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On adaptive estimation based on ranks in ARMA processes

Allali J., Kaaouachi A., Melhaoui S.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 2

293--302

This paper describes the adaptive estimation problem based on ranks for the parameter of an ARMA process. The local asymptotic normality property with a ranked based central sequence allows for the construction of estimators which are locally asymptotically minimax (LAM). By using a consistent estimate of the score function, we obtain the adaptive estimators which are LAM and which do not depend on the innovation density.

Limits of truncation experiments

Marohn F.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 1

71--88

Given n i.i.d. copies X1,…., Xn of a random variable X with distribution Pѵ, ѵ θ∈Θ⊂Rk, we are only interested in those observations that fall into some set D = D(n) ⊂ Rd having but a small probability of occurrence. The truncation set D is assumed to be known and non-random. Denoting the distribution of the truncated random variable X1D(X) by Pnѵ we consider the triangular array of experiments (Rd, ßd, (Pnѵθ ), n∈N, and investigate the asymptotic behavior of the n-fold products ((Rd)n, (ßd)n, (Pnѵ)nѵ∈Θ ). Under a suitable density expansion,Gaussian shifts as well as Poisson experiments occur in the limit, where the latter case typically occurs when the number of expected observations falling in D is bounded. Finally, we investigate invariance properties of the occurring Poisson limits.