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Mathematical Statistics and Stochastic Processes by Denis Bosq

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Chapter 7

Asymptotic Statistics

7.1. Generalities

Asymptotic statistics is the study of decision rules when the number of observations tends to infinity.

Theoretically, the asymptotic model may be described as follows: one considers a statistical model written as images, a sequence images of sub-σ-algebras of images, and a sequence (dn, n ≥ 1) of images-adapted decision rules (i.e. dn is images-measurable for all n ≥ 1).

The decision space being provided with a distance δ, we say that (dn) is convergent in probability if:

images

where aθ denotes the “correct” decision when the value of the parameter is θ.

In the usual case, where (Xn,n ≥ 1) is a sample from Pθ, θ ∈ Θ, we have images, images, and , and the convergence in probability may be rewritten ...

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