Mathematical Statistics and Stochastic Processes

Chapter 8

Non-Parametric Methods and Robustness

8.1. Generalities

A statistical model is said to be non-parametric if Θ is “vast”. When Θ is a vector space or a convex set, “vast” generally means “of infinite dimension”, otherwise the distinction between parametric and non-parametric models is not so clear.

EXAMPLE 8.1.–

1) A Gaussian or exponential model is parametric.

2) Let P0 be the set of probabilities on dominated by the Lebesgue measure λ. The model is non-parametric (we may set θ = dP/dλ, and Θ is a convex set of infinite dimension in L1(λ)).

3) Let P1 be the set of probabilities on which have a unique median. The model is non-parametric.

Non-parametric methods are interesting for three principal reasons:

1) They avoid errors due to the choice of a specific but often erroneous parametric model.

2) They guide the user in the choice of a parametric model.

3) In certain cases, they provide initial estimators for the parameters of a parametric model from which we may construct more ...