Jackknife Estimator Consistency for Nonlinear Mixture

Abstract

Extended Abstract This paper continues our studies of the jackknife (JK) technique application for estimation of estimators’ covariance matrices in models of mixture with varying concentrations (MVC) [2, 3]. On JK applications for homogeneous samples, see [1]. In MVC models one deals with a non-homogeneous sample, which consists of subjects belonging to 𝑀 different sub-populations (mixture components). One knows the probabilities with which a subject belongs to the mixture components and these probabilities are different for different subjects. Therefore, the considered observations are independent but not identically distributed. We consider objects from a mixture with various concentrations. All objects from the sample Ξ 𝑛 belongs to one of M different mixture components. Each object from the sample 𝛯 𝑛 = (𝜉 𝑗 ) 𝑗=1 𝑛 has observed characteristics 𝜉 𝑗 = (𝑋 𝑗 , 𝑌 𝑗 ) ∈ ℝ 𝐷 and one hidden 𝜅 𝑗 . 𝜅 𝑗 = 𝑚 if 𝑗 -th objects belongs to the 𝑚 -th component. These numbers are unknown, but we know the mixing probabilities 𝑝 𝑗;𝑛𝑚 = 𝑃{𝜅 𝑗 = 𝑚} . The 𝑋 𝑗 is a vector of regressors and 𝑌 𝑗 is a response in the regression model Here 𝑏 (𝑚) ∈ Θ ⊆ ℝ 𝑑 is a vector of unknown regression parameters for the 𝑚 -th component, the 𝑔: ℝ 𝐷−1 × Θ → ℝ is a known regression function, 𝜀 𝑗 is a regression error term. Random variables 𝑋 𝑗 and 𝜀 𝑗 are independent and their distribution is different