Minimum Divergence Estimators Based on Grouped Data

Abstract

The paper considers statistical models with real-valued observations i.i.d. by F(x, θ₀) from a family of distribution functions (F(x, θ); θ ε Θ), Θ ⊂ R ^s, s ≥ 1. For random quantizations defined by sample quantiles (F _n ⁻¹ (λ₁),θ, F _n ⁻¹ (λ _m−1)) of arbitrary fixed orders 0 < λ₁ θ < λ_m-1 < 1, there are studied estimators θ_φ,n of θ₀ which minimize φ-divergences of the theoretical and empirical probabilities. Under an appropriate regularity, all these estimators are shown to be as efficient (first order, in the sense of Rao) as the MLE in the model quantified nonrandomly by (F ⁻¹ (λ₁,θ₀),θ, F ⁻¹ (λ _m−1, θ₀)). Moreover, the Fisher information matrix I _m (θ₀, λ) of the latter model with the equidistant orders λ = (λ _j = j/m : 1 ≤ j ≤ m − 1) arbitrarily closely approximates the Fisher information J(θ₀) of the original model when m is appropriately large. Thus the random binning by a large number of quantiles of equidistant orders leads to appropriate estimates of the above considered type.