Probability Theory, 2nd Edition

8 Mathematical statistics

8.1 Statistical models

8.1.1 Nonparametric statistical models

The main objective of Probability Theory is to describe and analyze random experiments by means of a suitable probability space $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathcal{A}\mathrm{,}\mathbb{P}\mathrm{)}$. Here it is always assumed that the probability space is known, in particular, that the describing probability measure, $\mathbb{P}$, is identified.

Description of a random experiment and its properties by a probability space. The distribution of the outcomes is assumed to be known.

Mathematical Statistics deals mainly with the reverse question: one executes an experiment, that is, one draws a sample (e. g., one takes a series of measurements of an item or one questions several people), and, on the basis of the observed sample, ...