Minimum Scoring Rule Inference

Proper scoring rules are devices for encouraging honest assessment of probability distributions. Just like log-likelihood, which is a special case, a proper scoring rule can be applied to supply an unbiased estimating equation for any statistical model, and the theory of such equations can be applied to understand the properties of the associated estimator. In this paper, we discuss some novel applications of scoring rules to parametric inference. In particular, we focus on scoring rule test statistics, and we propose suitable adjustments to allow reference to the usual asymptotic chi-squared distribution. We further explore robustness and interval estimation properties, by both theory and simulations.