Estimation locale linéaire de la densité conditionnelle pour des données fonctionnelles

In this Note, we introduce the local linear estimation of the conditional density of a scalar response variable given a random variable taking values in a semi-metric space. Under some general conditions, we establish the pointwise and uniform almost complete convergences with rates of this estimator. Moreover, as an application, we use the obtained results to derive some asymptotic properties for the local linear estimator of the conditional mode.     

响应变量删失时函数型部分线性分位数回归模型的估计

最近几年,函数型数据分析的理论和应用飞速发展.在许多实际应用里,响应变量往往存在随机右删失的情况.考虑利用函数型部分线性分位数回归模型来刻画函数型和标量预测量与右删失响应变量之间的关系.基于函数型主成分基函数来逼近未知的斜率函数,通过极小化逆概率加权分位数损失函数得到未知系数的估计量.文章的估计方法容易通过加权分位数回归程序实现.在一定的假设条件下,给出了有限维参数估计量的渐近正态性与斜率函数估计量的收敛速度.最后,通过模拟计算与应用实例证明了所提方法的有效性.     

Partial functional linear quantile regression

This paper studies estimation in partial functional linear quantile regression in which the dependent variable is related to both a vector of finite length and a function-valued random variable as predictor variables. The slope function is estimated by the functional principal component basis. The asymptotic distribution of the estimator of the vector of slope parameters is derived and the global convergence rate of the quantile estimator of unknown slope function is established under suitable norm. It is showed that this rate is optimal in a minimax sense under some smoothness assumptions on the covariance kernel of the covariate and the slope function. The convergence rate of the mean squared prediction error for the proposed estimators is also be established. Finite sample properties of our procedures are studied through Monte Carlo simulations. A real data example about Berkeley growth data is used to illustrate our proposed methodology.     

Linear Least Squares Estimation of Regression Models for Two-Dimensional Random Fields

We consider the problem of estimating regression models of two-dimensional random fields. Asymptotic properties of the least squares estimator of the linear regression coefficients are studied for the case where the disturbance is a homogeneous random field with an absolutely continuous spectral distribution and a positive and piecewise continuous spectral density. We obtain necessary and sufficient conditions on the regression sequences such that a linear estimator of the regression coefficients is asymptotically unbiased and mean square consistent. For such regression sequences the asymptotic covariance matrix of the linear least squares estimator of the regression coefficients is derived.     

Estimation of the distribution of random shifts deformation

Consider discrete values of functions shifted by unobserved translation effects, which are independent realizations of a random variable with unknown distribution μ modeling the variability in the response of each individual. Our aim is to construct a nonparametric estimator of the density of these random translation deformations using semiparametric preliminary estimates of the shifts. Based on the results of Dalalyan et al. [7], semiparametric estimators are obtained in our discrete framework and their performance studied. From these estimates we construct a nonparametric estimator of the target density. Both rates of convergence and an algorithm to construct the estimator are provided.      

Partially linear varying coefficient models with missing at random responses

This paper considers partially linear varying coefficient models when the response variable is missing at random. The paper uses imputation techniques to develop an omnibus specification test. The test is based on a simple modification of a Cramer von Mises functional that overcomes the curse of dimensionality often associated with the standard Cramer von Mises functional. The paper also considers estimation of the mean functional under the missing at random assumption. The proposed estimator lies in between a fully nonparametric and a parametric one and can be used, for example, to obtain a novel estimator for the average treatment effect parameter. Monte Carlo simulations show that the proposed estimator and test statistic have good finite sample properties. An empirical application illustrates the applicability of the results of the paper.     

Semiparametric estimation of average treatment effect through a random coefficient dummy endogenous variable model

This paper provides an estimation procedure for average treatment effect through a random coefficient dummy endogenous variable model. A leading example of the model is estimating the effect of a training program on earnings. The model is composed of two equations: an outcome equation and a decision equation. Given the linear restriction in outcome and decision equations, Chen (1999) provided a distribution-free estimation procedure under conditional symmetric error distributions. In this paper we extend Chen’s estimator by relaxing the linear index into a nonparametric function, which greatly reduces the risk of model misspecification. A two-step approach is proposed: the first step uses a nonparametric regression estimator for the decision variable, and the second step uses an instrumental variables approach to estimate average treatment effect in the outcome equation. The proposed estimator is shown to be consistent and asymptotically normally distributed. Furthermore, we investigate the finite performance of our estimator by a Monte Carlo study and also use our estimator to study the return of college education in different periods of China. The estimates seem more reasonable than those of other commonly used estimators.     

Asymptotic theory of multiple-set linear canonical analysis

This paper deals with asymptotics for multiple-set linear canonical analysis (MSLCA). A definition of this analysis, that adapts the classical one to the context of Euclidean random variables, is given and properties of the related canonical coefficients are derived. Then, estimators of the MSLCA’s elements, based on empirical covariance operators, are proposed and asymptotics for these estimators is obtained. More precisely, we prove their consistency and we obtain asymptotic normality for the estimator of the operator that gives MSLCA, and also for the estimator of the vector of canonical coefficients. These results are then used to obtain a test for mutual non-correlation between the involved Euclidean random variables.     

Composite Quantile Regression for Functional Linear Models with Dependent Errors

n this paper, we propose composite quantile regression for functional linear model with dependent data, in which the errors are from a short-range dependent and strictly stationary linear process. The functional principal component analysis is employed to approximate the slope function and the functional predictive variable respectively to construct an estimator of the slope function, and the convergence rate of the estimator is obtained under some regularity conditions. Simulation studies and a real data analysis are presented for illustration of the performance of the proposed estimator.     

�����������ĺ���������ģ�͵ĸ��Ϸ�λ������

??n this paper, we propose composite quantile regression for functional linear model with dependent data, in which the errors are from a short-range dependent and strictly stationary linear process. The functional principal component analysis is employed to approximate the slope function and the functional predictive variable respectively to construct an estimator of the slope function, and the convergence rate of the estimator is obtained under some regularity conditions. Simulation studies and a real data analysis are presented for illustration of the performance of the proposed estimator.     

Adaptive estimation of linear functionals in functional linear models

We consider the estimation of the value of a linear functional of the slope parameter in functional linear regression, where scalar responses are modeled in dependence of randomfunctions. In Johannes and Schenk [2010] it has been shown that a plug-in estimator based on dimension reduction and additional thresholding can attain minimax optimal rates of convergence up to a constant. However, this estimation procedure requires an optimal choice of a tuning parameter with regard to certain characteristics of the slope function and the covariance operator associated with the functional regressor. As these are unknown in practice, we investigate a fully data-driven choice of the tuning parameter based on a combination of model selection and Lepski??s method, which is inspired by the recent work of Goldenshluger and Lepski [2011]. The tuning parameter is selected as theminimizer of a stochastic penalized contrast function imitating Lepski??smethod among a random collection of admissible values. We show that this adaptive procedure attains the lower bound for the minimax risk up to a logarithmic factor over a wide range of classes of slope functions and covariance operators. In particular, our theory covers pointwise estimation as well as the estimation of local averages of the slope parameter.     

Varying coefficients partially linear models with randomly censored data

This paper considers the problem of estimation and inference in semiparametric varying coefficients partially linear models when the response variable is subject to random censoring. The paper proposes an estimator based on combining inverse probability of censoring weighting and profile least squares estimation. The resulting estimator is shown to be asymptotically normal. The paper also proposes a number of test statistics that can be used to test linear restrictions on both the parametric and nonparametric components. Finally, the paper considers the important issue of correct specification and proposes a nonsmoothing test based on a Cramer von Mises type of statistic, which does not suffer from the curse of dimensionality, nor requires multidimensional integration. Monte Carlo simulations illustrate the finite sample properties of the estimator and test statistics.     

Asymptotic distributions of two “synthetic data” estimators for censored single-index models

The censored single-index model provides a flexible way for modelling the association between a response and a set of predictor variables when the response variable is randomly censored and the link function is unknown. It presents a technique for “dimension reduction” in semiparametric censored regression models and generalizes the existing accelerated failure time models for survival analysis. This paper proposes two methods for estimation of single-index models with randomly censored samples. We first transform the censored data into synthetic data or pseudo-responses unbiasedly, then obtain estimates of the index coefficients by the rOPG or rMAVE procedures of Xia (2006) [1]. Finally, we estimate the unknown nonparametric link function using techniques for univariate censored nonparametric regression. The estimators for the index coefficients are shown to be root-n consistent and asymptotically normal. In addition, the estimator for the unknown regression function is a local linear kernel regression estimator and can be estimated with the same efficiency as the parameters are known. Monte Carlo simulations are conducted to illustrate the proposed methodologies.     

Safe and tight linear estimators for global optimization

Global optimization problems are often approached by branch and bound algorithms which use linear relaxations of the nonlinear constraints computed from the current variable bounds. This paper studies how to derive safe linear relaxations to account for numerical errors arising when computing the linear coefficients. It first proposes two classes of safe linear estimators for univariate functions. Class-1 estimators generalize previously suggested estimators from quadratic to arbitrary functions, while class-2 estimators are novel. When they apply, class-2 estimators are shown to be tighter theoretically (in a certain sense) and almost always tighter numerically. The paper then generalizes these results to multivariate functions. It shows how to derive estimators for multivariate functions by combining univariate estimators derived for each variable independently. Moreover, the combination of tight class-1 safe univariate estimators is shown to be a tight class-1 safe multivariate estimator. Finally, multivariate class-2 estimators are shown to be theoretically tighter (in a certain sense) than multivariate class-1 estimators.     

Use of prior information in the consistent estimation of regression coefficients in measurement error models

A multivariate ultrastructural measurement error model is considered and it is assumed that some prior information is available in the form of exact linear restrictions on regression coefficients. Using the prior information along with the additional knowledge of covariance matrix of measurement errors associated with explanatory vector and reliability matrix, we have proposed three methodologies to construct the consistent estimators which also satisfy the given linear restrictions. Asymptotic distribution of these estimators is derived when measurement errors and random error component are not necessarily normally distributed. Dominance conditions for the superiority of one estimator over the other under the criterion of Löwner ordering are obtained for each case of the additional information. Some conditions are also proposed under which the use of a particular type of information will give a more efficient estimator.     

On estimation in a spatial functional linear regression model with derivatives

This paper deals with functional linear regression for spatial data. We study the asymptotic properties of an estimator of a linear model where a spatial scalar response variable is related to a spatial functional explanatory variable and to its derivative. Convergence results with rate of this estimator are derived.     

Randomly censored partially linear single-index models

This paper proposes a method for estimation of a class of partially linear single-index models with randomly censored samples. The method provides a flexible way for modelling the association between a response and a set of predictor variables when the response variable is randomly censored. It presents a technique for “dimension reduction” in semiparametric censored regression models and generalizes the existing accelerated failure-time models for survival analysis. The estimation procedure involves three stages: first, transform the censored data into synthetic data or pseudo-responses unbiasedly; second, obtain quasi-likelihood estimates of the regression coefficients in both linear and single-index components by an iteratively algorithm; finally, estimate the unknown nonparametric regression function using techniques for univariate censored nonparametric regression. The estimators for the regression coefficients are shown to be jointly root-n consistent and asymptotically normal. In addition, the estimator for the unknown regression function is a local linear kernel regression estimator and can be estimated with the same efficiency as all the parameters are known. Monte Carlo simulations are conducted to illustrate the proposed methodology.     

Asymptotic properties of Lasso in high-dimensional partially linear models

We study the properties of the Lasso in the high-dimensional partially linear model where the number of variables in the linear part can be greater than the sample size. We use truncated series expansion based on polynomial splines to approximate the nonparametric component in this model. Under a sparsity assumption on the regression coefficients of the linear component and some regularity conditions, we derive the oracle inequalities for the prediction risk and the estimation error. We also provide sufficient conditions under which the Lasso estimator is selection consistent for the variables in the linear part of the model. In addition, we derive the rate of convergence of the estimator of the nonparametric function. We conduct simulation studies to evaluate the finite sample performance of variable selection and nonparametric function estimation.     

Simultaneous estimation and variable selection in median regression using Lasso-type penalty

We consider the median regression with a LASSO-type penalty term for variable selection. With the fixed number of variables in regression model, a two-stage method is proposed for simultaneous estimation and variable selection where the degree of penalty is adaptively chosen. A Bayesian information criterion type approach is proposed and used to obtain a data-driven procedure which is proved to automatically select asymptotically optimal tuning parameters. It is shown that the resultant estimator achieves the so-called oracle property. The combination of the median regression and LASSO penalty is computationally easy to implement via the standard linear programming. A random perturbation scheme can be made use of to get simple estimator of the standard error. Simulation studies are conducted to assess the finite-sample performance of the proposed method. We illustrate the methodology with a real example.     

Global adapted solution of one-dimensional backward stochastic Riccati equations,with application to the mean–variance hedging

Backward stochastic Riccati equations are motivated by the solution of general linear quadratic optimal stochastic control problems with random coefficients, and the solution has been open in the general case. One distinguishing difficult feature is that the drift contains a quadratic term of the second unknown variable. In this paper, we obtain the global existence and uniqueness result for a general one-dimensional backward stochastic Riccati equation. This solves the one-dimensional case of Bismut–Peng's problem which was initially proposed by Bismut (Lecture Notes in Math. 649 (1978) 180). We use an approximation technique by constructing a sequence of monotone drifts and then passing to the limit. We make full use of the special structure of the underlying Riccati equation. The singular case is also discussed. Finally, the above results are applied to solve the mean–variance hedging problem with general random market conditions.