Mathematical Statistics and Stochastic Processes

Chapter 2

Principles of Decision Theory

2.1. Generalities

The examples given in Chapter 1 show that a statistical problem may be characterized by the following elements:

– a probability distribution that is not entirely known;

– observations;

– a decision to be made.

Formalization: Wald’s decision theory provided a common framework for statistics problems. We take:

1) A triplet where P is a family of probabilities on the measurable space is called a statistical model. We often set P = {Pθ, θ ∈ Θ} where we suppose that is injective, and Θ is called the parameter space.

2) A measurable space is called a decision (or action) space.

3) A set of measurable mappings is called a set of decision functions (d.f.) (or decision rules).

Description: From an observation that follows an unknown distribution P ∈ P, a statistician chooses an element a ∈ D using an element d from .

Preference relation ...