A quadratic and linear metric characterizing the sampling design with fixed sample size considered from a geometric viewpoint

The first-order and second-order inclusion probabilities are chosen by the statistician. They are subjective probabilities. We innovatively define univariate and bivariate random quantities whose logically possible values are samples of a given size in order to obtain the first-order and secondorder inclusion probabilities by means of their coherent previsions. We consider linear maps connected with univariate random quantities as well as bilinear maps connected with bivariate random quantities. The covariance of two univariate random quantities that are the components of a bivariate random quantity has been expressed by means of two bilinear maps. We show that a univariate random quantity denoted by S is complementary to the univariate Horvitz-Thompson estimator. We identify a quadratic and linear metric with regard to two univariate random quantities representing deviations that we innovatively define. We use the α-criterion of concordance introduced by Gini in order to identify it. It is a statistical criterion that we innovatively apply to probability.