ABSTRACT
A random experiment leads to some observed values; denote them X 1,…,Xn . To make a formal analysis of the experiment and its results, we include everything in the frame of a statistical model. The classical statistical model assumes that the vector X = (X 1,…,Xn ) can attain the values in a sample space https://www.w3.org/1998/Math/MathML"> X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315268125/d57fa0be-4eee-4d70-bfcd-ee2d3690ba91/content/inline-math1_1.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> (or https://www.w3.org/1998/Math/MathML"> X n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315268125/d57fa0be-4eee-4d70-bfcd-ee2d3690ba91/content/inline-math1_2.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> ) and the subsets of https://www.w3.org/1998/Math/MathML"> X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315268125/d57fa0be-4eee-4d70-bfcd-ee2d3690ba91/content/inline-math1_3.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> are random events of our interest. If https://www.w3.org/1998/Math/MathML"> X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315268125/d57fa0be-4eee-4d70-bfcd-ee2d3690ba91/content/inline-math1_4.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> is finite, then there is no problem working with the family of all its subsets. However, some space https://www.w3.org/1998/Math/MathML"> X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315268125/d57fa0be-4eee-4d70-bfcd-ee2d3690ba91/content/inline-math1_5.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> can be too rich, as, e.g., the n-dimensional Euclidean space; then we do not consider all its subsets, but restrict our considerations only to some properly selected subsets/events. In order to describe the experiments and the events mathematically, we consider the family of events that creates a σ-field, i.e., that is closed with respect to the countable unions and the complements of its elements. Let us denote it as https://www.w3.org/1998/Math/MathML"> B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315268125/d57fa0be-4eee-4d70-bfcd-ee2d3690ba91/content/inline-math1_6.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> (or https://www.w3.org/1998/Math/MathML"> B n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315268125/d57fa0be-4eee-4d70-bfcd-ee2d3690ba91/content/inline-math1_7.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> ). The probabilistic behavior of random vector X is described by the probability distribution P. which is a set function defined on https://www.w3.org/1998/Math/MathML"> B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315268125/d57fa0be-4eee-4d70-bfcd-ee2d3690ba91/content/inline-math1_8.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> . The classical statistical model is a family https://www.w3.org/1998/Math/MathML"> P = { P θ , θ ∈ Θ } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315268125/d57fa0be-4eee-4d70-bfcd-ee2d3690ba91/content/inline-math1_9.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> of probability distributions, to which our specific distribution P also belongs. While we can observe X 1,…,Xn , the parameter θ is unobservable. It is a real number or a vector and can take on any value in the parametric space https://www.w3.org/1998/Math/MathML"> Θ ⊆ R p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315268125/d57fa0be-4eee-4d70-bfcd-ee2d3690ba91/content/inline-math1_10.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where p is a positive integer.
