Testing for Normality in Geostatistics. A New Approach Based on the Mahalanobis Distance

Abstract

Simple kriging is a best linear predictor (BLP) and ordinary kriging is a best linear unbiased predictor (BLUP). When the underlying process is normal, simple kriging is not only a BLP but a best predictor (BT) as well, that is, under squared loss, this predictor coincides with the conditional expectation of the predictor given the information. In this scenario, ordinary kriging provides an approximation to the BP. For this reason, in applied geostatistics, it is important to test for normality. Given a realization of a spatial random process, the simple kriging predictor will be optimal if the random vector follows a multivariate normal distribution. Some classical tests, such as Shapiro-Wilk (SW), Shapiro-Francia (SF), or Anderson-Darling (AD) are frequently used to evaluate the normality assumption. Such approaches assume independence and hence are not effective for at least two reasons. On the one hand, observations in a geostatistical analysis are typically spatially correlated. On the other hand, kriging optimality as mentioned above is based on multivariate rather than univariate normality. In this work, we provide a simulation study to describe the negative effect of using normality univariate tests with geostatistical data. We also show how the Mahalanobis distance can be adapted to the geostatistical context to test for normality.