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

The goal of the present article is to establish the asymptotic normality of L ₂-deviations of the kernel estimators of the distribution function F _n (x), defined as M _n = ∫ (F _n (x) − R(x))² ω(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.