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AbstractWe, in this article, focus on functional partially linear quantile regression, where the observations are missing at random, which allows the response or covariates or response and covariates simultaneously missing. Estimation of the unknown function is done based on reproducing kernel method. Under suitable assumptions, we discuss consistency with rates of the estimators, and establish asymptotic normality of the estimator for the parameter. At the same time, we study hypothesis test of the parameter, and prove asymptotic distributions of restricted estimators of the parameter and test statistic under null hypothesis and local alternative hypothesis, respectively. Also, we study variable selection of the linear part of the model. By simulation and real data, finite sample performance of the proposed methods is analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the empirical likelihood in the single-index partially functional linear model. When the response variables or/and part of the covariates are missing at random, we construct the empirical likelihood ratio of the parameter in the model based on B-spline approximation for the link and slope functions, and define maximum empirical likelihood (MEL) estimator of the parameter. Under suitable assumptions, the asymptotic distributions of the proposed empirical log-likelihood ratio and MEL estimator are established. At the same time, based on penalized empirical likelihood (PEL) approach, we define the PEL estimator of the parameter and investigate variable selection of the model. A simulation study is done to evaluate the finite sample performance for the proposed methods. [ABSTRACT FROM AUTHOR]

Based on Bayesian method, we investigate single-index quantile regression with missing observation. In particular, using spline approximation for the link function, we construct quasi-posterior distribution of the index vector based on asymmetric Laplace likelihood with missing observation, and establish asymptotically normality of the posterior estimator of the index parameters. At the same time, we use a hierarchical model based on spike and slab Gaussian priors to do variable selection and study consistency of the variable selection. Finite sample performance of the proposed methods is analyzed via simulation and real data too. [ABSTRACT FROM AUTHOR]

Based on the approach of left and right kernel smoothing with unilateral kernel function, we, in this paper, define estimators of change point and jump size in nonparametric regression model with response missing at random. It is shown that the change point estimator is n-consistent and converges to the smallest maximizer of one-dimensional bilateral compound Poisson process, the jump size estimator is asymptotically normal. A simulation study is conducted to investigate the finite sample behavior for the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the partially linear varying-coefficient quantile regression model with observations missing at random (MAR), which include the responses or the responses and covariates MAR. Based on the local linear estimation of the varying-coefficient function in the model, we construct empirical log-likelihood ratio functions for unknown parameter in the linear part of the model, which are proved to be asymptotically weighted chi-squared distributions, further the adjusted empirical log-likelihood ratio functions are verified to converge to standard chi-squared distribution. The asymptotic normality of maximum empirical likelihood estimator for the parameter is also established. In order to do variable selection, we consider penalized empirical likelihood by using smoothly clipped absolute deviationv (SCAD) penalty, and the oracle property of the penalized likelihood estimator of the parameter is proved. Furthermore, Monte Carlo simulation and a real data analysis are undertaken to test the performance of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the left-truncated and right-censored model, and construct the local linear and Nadaraya-Watson type estimators of the conditional density. Under suitable conditions, we establish the asymptotic normality of the proposed estimators when the observations are assumed to be a stationary α-mixing sequence. Finite sample behavior of the estimators is investigated via simulations too. [ABSTRACT FROM AUTHOR]

Based on the inverse probability weight method, we, in this article, construct the empirical likelihood (EL) and penalized empirical likelihood (PEL) ratios of the parameter in the linear quantile regression model when the covariates are missing at random, in the presence and absence of auxiliary information, respectively. It is proved that the EL ratio admits a limiting Chi-square distribution. At the same time, the asymptotic normality of the maximum EL and PEL estimators of the parameter is established. Also, the variable selection of the model in the presence and absence of auxiliary information, respectively, is discussed. Simulation study and a real data analysis are done to evaluate the performance of the proposed methods. [ABSTRACT FROM PUBLISHER]

In this article, we study globalL2error of non linear wavelet estimator of density in the Besov spaceBspqfor missing data model when covariables are present and prove that the estimator can achieve the optimal rate of convergence, which is similar to the result studied by Donoho et al. (1996) in complete independent data case with term-by-term thresholding of the empirical wavelet coefficients. Finite-sample behavior of the proposed estimator is explored via simulations. [ABSTRACT FROM PUBLISHER]

In order to investigate the convergence rate of the asymptotic normality for the estimator of the conditional mode function for the left-truncation model, we derive a Berry–Esseen type bound of the estimator when the lifetime observations with multivariate covariates form a stationary α-mixing sequence. The finite sample performance of the estimator of the conditional mode function is explored through simulations. [ABSTRACT FROM PUBLISHER]

Consider the heteroscedastic partially linear errors-in-variables (EV) modelyi=xiβ +g(ti) + εi, ξi=xi+ μi(1 ⩽i⩽n), where εi= σieiare random errors with mean zero, σ2i=f(ui), (xi,ti,ui) are non random design points,xiare observed with measurement errors μi. Whenf( · ) is known, we derive the Berry–Esseen type bounds for estimators of β andg( · ) under {ei, 1 ⩽i⩽n} is a sequence of stationary α-mixing random variables, whenf( · ) is unknown, the Berry–Esseen type bounds for estimators of β,g( · ), andf( · ) are discussed under independent errors. [ABSTRACT FROM PUBLISHER]

Consider the heteroscedastic semi-parametric model y i  = x i β +g(t i ) + σ i e i (1 ≤ i ≤ n), where , the design points (x i , t i , u i ) are known and non random, g(·) and f(·) are unknown functions defined on closed interval [0, 1], and the random errors e i are assumed to be a sequence of stationary α-mixing random variables with mean zero. Under appropriate conditions, we study the asymptotic normality of least-squares estimator and two weighted least-squares estimators of β. Also, the asymptotic normality of the estimators of g(·) and f(·) is considered. Finite sample behavior of the estimators is investigated via simulations as well. [ABSTRACT FROM AUTHOR]

The purpose of this article is to use the empirical likelihood method to study construction of the confidence region for the parameter of interest in heteroscedastic partially linear errors-in-variables model with martingale difference errors. When the variance functions of the errors are known or unknown, we propose the empirical log-likelihood ratio statistics for the parameter of interest. For each case, a nonparametric version of Wilks’ theorem is derived. The results are then used to construct confidence regions of the parameter. A simulation study is carried out to assess the performance of the empirical likelihood method. [ABSTRACT FROM PUBLISHER]

In this article, we establish strong uniform convergence and asymptotic normality of estimators of conditional quantile and conditional distribution function for a left truncated model when the data exhibit some kind of dependence. It is assumed that the observations form a stationary α-mixing sequence. The results of Lemdani et al. (2009) are relaxed from the i.i.d. assumption to α-mixing setting. Finite sample behavior of the estimators is investigated via simulations as well. [ABSTRACT FROM AUTHOR]

In this article, we employ the method of empirical likelihood to construct confidence intervals of conditional density for a left-truncation model. It is proved that the empirical likelihood ratio admits a limiting chi-square distribution with one degree of freedom when the lifetime observations with multivariate covariates form a stationary α-mixing sequence. [ABSTRACT FROM AUTHOR]

In this article, we study the consistency of the error density estimator in nonparametric regression models when the errors form a stationary α-mixing sequences. These results improve the results of Cheng (2004) from the i.i.d. assumption to α-mixing condition and weaken the restrictions for the bandwidths an. Also, the rates of strong convergence for the estimator are investigated. Furthermore, we derive the asymptotic normality and the law of the iterated logarithm of the histogram-type error density estimator, which complement the conclusion in Cheng (2002) in α-mixing setting. [ABSTRACT FROM AUTHOR]