Asymptotic Distributions of Integrated Square Errors of Nonparametric Estimators Based on Indirect Observations Under Dose-Effect Dependence

Abstract

The goal of the present article is to establish the asymptotic normality of L₂-deviations of the kernel estimators of the distribution function F_n(x), defined as M_n = ∫ (F_n(x) − R(x))²ω(x)dx, where R(x) is a conditional average distribution function of a random variable X, ω(x) is a weight function under dose-effect dependence based on the sample U⁽ⁿ⁾ = {(W_i, Y_i), 1 ≤ i ≤ n}, W_i = I(X_i < U_i) is an indicator of the event (X_i < U_i), and Y is a random variable that depends on U and defines the measurement error in the injected random dose. These results may be used to construct goodness-of-fit and homogeneity tests under dose-effect dependence.