We introduce and study a local linear nonparametric regression estimator for censorship model. The main goal of this paper is, to establish the uniform almost sure consistency result with rate over a compact set for the new estimate. To support our theoretical result, a simulation study has been done to make comparison with the classical regression estimator.
In this paper, we built a new nonparametric regression estimator with the local linear method by using the mean squared relative error as a loss function when the data are subject to random right censoring. We establish the uniform almost sure consistency with rate over a compact set of the proposed estimator. Some simulations are given to show the asymptotic behavior of the estimate in different cases.
We reexamine the classical linear regression model when the model is subject to two types of uncertainty: (i) some of covariates are either missing or completely inaccessible, and (ii) the variance of the measurement error is undetermined and changing according to a mechanism unknown to the statistician. By following the recent theory of sublinear expectation, we propose to characterize such mean and variance uncertainty in the response variable by two specific nonlinear random variables, which encompass an infinite family of probability distributions for the response variable in the sense of (linear) classical probability theory. The approach enables a family of estimators under various loss functions for the regression parameter and the parameters related to model uncertainty. The consistency of the estimators is established under mild conditions on the data generation process. Three applications are introduced to assess the quality of the approach including a forecasting model for the S&P Index.
Principled nonparametric tests for regression curvature in $mathbb{R}^{d}$ are often statistically and computationally challenging. This paper introduces the stratified incomplete local simplex (SILS) tests for joint concavity of nonparametric multiple regression. The SILS tests with suitable bootstrap calibration are shown to achieve simultaneous guarantees on dimension-free computational complexity, polynomial decay of the uniform error-in-size, and power consistency for general (global and local) alternatives. To establish these results, a general theory for incomplete $U$-processes with stratified random sparse weights is developed. Novel technical ingredients include maximal inequalities for the supremum of multiple incomplete $U$-processes.
The coefficient function of the leading differential operator is estimated from observations of a linear stochastic partial differential equation (SPDE). The estimation is based on continuous time observations which are localised in space. For the asymptotic regime with fixed time horizon and with the spatial resolution of the observations tending to zero, we provide rate-optimal estimators and establish scaling limits of the deterministic PDE and of the SPDE on growing domains. The estimators are robust to lower order perturbations of the underlying differential operator and achieve the parametric rate even in the nonparametric setup with a spatially varying coefficient. A numerical example illustrates the main results.
Fan, Gijbels and King [Ann. Statist. 25 (1997) 1661--1690] considered the estimation of the risk function $psi (x)$ in the proportional hazards model. Their proposed estimator is based on integrating the estimated derivative function obtained through a local version of the partial likelihood. They proved the large sample properties of the derivative function, but the large sample properties of the estimator for the risk function itself were not established. In this paper, we consider direct estimation of the relative risk function $psi (x_2)-psi (x_1)$ for any location normalization point $x_1$. The main novelty in our approach is that we select observations in shrinking neighborhoods of both $x_1$ and $x_2$ when constructing a local version of the partial likelihood, whereas Fan, Gijbels and King [Ann. Statist. 25 (1997) 1661--1690] only concentrated on a single neighborhood, resulting in the cancellation of the risk function in the local likelihood function. The asymptotic properties of our estimator are rigorously established and the variance of the estimator is easily estimated. The idea behind our approach is extended to estimate the differences between groups. A simulation study is carried out.
Feriel Bouhadjera
,Riad Remita
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(2020)
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"Strong consistency of the nonparametric local linear regression estimation under censorship model"
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Feriel Bouhadjera
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