The article contributes to the decision-theoretic framework for the evaluation of point forecasts. Let \(Y \in \mathbb{R}^d\) be a random variable observed by a forecaster, \(F\) be its cumulative distribution function, \(T(F) \in \mathbb{R}^k\) be a functional and \(S(x,y)\) be a scoring function (\(x \in \mathbb{R}^k\), \(y \in \mathbb{R}^d\)). A scoring function is said to be strictly consistent for \(T(F)\) if \(x=T(F)\) is the unique minimizer for \(E_F(S(x,Y))\) for all \(F\). A functional \(T(F)\) is called elicitable if there exists a strictly consistent scoring function for it. In case \(d=k=1\), the examples are moments, ratios of moments, quantiles and expectiles. However, the variance is not an elicitable functional if \(k=1\) but may be a component of an elicitable functional if \(k \geq 2\).
In the paper, the necessary and sufficient conditions for strictly consistent scoring functions are given. Some new examples of one-dimensional functionals which are not elicitable but may be a component of multi-dimensional elicitable functional are considered.