectively. As the construction of the bands does not rely on asymptotic results, we emphasize that our tests can be used for any sample size.