ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: 60,62 Moldavskaya, Elina, Cybernetical Institute of National Academy of Science of Ukraine Minimum Contrast Estimators of Parameter of Spectral Density of Random Fields in Continuous Case. Let ξ(x), x ∈ R^{n} be a random field with continuous parameter, mean zero and spectral density f(λ; θ) = σ^{2}(θ)ψ(λ; θ), λ ∈ R^{n}, θ ∈ Θ ⊆ R^{m}

∫_{R^{n}} \frac{ψ(λ;θ)}{1+∥λ∥^{2}} dλ = 1, let I_{T} (λ), λ ∈ R^{n}, T = (T_{1,...,}T_{n})∈ R^{n} be a peri-

odogramm of the observations. Consider Ibragimov’s contrast function

K(θ_{0}; θ) = ∫_{R^{n}} \frac{f(λ;θ)}{1+∥λ∥^{2}} log ψ(λ; θ)dλ , θ_{0}, θ ∈ Θ, where θ_{0} is true value of

parameter and corresponding contrast field

U_{T} (θ) = ∫_{R^{n}} \frac{I(λ)}{1 + ∥λ∥^{2}} log ψ(λ; θ)dλ,

θ ∈ Θ, Define minimum contrast estimators θ̂_{T} = arg min U_{T} (θ) and let σ̂_{T}^{2} = ∫_{R^{n}} \frac{I(λ)}{1+∥λ∥^{2}} dλ Ibragimov’s estimators are interpretated as minimum

contrast estimators and conditions of consistency and asymptotic normality of θ̂_{T} and σ̂_{T}^{2} as T̃ → ∞ , T̃ = min{T_{i}, i = 1, ..., n} are given. This is joint result with prof. N. N. Leonenko.