  In statistical analysis, measuring a score of predictive performance is an
important task. In many scientific fields, appropriate scores were tailored to
tackle the problems at hand. A proper score is a popular tool to obtain
statistically consistent forecasts. Furthermore, a mathematical
characterization of the proper score was studied. As a result, it was revealed
that the proper score corresponds to a Bregman divergence, which is an
extension of the squared distance over the set of probability distributions. In
the present paper, we introduce composite scores as an extension of the typical
scores in order to obtain a wider class of probabilistic forecasting. Then, we
propose a class of composite scores, named Holder scores, that induce
equivariant estimators. The equivariant estimators have a favorable property,
implying that the estimator is transformed in a consistent way, when the data
is transformed. In particular, we deal with the affine transformation of the
data. By using the equivariant estimators under the affine transformation, one
can obtain estimators that do no essentially depend on the choice of the system
of units in the measurement. Conversely, we prove that the Holder score is
characterized by the invariance property under the affine transformations.
Furthermore, we investigate statistical properties of the estimators using
Holder scores for the statistical problems including estimation of regression
functions and robust parameter estimation, and illustrate the usefulness of the
newly introduced scores for statistical forecasting.
