Calibration tests for multivariate Gaussian forecasts

Forecasts by nature should take the form of probabilistic distributions. Calibration, the statistical consistency of forecast distributions and observations, is a central property of good probabilistic forecasts. Calibration of univariate forecasts has been widely discussed, and significance tests are commonly used to investigate whether a prediction model is miscalibrated. However, calibration tests for multivariate forecasts are rare. In this paper, we propose calibration tests for multivariate Gaussian forecasts based on two types of the Dawid-Sebastiani score (DSS): the multivariate DSS (mDSS) and the individual DSS (iDSS). Analytic results and simulation studies show that the tests have sufficient power to detect miscalibrated forecasts with incorrect mean or incorrect variance. But for forecasts with incorrect correlation coefficients, only the tests based on mDSS are sensitive to miscalibration. As an illustration, we apply the methodology to weekly data on Norovirus disease incidence among males and females in Germany, in 2011-2014. The results further show that tests for multivariate forecasts are useful tools and superior to univariate calibration tests for correlated multivariate forecasts.