Quadratic error of the conditional hazard function in the local linear estimation for functional data

Abstract

In this paper we investigate the asymptotic mean square error and the rates of convergence of the estimator based on the local linear method of the conditional hazard function. Under some general conditions, the expressions of the bias and variance are given. The efficiency of our estimator is evaluated through a simulation study. We proved, theoretically and on the scope of a simulation study, that our proposed estimator has better performance than the estimator based on the standard kernel method.

Limit Law of the Local Linear Estimate of the Conditional Hazard Function for Functional Data

“…and the following auxiliary result which play a main role in the peruve of our Theorem 1 Lemma 4. (see Merouan et al [30]) Under the assumptions of Theorem 1, we have…”

“…The asymptotic mean square error of the conditional hazard function was studied by Rabhi et al [33]. Recently, Merouan et al [30]. They established the mean square error and the rates of the convergence of the estimator based on the local modeling approach of the conditional hazard function.…”

“…Our paper presents the result of the asymptotic normality of the nonparametric local linear estimator proposed by Massim and Mechab [29] and Merouan et al [30] for independent and identically distributed observations. It is well clear that the importance of this result is the construction of confidence intervals.…”

Uniform Convergence of Nonparametric Conditional Hazard Function in the Single Functional Index Modeling for Dependent Data

“…Proof: The proof is based on the decomposition in Theorem of [15] and the results below. Through these results, we observe that's the same rate of convergence as [15], where the data are independent.…”

“…So, it requires to prove (10), (11) and (12). Concerning (12), we consider the following decomposition Applying the Fuck-Nagaev exponential inequality (Proposition A.11(ii) in [14]), for all , we have (13) where Again, by the lemma in [8], we obtain ( 14) and (15). In other hand for (18), by the same manner in Lemma 2 for , we have We have that is proved in Lemma 2, and taking the same in pervious calculs , we get Secondly, we use the same strides for to get the required.…”