A note on the integrated square errors of kernel density estimators under random censorship (Q1805767)

The author considers the kernel density estimator \[ f_n(t) = h^{-1}_n \int K((t-s)/h_n)dF_n(t) \] when the nonnegative real valued i.i.d.\ data are censored and the classical empirical d.f.\ \(F_n(t)\) is replaced by \(1-S_n(t)\), where \(S_n(t)\) is the Kaplan-Meier estimator of the survival function \(S(t) = 1-F(t)\). Under standard conditions on the kernel \(K(t)\) with the support \((-1,1)\), and under the assumption that the estimated density is twice continuously differentiable and \(h_n = O(n^{-1/5})\), he proves an asymptotic formula for the error \(I_n(t_0) = \int ^{t_0}_0 [f_n(t)-f(t)]^2dt\) accurate up to \(O_p(h^4_n)+O_p(1/(nh_n))\). Combining this with a former formula for \(EI_n(t_0)\), he deduces the corollary that \(I_n(t_0)/EI_n(t_0)\) converges to one in probability.