A central limit theorem for the weighted integrated squared error of the kernel type density estimator under the proportional hazard model

Abstract

The problem of asymptotic distribution of the weighted integrated squared error of the kernel type density estimator under the proportional hazard model of random censorship is treated in this paper. It is shown that three different cases arise depending on whether nh ⁵→, ∞ or λ, 0< λ<∞ In each case it is shown that the asymptotic distribution is normal. A Martingale central limit theorem is used to prove these results.

Keywords:

• Martingale central limit theorem
• proportional hazard model of random censorship
• weighted integrated squared error
• U-statistic