GRADE ESTIMATION OF CHI-SQUARE DIVERGENCE

For F1, F2 being distribution functions such that F2 is absolutely continuous w.r.t. F1 and F1 is continous, the chi-square divergence between F2 and F1 equal to chi(2) = integral g2(s)ds - 1, g(s) = (F2-degrees F1(-1)), (s), is considered. Its estimator chi(2) based on the modified kernel estimate of the grade density g is introduced. Asymptotic normality of chi(2) is established, asymptotic variance turns out to be larger than for the estimator of chi(2) in the case when the i.i.d. sample from g is observable. Finally, choice of smoothing parameter is discussed based on calculation of the asymptotic mean square error for some approximation of chi(2).