On Efficient Estimation in Continuous Models Based on Finitely Quantized Observations

Abstract

We consider minimum φ-divergence estimators of parameters θ of arbitrary dominated models μ_θ ≪ λ on the real line, based on finite quantizations of i.i.d. observations X ₁,…, X _n from these models. The quantizations are represented by finite interval partitions 𝒫 _n  = (A _n1,…, A _{nm n} ) of the real line, where m _n is allowed to increase to infinity for n → ∞. The models with densities f _θ = dμ_θ/dλ are assumed to be regular in the sense that they admit finite Fisher informations 𝒥_θ. In the first place we have in mind continuous models dominated by the Lebesgue measure λ. Owing to the quantizations, are discrete-model estimators for which the desirable properties (computation complexity, robustness, etc.) can be controlled by a suitable choice of functions φ. We formulate conditions under which these estimators are consistent and efficient in the original models μ_θ in the sense that as n → ∞.

Keywords:

• Asymptotic normality
• Consistency
• Efficiency
• Minimum disparity estimators
• Quantized observations

Mathematics Subject Classification:

• 62F12
• 62B10

Acknowledgments

This work has been supported by the Spanish grants BMF2003-00892, BMF2003-04820, and GV04B-670 and by the Czech grants ASCR A1075403 and MSMTV 1M 0572.