Local Polynomial Estimation of Regression Functions for Mixing Processes

Local polynomial fitting has many exciting statistical properties which where established under i.i.d. setting. However, the need for non-linea r time series modeling, constructing predictive intervals, understanding divergence of non-linear time series requires the development of the theory of local polynomial fitting for dependent data. In this paper, we study the problem of estimating conditional mean functions and their derivatives via a local polynomial fit. The functions include conditional moments, conditional distribution as well as conditional density functions. Joint asymptotic normality for derivative estimation is established for both strongly mixing and ρ-mixing processes.     

Quantile Regression for Location‐Scale Time Series Models with Conditional Heteroscedasticity

This paper considers quantile regression for a wide class of time series models including autoregressive and moving average (ARMA) models with asymmetric generalized autoregressive conditional heteroscedasticity errors. The classical mean‐variance models are reinterpreted as conditional location‐scale models so that the quantile regression method can be naturally geared into the considered models. The consistency and asymptotic normality of the quantile regression estimator is established in location‐scale time series models under mild conditions. In the application of this result to ARMA‐generalized autoregressive conditional heteroscedasticity models, more primitive conditions are deduced to obtain the asymptotic properties. For illustration, a simulation study and a real data analysis are provided.     

Estimation and Test for Multi-Dimensional Regression Models

This work is concerned with the estimation of multi-dimensional regression and the asymptotic behavior of the test involved in selecting models. The main problem with such models is that we need to know the covariance matrix of the noise to get an optimal estimator. We show in this article that if we choose to minimize the logarithm of the determinant of the empirical error covariance matrix, then we get an asymptotically optimal estimator. Moreover, under suitable assumptions, we show that this cost function leads to a very simple asymptotic law for testing the number of parameters of an identifiable and regular regression model. Numerical experiments confirm the theoretical results.     

Penalized regression models with autoregressive error terms

Penalized regression methods have recently gained enormous attention in statistics and the field of machine learning due to their ability of reducing the prediction error and identifying important variables at the same time. Numerous studies have been conducted for penalized regression, but most of them are limited to the case when the data are independently observed. In this paper, we study a variable selection problem in penalized regression models with autoregressive (AR) error terms. We consider three estimators, adaptive least absolute shrinkage and selection operator, bridge, and smoothly clipped absolute deviation, and propose a computational algorithm that enables us to select a relevant set of variables and also the order of AR error terms simultaneously. In addition, we provide their asymptotic properties such as consistency, selection consistency, and asymptotic normality. The performances of the three estimators are compared with one another using simulated and real examples.     

An Orthogonality‐Based Estimation of Moments for Linear Mixed Models

Abstract. Estimating higher‐order moments, particularly fourth‐order moments in linear mixed models is an important, but difficult issue. In this article, an orthogonality‐based estimation of moments is proposed. Under only moment conditions, this method can easily be used to estimate the model parameters and moments, particularly those of higher order than the second order, and in the estimators the random effects and errors do not affect each other. The asymptotic normality of all the estimators is provided. Moreover, the method is readily extended to handle non‐linear, semiparametric and non‐linear models. A simulation study is carried out to examine the performance of the new method.     

Quantile Regression Estimator for GARCH Models

Abstract. In this article, we study the quantile regression estimator for GARCH models. We formulate the quantile regression problem by a reparametrization method and verify that the obtained quantile regression estimator is strongly consistent and asymptotically normal under certain regularity conditions. We also present our simulation results and a real data analysis for illustration.     

Jackknife Estimation for Smooth Functions of the Parametric Component in Partially Linear Regression Models

Abstract

It is known that due to the existence of the nonparametric component, the usual estimators for the parametric component or its function in partially linear regression models are biased. Sometimes this bias is severe. To reduce the bias, we propose two jackknife estimators and compare them with the naive estimator. All three estimators are shown to be asymptotically equivalent and asymptotically normally distributed under some regularity conditions. However, through simulation we demonstrate that the jackknife estimators perform better than the naive estimator in terms of bias when the sample size is small to moderate. To make our results more useful, we also construct consistent estimators of the asymptotic variance, which are robust against heterogeneity of the error variances.     

Integrated Estimation of Functional-Coefficient Regression Models with Different Smoothing Variables

In this article, the functional-coefficient regression models with different smoothing variables in different coefficient functions are discussed. The integrated estimates of the coefficient functions are defined by marginal integration on the initial value obtained by local linear technique. Their asymptotical normalities are studied.     

Rank Estimation of Log-Linear Regression with Interval-Censored Data

Interval-censored data arise in a wide variety of research and application fields such as cancer and AIDS studies. In this paper, we study a log-linear regression model when data are subject to interval censoring. We use a U-statistic based on ranks to estimate regression coefficients and establish large sample properties of the estimator. We illustrate the performance of the proposed estimate with simulations and a numerical example.     

Asymptotic Properties of Quasi-Maximum Likelihood Estimates in Generalized Linear Models

This article introduces a novel method, named JC ₁, for obtaining G-efficient mixture design to fit quadratic models. The advantage of JC ₁ method over existing algorithms is that it gives G-efficient designs without need of generating all the extreme vertices, edge centroids and constraint plane centroids of the mixture experimental region. The performance of the new method is illustrated and its comparison is given with popularly used algorithms—Snee (1975) algorithm and Welch (1985 Welch , W. J. ( 1985 ). ACED: Algorithms for the construction of experimental designs . Amer. Statistician 39 : 146 .[Crossref] , [Google Scholar]) ACED algorithm for second-order (quadratic model) designs and it is observed that JC ₁ method performs as well as the existing methods or sometimes better than those with additional advantage of large savings in computational efforts.     

Partially Linear Additive Models with Unknown Link Functions

In this paper, we consider partially linear additive models with an unknown link function, which include single‐index models and additive models as special cases. We use polynomial spline method for estimating the unknown link function as well as the component functions in the additive part. We establish that convergence rates for all nonparametric functions are the same as in one‐dimensional nonparametric regression. For a faster rate of the parametric part, we need to define appropriate ‘projection’ that is more complicated than that defined previously for partially linear additive models. Compared to previous approaches, a distinct advantage of our estimation approach in implementation is that estimation directly reduces estimation in the single‐index model and can thus deal with much larger dimensional problems than previous approaches for additive models with unknown link functions. Simulations and a real dataset are used to illustrate the proposed model.     

模糊线性回归模型的最小二乘方法

针对自变量和因变量皆模糊的数据系统中的回归分析问题,为避免自变量退化成数值变量时可能引致的估计误差增大而带来的问题,提出系统中引入模糊调整项的回归模型的一般结构,并运用基于模糊数间完备距离的最小二乘法研究模型解析表达式;利用水平截集概念将模糊多元回归模型转化成两个传统回归模型,根据模糊数间距离采用最小二乘法得到参数估计,给出员工工作绩效评估的算例说明方法的有效性,并结合Bootstrap方法的应用,研究回归参数所具有的随机不确定性动态变化。     

Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects

Abstract. We consider N independent stochastic processes (X _i (t), t ∈ [0,T _i ]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φ _i . The distribution of the random effect φ _i depends on unknown parameters which are to be estimated from the continuous observation of the processes X_i. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ _i and φ _i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T _i =T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.     

Estimation and Asymptotic Properties in Periodic GARCH(1, 1) Models

This article studies the probabilistic structure and asymptotic inference of the first-order periodic generalized autoregressive conditional heteroscedasticity (PGARCH(1, 1)) models in which the parameters in volatility process are allowed to switch between different regimes. First, we establish necessary and sufficient conditions for a PGARCH(1, 1) process to have a unique stationary solution (in periodic sense) and for the existence of moments of any order. Second, using the representation of squared PGARCH(1, 1) model as a PARMA(1, 1) model, we then consider Yule-Walker type estimators for the parameters in PGARCH(1, 1) model and derives their consistency and asymptotic normality. The estimator can be surprisingly efficient for quite small numbers of autocorrelations and, in some cases can be more efficient than the least squares estimate (LSE). We use a residual bootstrap to define bootstrap estimators for the Yule-Walker estimates and prove the consistency of this bootstrap method. A set of numerical experiments illustrates the practical relevance of our theoretical results.     

Consistency of the Semi-parametric MLE in Linear Regression Models with Interval-censored Data

Abstract.  Consider the model Y = β ^' X + ε . Let F ₀ be the unknown cumulative distribution function of the random variable ε . Consistency of the semi-parametric Maximum likelihood estimator of ( β , F ₀), denoted by     , has not been established under any interval censorship (IC) model. We prove in this paper that     is consistent under the mixed case IC model and some mild assumptions.     

Robust estimation and selection for single-index regression model

In this paper, we consider a single-index regression model for which we propose a robust estimation procedure for the model parameters and an efficient variable selection of relevant predictors. The proposed method is known as the penalized generalized signed-rank procedure. Asymptotic properties of the proposed estimator are established under mild regularity conditions. Extensive Monte Carlo simulation experiments are carried out to study the finite sample performance of the proposed approach. The simulation results demonstrate that the proposed method dominates many of the existing ones in terms of robustness of estimation and efficiency of variable selection. Finally, a real data example is given to illustrate the method.     

Variable Selection for Semiparametric Partially Linear Covariate-Adjusted Regression Models

In this article, the partially linear covariate-adjusted regression models are considered, and the penalized least-squares procedure is proposed to simultaneously select variables and estimate the parametric components. The rate of convergence and the asymptotic normality of the resulting estimators are established under some regularization conditions. With the proper choices of the penalty functions and tuning parameters, it is shown that the proposed procedure can be as efficient as the oracle estimators. Some Monte Carlo simulation studies and a real data application are carried out to assess the finite sample performances for the proposed method.     

Estimation of the Variance Function in Heteroscedastic Linear Regression Models

Heteroscedasticity generally exists when a linear regression model is applied to analyzing some real-world problems. Therefore, how to accurately estimate the variance functions of the error term in a heteroscedastic linear regression model is of great importance for obtaining efficient estimates of the regression parameters and making valid statistical inferences. A method for estimating the variance function of heteroscedastic linear regression models is proposed in this article based on the variance-reduced local linear smoothing technique. Some simulations and comparisons with other method are conducted to assess the performance of the proposed method. The results demonstrate that the proposed method can accurately estimate the variance functions and therefore produce more efficient estimates of the regression parameters.     

Linear-representation Based Estimation of Stochastic Volatility Models

Abstract.  A new way of estimating stochastic volatility models is developed. The method is based on the existence of autoregressive moving average (ARMA) representations for powers of the log-squared observations. These representations allow to build a criterion obtained by weighting the sums of squared innovations corresponding to the different ARMA models. The estimator obtained by minimizing the criterion with respect to the parameters of interest is shown to be consistent and asymptotically normal. Monte-Carlo experiments illustrate the finite sample properties of the estimator. The method has potential applications to other non-linear time-series models.