Nonparametric Part Research Articles

A partially linear additive model for clustered proportion data.

Proportion data with support lying in the interval [0,1] are a commonplace in various domains of medicine and public health. When these data are available as clusters, it is important to correctly incorporate the within-cluster correlation to improve the estimation efficiency while conducting regression-based risk evaluation. Furthermore, covariates may exhibit a nonlinear relationship with the (proportion) responses while quantifying disease status. As an alternative to various existing classical methods for modeling proportion data (such as augmented Beta regression) that uses maximum likelihood, or generalized estimating equations, we develop a partially linear additive model based on the quadratic inference function. Relying on quasi-likelihood estimation techniques and polynomial spline approximation for unknown nonparametric functions, we obtain the estimators for both parametric part and nonparametric part of our model and study their large-sample theoretical properties. We illustrate the advantages and usefulness of our proposition over other alternatives via extensive simulation studies, and application to a real dataset from a clinical periodontal study.

Efficient parameter estimation and variable selection in partial linear varying coefficient quantile regression model with longitudinal data

Efficient estimation and variable selection in partial linear varying coefficient quantile regression model with longitudinal data is concerned in this paper. To improve estimation efficiency in quantile regression, based on B-spline basis approximation for nonparametric parts, we propose a new estimating function, which can incorporate the correlation structure between repeated measures. In order to reduce computational burdens, the induced smoothing method is used. The new method is empirically shown to be much more efficient and robust than the popular generalized estimating equations based methods. Under mild conditions, the asymptotically normal distribution of the estimators for the parametric components and the optimal convergence rate of the estimators for the nonparametric functions are established. Furthermore, to do variable selection, a smooth-threshold estimating equation is proposed, which can use the correlation structure and select the nonparametric and parametric parts simultaneously. Theoretically, the variable selection procedure works beautifully, including consistency in variable selection and oracle property in estimation. Simulation studies and real data analysis are included to show the finite sample performance.

Estimation and inference for varying coefficient partially nonlinear errors-in-variables models

ABSTRACTIn this article, we study the varying coefficient partially nonlinear model with measurement errors in the nonparametric part. A local corrected profile nonlinear least-square estimation procedure is proposed and the asymptotic properties of the resulting estimators are established. Further, a generalized likelihood ratio (GLR) statistic is proposed to test whether the varying coefficients are constant. The asymptotic null distribution of the statistic is obtained and a residual-based bootstrap procedure is employed to compute the p-value of the statistic. Some simulations are conducted to evaluate the performance of the proposed methods. The results show that the estimating and testing procedures work well in finite samples.

Sequential profile Lasso for ultra-high-dimensional partially linear models

ABSTRACTIn this paper, we study ultra-high-dimensional partially linear models when the dimension of the linear predictors grows exponentially with the sample size. For the variable screening, we propose a sequential profile Lasso method (SPLasso) and show that it possesses the screening property. SPLasso can also detect all relevant predictors with probability tending to one, no matter whether the ultra-high models involve both parametric and nonparametric parts. To select the best subset among the models generated by SPLasso, we propose an extended Bayesian information criterion (EBIC) for choosing the final model. We also conduct simulation studies and apply a real data example to assess the performance of the proposed method and compare with the existing method.

Treatment recommendation and parameter estimation under single-index contrast function

ABSTRACTIn this article, we consider a semiparametric model for contrast function which is defined as the conditional expected outcome difference under comparative treatments. The contrast function can be used to recommend treatment for better average outcomes. Existing approaches model the contrast function either parametrically or nonparametrically. We believe our approach improves interpretability over the non-parametric approach while enhancing robustness over the parametric approach. Without explicit estimation of the nonparametric part of our model, we show that a kernel-based method can identify the parametric part up to a multiplying constant. Such identification suffices for treatment recommendation. Our method is also extended to high-dimensional settings. We study the asymptotics of the resulting estimation procedure in both low- and high-dimensional cases. We also evaluate our method in simulation studies and real data analyses.

Short-term flood forecasting with an amended semi-parametric regression ensemble model

Flood forecasting is very important research topic in disaster prevention and reduction. The characteristics of flood involve a rather complex systematic dynamic under the influence of different meteorological factors including linear and non-linear patterns. Recently there are many novel forecasting methods of improving the forecasting accuracy. This paper explores the potential and effect of the semiparametric regression to modelize flood water-level and to forecast the inundation of Mekong Delta in Vietnam. The semi-parametric regression technique is a combination of a parametric regression approach and a non-parametric regression concept. In the process of model building, three altered linear regression models are applied for the parametric component. They are stepwise multiple linear regression, partial least squares solution and multirecursive regression method. They are used to capture flood’s linear characteristics. The nonparametric part is solved by a modified estimation of a smooth function. Furthermore, some justified nonlinear regression models based on artificial neural network are also able to obtain flood’s non-linear characteristics. They help us to smooth the model's non-parametric constituent easily and quickly. The last element is the model's error. Then the semiparametric regression is used for ensemble model based on the principle component analysis technique. Flood water-level forecasting, with a lead time of one and more days, has been made by using a selected sequence of past water-level values and some relevant factors observed at a specific location. Time-series analytical method is utilized to build the model. Obtained empirical results indicate that the prediction by using the amended semi-parametric regression ensemble model is generally better than those obtained by using the other models presented in this study in terms of the same evaluation measurements. Our findings reveal that the estimation power of the modern statistical model is reliable and auspicious. The proposed model here can be used as a promising alternative forecasting tool for flood to achieve better forecasting accuracy and to optimize prediction quality further.

Estimation and variable selection in single-index composite quantile regression

ABSTRACTIn this article, a new composite quantile regression estimation approach is proposed for estimating the parametric part of single-index model. We use local linear composite quantile regression (CQR) for estimating the nonparametric part of single-index model (SIM) when the error distribution is symmetrical. The weighted local linear CQR is proposed for estimating the nonparametric part of SIM when the error distribution is asymmetrical. Moreover, a new variable selection procedure is proposed for SIM. Under some regularity conditions, we establish the large sample properties of the proposed estimators. Simulation studies and a real data analysis are presented to illustrate the behavior of the proposed estimators.

Counting process-based dimension reduction methods for censored outcomes

We propose a class of dimension reduction methods for right censored survival data using a counting process representation of the failure process. Semiparametric estimating equations are constructed to estimate the dimension reduction subspace for the failure time model. The proposed method addresses two fundamental limitations of existing approaches. First, using the counting process formulation, it does not require any estimation of the censoring distribution to compensate the bias in estimating the dimension reduction subspace. Second, the nonparametric part in the estimating equations is adaptive to the structural dimension, hence the approach circumvents the curse of dimensionality. Asymptotic normality is established for the obtained estimators. We further propose a computationally efficient approach that simplifies the estimation equation formulations and requires only a singular value decomposition to estimate the dimension reduction subspace. Numerical studies suggest that our new approaches exhibit significantly improved performance for estimating the true dimension reduction subspace. We further conduct a real data analysis on a skin cutaneous melanoma dataset from The Cancer Genome Atlas. The proposed method is implemented in the R package "orthoDr".

Quantile regression for robust estimation and variable selection in partially linear varying-coefficient models

ABSTRACTIn this paper, we develop a new estimation procedure based on quantile regression for semiparametric partially linear varying-coefficient models. The proposed estimation approach is empirically shown to be much more efficient than the popular least squares estimation method for non-normal error distributions, and almost not lose any efficiency for normal errors. Asymptotic normalities of the proposed estimators for both the parametric and nonparametric parts are established. To achieve sparsity when there exist irrelevant variables in the model, two variable selection procedures based on adaptive penalty are developed to select important parametric covariates as well as significant nonparametric functions. Moreover, both these two variable selection procedures are demonstrated to enjoy the oracle property under some regularity conditions. Some Monte Carlo simulations are conducted to assess the finite sample performance of the proposed estimators, and a real-data example is used to illustrate the application of the proposed methods.

Adaptive varying-coefficient linear quantile model: a profiled estimating equations approach

We consider an estimating equations approach to parameter estimation in adaptive varying-coefficient linear quantile model. We propose estimating equations for the index vector of the model in which the unknown nonparametric functions are estimated by minimizing the check loss function, resulting in a profiled approach. The estimating equations have a bias-corrected form that makes undersmoothing of the nonparametric part unnecessary. The estimating equations approach makes it possible to obtain the estimates using a simple fixed-point algorithm. We establish asymptotic properties of the estimator using empirical process theory, with additional complication due to the nuisance nonparametric part. The finite sample performance of the new model is illustrated using simulation studies and a forest fire dataset.

Bias-corrected estimations in varying-coefficient partially nonlinear models with measurement error in the nonparametric part

ABSTRACTIn this paper, we consider the statistical inference for the varying-coefficient partially nonlinear model with additive measurement errors in the nonparametric part. The local bias-corrected profile nonlinear least-squares estimation procedure for parameter in nonlinear function and nonparametric function is proposed. Then, the asymptotic normality properties of the resulting estimators are established. With the empirical likelihood method, a local bias-corrected empirical log-likelihood ratio statistic for the unknown parameter, and a corrected and residual adjusted empirical log-likelihood ratio for the nonparametric component are constructed. It is shown that the resulting statistics are asymptotically chi-square distribution under some suitable conditions. Some simulations are conducted to evaluate the performance of the proposed methods. The results indicate that the empirical likelihood method is superior to the profile nonlinear least-squares method in terms of the confidence regions of parameter and point-wise confidence intervals of nonparametric function.

Focused Information Criterion for Series Estimation in Partially Linear Models

This paper proposes a focused information criterion for variable selection in partially linear models. Our criterion is designed to select an optimal model for estimating a focus parameter, which is a parameter of interest. We estimate the model using the series method and jointly select the variables in the linear part and the series length in the nonparametric part. A Monte Carlo simulation shows that the proposed focused information criterion successfully selects the model that has a relatively small mean squared error of the estimator for the focus parameter.

Quantile regression and variable selection for partially linear model with randomly truncated data

This paper focuses on the problem of estimation and variable selection for quantile regression (QR) of partially linear model (PLM) where the response is subject to random left truncation. We propose a three-stage estimation procedure for parametric and nonparametric parts based on the weights which are random quantities and determined by the product-limit estimates of the distribution function of truncated variable. The estimators obtained in the second and third stages are more efficient than the initial estimators in the first stage. Furthermore, we propose a variable selection procedure for the QR of PLM by combining the estimation method with the smoothly clipped absolute deviation penalty to get sparse estimation of the regression parameter. The oracle properties of the variable selection approach are established. Simulation studies are conducted to examine the performance of our estimators and variable selection method.

Simultaneous Inference for the Partially Linear Model with a Multivariate Unknown Function when the Covariates are Measured with Errors

In this paper, we analyze the nonparametric part of a partially linear model when the covariates in parametric and non-parametric parts are subject to measurement errors. Based on a two-stage semi-parametric estimate, we construct a uniform confidence surface of the multivariate function for simultaneous inference. The developed methodology is applied to perform inference for the U.S. gasoline demand where the income and price variables are measured with errors. The empirical results strongly suggest that the linearity of the U.S. gasoline demand is rejected.

Simultaneous Inference for the Partially Linear Model with A Multivariate Unknown Function When the Covariates are Measured With Errors

In this paper, we analyze the nonparametric part of a partially linear model when the covariates in parametric and non-parametric parts are subject to measurement errors. Based on a two-stage semi-parametric estimate, we construct a uniform confidence surface of the multivariate function for simultaneous inference. The developed methodology is applied to perform inference for the U.S. gasoline demand where the income and price variables are measured with errors. The empirical results strongly suggest that the linearity of the U:S: gasoline demand is rejected.

A kernel-based mixed effect regression model for earthquake ground motions

This paper presents a semi-parametric mixed-effect regression approach for analyzing and modeling earthquake ground motions, taking into account the correlations between records. Using kernels, the proposed method extends the classical mixed model equations to complicated relationships. The predictive equation is composed of parametric and nonparametric parts. The parametric part incorporates known relationships into the model, while the nonparametric part captures the relationships which cannot be cast into a simple parametric form. A least squares kernel machine is used to infer the nonparametric part of the model. The resulting semi-parametric model combines the strengths of parametric and nonparametric approaches, allowing incorporation of prior, well-justified knowledge into the model while retaining flexibility with respect to the explanatory variables for which the functional form is uncertain. Equations for pointwise confidence and prediction intervals around the conditional mean are provided. The validity of the proposed method is demonstrated through numerical simulations and using recorded ground motions.

A new local estimation method for single index models for longitudinal data

Single index models are natural extensions of linear models and overcome the so-called curse of dimensionality. They are very useful for longitudinal data analysis. In this paper, we develop a new efficient estimation procedure for single index models with longitudinal data, based on Cholesky decomposition and local linear smoothing method. Asymptotic normality for the proposed estimators of both the parametric and nonparametric parts will be established. Monte Carlo simulation studies show excellent finite sample performance. Furthermore, we illustrate our methods with a real data example.

Statistical inference for a varying-coefficient partially nonlinear model with measurement errors

In this study a varying-coefficient partially nonlinear model with measurement errors in the nonparametric part is proposed. Based on the corrected profile least-squared estimation methodology, we define the estimates of the unknowns of the current models, and check whether the coefficient functions are a constant or not by using the popular generalized likelihood ratio (GLR) test method. Further, the corresponding asymptotic distribution is established and a bootstrap procedure is also employed to implement the proposed methodology. Simulated and real examples are given to illustrate our proposed methodology.

Weighted composite quantile regression for single-index models

In this paper we propose a weighted composite quantile regression (WCQR) estimation for single-index models. For parametric part, the WCQR is augmented using a data-driven weighting scheme. With the error distribution unspecified, the proposed estimators share robustness from quantile regression and achieve nearly the same efficiency as the semiparametric maximum likelihood estimator for a variety of error distributions including the Normal, Student’s t, Cauchy distributions, etc. Furthermore, based on the proposed WCQR, we use the adaptive-LASSO to study variable selection for parametric part in the single-index models. For nonparametric part, the WCQR is augmented combining the equal weighted estimators with possibly different weights. Because of the use of weights, the estimation bias is eliminated asymptotically. By comparing asymptotic relative efficiency theoretically and numerically, WCQR estimation all outperforms the CQR estimation and some other estimate methods. Under regularity conditions, the asymptotic properties of the proposed estimations are established. The simulation studies and two real data applications are conducted to illustrate the finite sample performance of the proposed methods.

Minimum distance partial linear regression model checking with Berkson measurement errors

We propose a class of tests for fitting a parametric model to the nonparametric part in partial linear regression models in the presence of Berkson measurement errors in the covariates. The proposed tests are based on certain minimized L2 distances between a semi-parametric regression function estimator and the parametric regression model being fitted. We establish asymptotic normality of the proposed test statistics and that of the corresponding minimum distance estimators under the fitted model. The consistency of the tests and their asymptotic power against certain local alternatives are also investigated. Simulation and comparison studies are included to evaluate the finite sample performance of the proposed tests.