Geometrical foundations of the sampling design with fixed sample size

• Main Author: Pierpaolo Angelini
• Format: Article
• Language: English
• Published: Accademia Piceno Aprutina dei Velati 2020-06-01
• Series: Ratio Mathematica
• Subjects: tensor product; linear map; bilinear map; quadratic and linear metric; α-product; α-norm
• Online Access: http://eiris.it/ojs/index.php/ratiomathematica/article/view/511

We study the sampling design with fixed sample size from a geometric point of view. The first-order and second-order inclusion probabilities are chosen by the statistician. They are subjective probabilities. It is possible to study them inside of linear spaces provided with a quadratic and linear metric. We define particular random quantities whose logically possible values are all logically possible samples of a given size. In particular, we define random quantities which are complementary to the Horvitz-Thompson estimator. We identify a quadratic and linear metric with regard to two univariate random quantities representing deviations. We use the α-criterion of concordance introduced by Gini in order to identify it. We innovatively apply to probability this statistical criterion.