Mathematical Statistics and Stochastic Processes

Chapter 4

Statistics and Sufficiency

4.1. Samples and empirical distributions

DEFINITION 4.1.– Let (E, , P) be a statistical model. We call a measurable map from (E, ) to (E′, ′) a statistic with values in (E′, ′).

COMMENT 4.1.– It is important to emphasize the fact that a statistic does not depend on P ∈ P. The measurable map is not a statistic. A decision function, on the other hand, is always a statistic.

In this section, we indicate certain properties of some common statistics. We first give some definitions:

– Let μ be a probability on a measurable space (E0, 0). A sequence X1,…,Xn of n independent random variables with distribution μ is called a sample of size n of the distribution μ. The result of n independent draws following μ is called a realization of this sample.

– The measure (where δ(a) denotes the Dirac measure at the point a) is called the empirical distribution associated ...