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 , a sequence of sub-σ-algebras of , and a sequence (dn, n ≥ 1) of -adapted decision rules (i.e. dn is -measurable for all n ≥ 1).
The decision space being provided with a distance δ, we say that (dn) is convergent in probability if:
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 , , and , and the convergence in probability may be rewritten ...