Mathematical Statistics and Stochastic Processes

Chapter 5

Point Estimation

5.1. Generalities

5.1.1. Definition – examples

Let be a statistical model and let g be a map from Θ to a set A with a σ-algebra .

DEFINITION 5.1.– An estimator of g(θ) is a measurable map from E to A.

In an estimation problem, the set of decisions is therefore the set where the function g of the parameter takes its values. It acts to, in light of the observations, admit a value for g(θ) (a “function of the parameter”).

COMMENT 5.1.– If the model is written in the form , we may consider g to be a function of P.

EXAMPLE 5.1.– In all of the examples below, .

1) .

2) .

3) , where uθ denotes a uniform distribution on [0, θ], ; g(θ) = θ.

4) P = a set of distributions of the ...