Saddlepoint approximations for the distribution of some robust estimators of the variogram

Abstract

In this paper, we obtain a saddlepoint approximation for the small sample distribution of several variogram estimators such as the classical Matheron’s estimator, some M-estimators like the robust Huber’s variogram estimator, and also the $$\alpha $$-trimmed variogram estimator. The tail probability approximation obtained is very accurate even for small sample sizes. In the approximations we consider that the observations follow a distribution close to the normal, specifically, a scale contaminated normal model. To obtain the approximations we transform the original observations into a new ones, which can be considered independent if a linearized variogram can be accepted as model for them. To check this, a goodness of fit test for a variogram model is defined in the last part of the paper.