Confidence bands in nonparametric regression with length biased data

In this paper we deduce a confidence bands construction for the nonparametric estimation of a regression curve from length biased data, where a result from Bickel and Rosenblatt (1973,The Annals of Statistics,1, 1071–1095) is adapted to this new situation. The construction also involves the estimation of the variance of the local linear estimator of the regression, where we use a finite sample modification in order to improve the performance of these confidence bands in the case of finite samples.     

Strong convergence of LAD estimates in a censored regression model

A regression model with a nonnegativity constraint on the dependent variable, known as censored median regression model, is considered. Under some mild conditions, the LAD estimate of the regression coefficient is shown to be strongly consistent. Furthermore, its convergence rate and Bahadur strong representation are also obtained.     

Global nonparametric estimation of conditional quantile functions and their derivatives

Let (X, Y) be a random vector such that X is d-dimensional, Y is real valued, and θ(X) is the conditional αth quantile of Y given X, where α is a fixed number such that 0 < α < 1. Assume that θ is a smooth function with order of smoothness p > 0, and set r = (p − m)/(2p + d), where m is a nonnegative integer smaller than p. Let T(θ) denote a derivative of θ of order m. It is proved that there exists estimate of T(θ), based on a set of i.i.d. observations (X₁, Y₁), …, (X_n, Y_n), that achieves the optimal nonparametric rate of convergence n^−r in L_q-norms (1 ≤ q < ∞) restricted to compacts under appropriate regularity conditions. Further, it has been shown that there exists estimate of T(θ) that achieves the optimal rate (n/log n)^−r in L_∞-norm restricted to compacts.     

A class of estimators of the mean survival time from interval censored data with application to linear regression

A class of estimators of the mean survival time with interval censored data are studied by unbiased transformation method. The estimators are constructed based on the observations to ensure unbiasedness in the sense that the estimators in a certain class have the same expectation as the mean survival time. The estimators have good properties such as strong consistency （with the rate of O（n^-1/1 （log log n）^1/2）） and asymptotic normality. The application to linear regression is considered and the simulation reports are given.     

LAD estimation for nonlinear regression models with randomly censored data

The least absolute deviations (LAD) estimation for nonlinear regression models with randomly censored data is studied and the asymptotic properties of LAD estimators such as consistency, boundedness in probability and asymptotic normality are established. Simulation results show that for the problems with censored data, LAD estimation performs much more robustly than the least squares estimation.     

Uniform limit laws of the logarithm for nonparametric estimators of the regression function in presence of censored data

In this paper, we establish uniform-in-bandwidth limit laws of the logarithm for nonparametric Inverse Probability of Censoring Weighted (I.P.C.W.) estimators of the multivariate regression function under random censorship. A similar result is deduced for estimators of the conditional distribution function. The uniform-in-bandwidth consistency for estimators of the conditional density and the conditional hazard rate functions are also derived from our main result. Moreover, the logarithm laws we establish are shown to yield almost sure simultaneous asymptotic confidence bands for the functions we consider. Examples of confidence bands obtained from simulated data are displayed.      

Asymptotic properties of plug-in level set estimators for right censored data

We assume T1,...,Tn are i.i.d.data sampled from distribution function F with density function f and C1,...,Cn are i.i.d.data sampled from distribution function G.Observed data consists of pairs(Xi,δi),i=1,...,n,where Xi=min{Ti,Ci},δi=I(Ti Ci),I(A)denotes the indicator function of the set A.Based on the right censored data{Xi,δi},i=1,...,n,we consider the problem of estimating the level set{f c}of an unknown one-dimensional density function f and study the asymptotic behavior of the plug-in level set estimators.Under some regularity conditions,we establish the asymptotic normality and the exact convergence rate of theλg-measure of the symmetric difference between the level set{f c}and its plug-in estimator{fn c},where f is the density function of F,and fn is a kernel-type density estimator of f.Simulation studies demonstrate that the proposed method is feasible.Illustration with a real data example is also provided.     

Variable selection and coefficient estimation via composite quantile regression with randomly censored data

Composite quantile regression with randomly censored data is studied. Moreover, adaptive LASSO methods for composite quantile regression with randomly censored data are proposed. The consistency, asymptotic normality and oracle property of the proposed estimators are established. The proposals are illustrated via simulation studies and the Australian AIDS dataset.     

Change-point estimation for censored regression model

In this paper, we consider the change-point estimation in the censored regression model assuming that there exists one change point. A nonparametric estimate of the change-point is proposed and is shown to be strongly consistent. Furthermore, its convergence rate is also obtained.     

Asymptotic properties for the semiparametric regression model with randomly censored data

Suppose that the patients’ survival times.Y, are random variables following the semiparametric regression modelY = Xβ +g(T) + ε, where (X,T) is a radom vector taking values inR×[0,1],βis an unknown parameter,g (*) is an unknown smooth regression function andE is the random error with zero mean and variance σ². It is assumed that (X,T) is independent of E. The estimators andg _n (*) of P andg(*) are defined, respectively, when the observations are randomly censored on the right and the censoring distribution is unknown. Moreover, it is shown that is asymptotically normal andg _n (*) is weak consistence with rateO _p(n-1/3). Project supported by China Postdoctoral Science Foundation and the National Natural Science Foundation of China.     

L1-optimal estimates for a regression type function in R

Let X₁, X₂, …, X_n be random vectors that take values in a compact set in R^d, d ≥ 1. Let Y₁, Y₂, …, Y_n be random variables (“the responses”) which conditionally on X₁ = x₁, …, X_n = x_n are independent with densities f(y | x_i, θ(x_i)), i = 1, …, n. Assuming that θ lives in a sup-norm compact space Θ_q,d of real valued functions, an optimal L₁-consistent estimator of θ is constructed via empirical measures. The rate of convergence of the estimator to the true parameter θ depends on Kolmogorov's entropy of Θ_q,d.     

小波检测并估计非参函数变点

研究随机设计下非参函数变点的小波检测与估计问题.将小波方法与设计点转化方法相结合给出变点的检测统计量并研究检测的一致性.给出了变点个数和变点位置的估计量,证明了变点个数估计量的相合性并得到变点位置估计量的收敛速度.     

Testing strict monotonicity in nonparametric regression

A new test for strict monotonicity of the regression function is proposed which is based on a composition of an estimate of the inverse of the regression function with a common regression estimate. This composition is equal to the identity if and only if the “true” regression function is strictly monotone, and a test based on an L ²-distance is investigated. The asymptotic normality of the corresponding test statistic is established under the null hypothesis of strict monotonicity.      

Location estimation in nonparametric regression with censored data

Consider the heteroscedastic model Y=m(X)+σ(X)?, where ? and X are independent, Y is subject to right censoring, m(·) is an unknown but smooth location function (like e.g. conditional mean, median, trimmed mean…) and σ(·) an unknown but smooth scale function. In this paper we consider the estimation of m(·) under this model. The estimator we propose is a Nadaraya-Watson type estimator, for which the censored observations are replaced by ‘synthetic’ data points estimated under the above model. The estimator offers an alternative for the completely nonparametric estimator of m(·), which cannot be estimated consistently in a completely nonparametric way, whenever high quantiles of the conditional distribution of Y given X=x are involved.We obtain the asymptotic properties of the proposed estimator of m(x) and study its finite sample behaviour in a simulation study. The method is also applied to a study of quasars in astronomy.     

Estimation in nonparametric location-scale regression models with censored data

Consider the random vector (X, Y), where X is completely observed and Y is subject to random right censoring. It is well known that the completely nonparametric kernel estimator of the conditional distribution ${F(\cdot|x)}$ of Y given X = x suffers from inconsistency problems in the right tail (Beran 1981, Technical Report, University of California, Berkeley), and hence any location function m(x) that involves the right tail of ${F(\cdot|x)}$ (like the conditional mean) cannot be estimated consistently in a completely nonparametric way. In this paper, we propose an alternative estimator of m(x), that, under certain conditions, does not share the above inconsistency problems. The estimator is constructed under the model Y = m(X) + σ(X)ε, where ${\sigma(\cdot)}$ is an unknown scale function and ε (with location zero and scale one) is independent of X. We obtain the asymptotic properties of the proposed estimator of m(x), we compare it with the completely nonparametric estimator via simulations and apply it to a study of quasars in astronomy.     

Robust nonparametric regression estimation

In this paper we define a robust conditional location functional without requiring any moment condition. We apply the nonparametric proposals considered by C. Stone (Ann. Statist. 5 (1977), 595–645) to this functional equation in order to obtain strongly consistent, robust nonparametric estimates of the regression function. We give some examples by using nearest neighbor weights or weights based on kernel methods under no assumptions whatsoever on the probability measure of the vector (X,Y). We also derive strong convergence rates and the asymptotic distribution of the proposed estimates.     

Nonparametric time series regression

Consider ak-times differentiable unknown regression function(·) of ad-dimensional measurement variable. LetT() denote a derivative of(·) of orderm and setr=(k–m)/(2k+d). Given a bivariate stationary time series of lengthn, under some appropriate conditions, a sequence of local polynomial estimators of the functionT() can be chosen to achieve the optimal rate of convergencen ^–r inL ₂ norms restricted to compacts; and the optimal rate (n ^–1 logn) ^r in theL norms on compacts. These results generalize those by Stone (1982,Ann. Statist.,10, 1040–1053) which deals with nonparametric regression estimation for random (i.i.d.) samples. Applications of these results to nonlinear time series problems will also be discussed.This work was completed while the author was visiting Mathematical Sciences Research Institute at Berkeley, California. Research was supported in part by NSF Grant DMS-8505550, NC Board of Science and Technology Development Award 90SE06 and UNC Research Council.     

Asymptotic properties of some functions of nonparametric estimates of a density function

Let be a nonparametric estimate of a two-dimensional density f(x,y) constructed with the help of two-dimensional “window.” The main purpose of the paper is to study the asymptotic properties of the marginal moments estimates and differentiable functions , of these moments.     

Stochastic integrals of empirical-type processes with applications to censored regression

Motivated by the analysis of linear rank estimators and the Buckley-James nonparametric EM estimator in censored regression models, we study herein the asymptotic properties of stochastic integrals of certain two-parameter empirical processes. Applications of these results on empirical processes and their stochastic integrals to the asymptotic analysis of censored regression estimators are also given.     

Testing heteroscedasticity in nonparametric regression models based on residual analysis

The importance of detecting heteroscedasticity in regression analysis is widely recognized because efficient inference for the regression function requires that heteroscedasticity should be taken into account. In this paper, a simple test for heteroscedasticity is proposed in nonparametric regression based on residual analysis. Furthermore, some simulations with a comparison with Dette and Munk＇s method are conducted to evaluate the performance of the proposed test. The results demonstrate that the method in this paper performs quite satisfactorily and is much more powerful than Dette and Munk＇s method in some cases.