Local Linear Smoothing Technique Research Articles

Adaptive-weighted estimation of semi-varying coefficient models with heteroscedastic errors

An adaptive-weighted estimation procedure for parametric and nonparametric coefficients in semi-varying coefficient models with heteroscedastic errors is considered in this paper. Firstly, we present a consistent estimator of the variance function of the error term. In order to take the heteroscedasticity into consideration, we consider the weighted local linear smoothing technique. Asymptotic properties of the proposed estimators are established. Our theoretical results demonstrate that the adaptive-weighted estimator is more efficient than the unweighted profile least-squares estimators. The simulation results show that our adaptive-weighted estimators are more efficient, compared to profile least-squares estimators and re-weighted estimators under the finite sample size. Finally, our estimation procedure is applied to a real-world data.

Test for the covariance matrix in time-varying coefficients panel data models with fixed effects

This paper proposes tests for the null of sphericity and identity matrix for nonparametric time-varying coefficient panel data models with fixed effects. Firstly, based on the local linear smoothing technique, the estimators of the unknown coefficient functions and model residuals are obtained. Secondly, proper test statistics are proposed aiming at tests for sphericity or identity matrix with a large number of cross-sectional units and time series observations. In addition, the limiting distributions of the proposed test statistics are derived based on random matrix theory. At last, some simulation studies are conducted to examine the finite sample performance for the proposed test statistics and a real data example is analyzed.

Nonparametric Quantile Regression Estimation With Mixed Discrete and Continuous Data

In this article, we investigate the problem of nonparametrically estimating a conditional quantile function with mixed discrete and continuous covariates. A local linear smoothing technique combining both continuous and discrete kernel functions is introduced to estimate the conditional quantile function. We propose using a fully data-driven cross-validation approach to choose the bandwidths, and further derive the asymptotic optimality theory. In addition, we also establish the asymptotic distribution and uniform consistency (with convergence rates) for the local linear conditional quantile estimators with the data-dependent optimal bandwidths. Simulations show that the proposed approach compares well with some existing methods. Finally, an empirical application with the data taken from the IMDb website is presented to analyze the relationship between box office revenues and online rating scores. Supplementary materials for this article are available online.

Estimation of semi-varying coefficient error-in-variable models with surrogate data and validation sample

In this study, a semi-varying coefficient error-in-variable model with surrogate data and validation sample is proposed. Without specifying any error structure, we firstly use the local linear kernel smoothing technique to define the estimators and the proposed estimators are proved to be asymptotically normal. Then, we conduct generalized likelihood ratio (GLR) test on varying coefficient function. The data–driven bandwidth selection method is discussed. Finally, simulated studies are conducted to illustrate the finite sample properties of the proposed estimators and efficiency of the GLR methodology.

Analysis of longitudinal data with semiparametric varying-coefficient mean–covariance models

The efficient estimation of regression coefficients in the longitudinal data analysis requires a correct specification of the covariance structure. Existing approaches usually focus on modeling the mean with specification of certain covariance structures, which may lead to inefficient or biased estimators of parameters in the mean if misspecification occurs. In this article, we propose a novel data-driven approach based on semiparametric varying-coefficient models to model the mean and the covariance simultaneously, motivated by the modified Cholesky decomposition. An iterative estimation method is proposed, consisting of an orthogonality-based technique for parameters, an adaptive jump-preserving estimation method for varying coefficients, a modification of local linear smoothing technique for the autoregressive coefficient function, and a kernel smoothing technique for the variance function. Theoretical properties of the resulting estimators including uniform consistency and asymptotic normality are explicitly studied under certain mild conditions. Simulation studies are carried out to evaluate the efficacy of the proposed methods, and an analysis of a real data example is provided for illustration.

Nonparametric estimation of multivariate multiparameter conditional copulas

Nonparametric estimation of conditional copulas with one parameter has been investigated in Acar et al. (2011). The estimation for multivariate multiparameter conditional copulas, however, has not been considered so far. This paper adopts the local linear smoothing technique and Newton–Raphson method to estimate those copulas. Under some regularity conditions, the asymptotic normality of the estimators is obtained. Simulation work shows the efficiency of the proposed method. As an application, we analyze a life expectancies data set and show that the conditional t copula outperforms the conditional Clayton, Frank and Gumbel copulas.

Multi-dimensional functional principal component analysis

Functional principal component analysis is one of the most commonly employed approaches in functional and longitudinal data analysis and we extend it to analyze functional/longitudinal data observed on a general $d$-dimensional domain. The computational issues emerging in the extension are fully addressed with our proposed solutions. The local linear smoothing technique is employed to perform estimation because of its capabilities of performing large-scale smoothing and of handling data with different sampling schemes (possibly on irregular domain) in addition to its nice theoretical properties. Besides taking the fast Fourier transform strategy in smoothing, the modern GPGPU (general-purpose computing on graphics processing units) architecture is applied to perform parallel computation to save computation time. To resolve the out-of-memory issue due to large-scale data, the random projection procedure is applied in the eigendecomposition step. We show that the proposed estimators can achieve the classical nonparametric rates for longitudinal data and the optimal convergence rates for functional data if the number of observations per sample is of the order $(n/ \log n)^{d/4}$. Finally, the performance of our approach is demonstrated with simulation studies and the fine particulate matter (PM 2.5) data measured in Taiwan.

Local $M$-estimation for conditional variance function with dependent data

In this paper, a local $M$-estimation for the conditional variance function in heteroscedastic regression models under stationary $\alpha $-mixing dependent samples is developed. The local $M$-estimator is based on the local linear smoothing technique and the $M$-estimation technique, and it is shown to be not only asymptotically equivalent to the local linear estimator but also robust. The weak consistency as well as the asymptotic normality of the local $M$-estimator for the conditional variance function are obtained under mild conditions.

Nonparametric Estimation of Conditional Quantile Regression with Mixed Discrete and Continuous Data

In this paper, we investigate the nonlinear quantile regression with mixed discrete and continuous regressors. A local linear smoothing technique with the mixed continuous and discrete kernel function is proposed to estimate the conditional quantile regression function. Under some mild conditions, the asymptotic distribution is established for the proposed nonparametric estimators, which can be seen as a generalisation of some existing theory which only handles the case of purely continuous regressors. We further study the choice of the tuning parameters in the local quantile estimation procedure, and suggest using the cross-validation approach to choose the optimal bandwidths. A simulation study is provided to examine the finite sample behavior of the proposed method, which is also compared with the naive local linear quantile estimation without smoothing the discrete regressors and the nonparametric inverse-CDF method proposed by Li, Lin and Racine (2013).

Local M-estimation for Conditional Variance in Heteroscedastic Regression Models

In this article, we develop a local M-estimation for the conditional variance in heteroscedastic regression models. The estimator is based on the local linear smoothing technique and the M-estimation technique, and it is shown to be not only asymptotically equivalent to the local linear estimator but also robust. The consistency and asymptotic normality of the local M-estimator for the conditional variance in heteroscedastic regression models are obtained under mild conditions. The simulation studies demonstrate that the proposed estimators perform well in robustness.

A Varying-Coefficient Expectile Model for Estimating Value at Risk

This article develops a nonparametric varying-coefficient approach for modeling the expectile-based value at risk (EVaR). EVaR has an advantage over the conventional quantile-based VaR (QVaR) of being more sensitive to the magnitude of extreme losses. EVaR can also be used for calculating QVaR and expected shortfall (ES) by exploiting the one-to-one mapping from expectiles to quantiles, and the relationship between VaR and ES. Previous studies on conditional EVaR estimation only considered parametric autoregressive model set-ups, which account for the stochastic dynamics of asset returns but ignore other exogenous economic and investment related factors. Our approach overcomes this drawback and allows expectiles to be modeled directly using covariates that may be exogenous or lagged dependent in a flexible way. Risk factors associated with profits and losses can then be identified via the expectile regression at different levels of prudentiality. We develop a local linear smoothing technique for estimating the coefficient functions within an asymmetric least squares minimization set-up, and establish the consistency and asymptotic normality of the resultant estimator. To save computing time, we propose to use a one-step weighted local least squares procedure to compute the estimates. Our simulation results show that the computing advantage afforded by this one-step procedure over full iteration is not compromised by a deterioration in estimation accuracy. Real data examples are used to illustrate our method. Supplementary materials for this article are available online.

The Profile Likelihood Estimation for Single‐Index ARCH(p)‐M Model

We propose a class of single‐index ARCH(p)‐M models and investigate estimators of the parametric and nonparametric components. We first estimate the nonparametric component using local linear smoothing technique and then construct an estimator of parametric component by using profile quasimaximum likelihood method. Under regularity conditions, the asymptotic properties of our estimators are established.

Estimation and inference of semi-varying coefficient models with heteroscedastic errors

This article focuses on the estimation of the parametric component, which is of primary interest, in semi-varying coefficient models with heteroscedastic errors. Specifically, we first present a procedure for estimating the variance function of the error term and the resulting estimator is proved to be consistent. Then, by applying the local linear smoothing technique, and taking the estimated error heteroscedasticity into account, we suggest a re-weighting estimation of the constant coefficients and the resulting estimators are shown to have smaller asymptotic variances than the profile least-squares estimators that neglect the error heteroscedasticity while remaining the same biases. Some simulation experiments are conducted to evaluate the finite sample performance of the proposed methodologies. Finally, a real-world data set is analyzed to demonstrate the application of the methods.

A MODIFICATION ON RIDGE ESTIMATION FOR FUZZY NONPARAMETRIC REGRESSION

This paper deals with ridge estimation of fuzzy nonparametric regression models using triangular fuzzy numbers. This estimation method is obtained by implementing ridge regression learning algorithm in the La- grangian dual space. The distance measure for fuzzy numbers that suggested by Diamond is used and the local linear smoothing technique with the cross- validation procedure for selecting the optimal value of the smoothing param- eter is fuzzied to t the presented model. Some simulation experiments are then presented which indicate the performance of the proposed method.

Nonparametric estimation of varying coefficient error-in-variable models with validation sampling

In this paper, varying coefficient models are investigated with the response and covariate prone to measurement error. Without specifying any error structure equation, four estimators of the coefficient function vector are proposed by using the local linear kernel smoothing technique and also proved to be asymptotically normal. The data-driven bandwidth selection method is discussed. Simulation examples are conducted to evaluate the proposed estimation methods.

Statistical estimation in varying coefficient models with surrogate data and validation sampling

Varying coefficient error-in-covariables models are considered with surrogate data and validation sampling. Without specifying any error structure equation, two estimators for the coefficient function vector are suggested by using the local linear kernel smoothing technique. The proposed estimators are proved to be asymptotically normal. A bootstrap procedure is suggested to estimate the asymptotic variances. The data-driven bandwidth selection method is discussed. A simulation study is conducted to evaluate the proposed estimating methods.

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.

Assessing Ordinal Logistic Regression Models via Nonparametric Smoothing

A nonparametric local linear smoothing technique for testing goodness-of-fit of ordinal logistic regression models with continuous and categorical covariates is presented. This proposed test statistic is a generalization of Lin and Chen's statistic (2005) to ordinal response data. The extension of logistic regression models for binary responses to ordinal responses is usually involved by modeling cumulative logits. The expectation and variance of the proposed test statistic are derived, and the sampling distribution of the test statistic, which approximates a scaled chi-square distribution is evaluated by simulation. The power comparison between the proposed test and methods of Pulkstenis and Robinson (2004) are also provided. The simulations reveal that the proposed test has greater power to detect the omitted interaction term, incorrectly functional form of continuous covariates, or misspecification of the complimentary log-log link function for various sample sizes. The proposed testing procedure is illustrated by a numerical study.

Fuzzy nonparametric regression based on local linear smoothing technique

In a great deal of literature on fuzzy regression analysis, most of research has focused on some predefined parametric forms of fuzzy regression relationships, especially on the fuzzy linear regression models. In many practical situations, it may be unrealistic to predetermine a fuzzy parametric regression relationship. In this paper, a fuzzy nonparametric model with crisp input and LR fuzzy output is considered and, based on the distance measure for fuzzy numbers suggested by Diamond [P. Diamond, Fuzzy least squares, Information Sciences 46 (1988) 141–157], the local linear smoothing technique in statistics with the cross-validation procedure for selecting the optimal value of the smoothing parameter is fuzzified to fit this model. Some simulation experiments are conducted to examine the performance of the proposed method and three real-world datasets are analyzed to illustrate the application of the proposed method. The results demonstrate that the proposed method works quite well not only in producing satisfactory estimate of the fuzzy regression function, but also in reducing the boundary effect significantly.

Application of a local linear autoregressive model to BOD time series

In this paper, we analyze the biochemical oxygen demand data collected over two years from McDowell Creek, Charlotte, North Carolina, U.S.A., by fitting an autoregressive model with time-dependent coefficients. The local linear smoothing technique is developed and implemented to estimate the coefficient functions of the autoregressive model. A nonparametric version of the Akaike information criterion is developed to determine the order of the model and to select the optimal bandwidth. We also propose a hypothesis testing technique, based on the residual sum of squares and F-test, to detect whether certain coefficients in the model are really varying or whether any variables are significant. The approximate null distributions of the test are provided. The proposed model has some advantages, such as it is determined completely by data, it is easily implemented and it provides a better prediction. Copyright © 2000 John Wiley & Sons, Ltd.