Spatially varying dynamic coefficient models

In this work we present a flexible class of linear models to treat observations made in discrete time and continuous space, where the regression coefficients vary smoothly in time and space. This kind of model is particularly appealing in situations where the effect of one or more explanatory processes on the response present substantial heterogeneity in both dimensions. We describe how to perform inference for this class of models and also how to perform forecasting in time and interpolation in space, using simulation techniques. The performance of the algorithm to estimate the parameters of the model and to perform prediction in time is investigated with simulated data sets. The proposed methodology is used to model pollution levels in the Northeast of the United States.     

Estimation of random coefficient regression models

Linear regression models with coefficients across individual units regarded as random samples from some population are studied in this article from a Bayesian viewpoint. A prior distribution of the secondary parameters is derived following the Jeffreys rule. Posterior distribution of the primary and secondary parameters, and the predictive distribution of the future value are then examined. Computations of the parameter estimates are found to be rather straightforward. Data from a performance test on pigs is analysed and discussed. We also discuss the difficulties involved in using a Lindley and Smith (1972) prior in this problem.     

Quantile regression in varying coefficient models

This paper deals with the estimation of conditional quantiles in varying coefficient models by estimating the coefficients. Varying coefficient models are among popular models that have been proposed to alleviate the curse of dimensionality. Previous works on varying coefficient models deal with conditional means directly or indirectly. However, quantiles themselves can be defined without moment conditions and plotting several conditional quantiles would give us more understanding of the data than plotting just the conditional mean. Particularly, we estimate the conditional median by estimating varying coefficients by local L₁ regression.     

Statistical inference in partially time-varying coefficient models

A partially time-varying coefficient time series model is introduced to characterize the nonlinearity and trending phenomenon. To estimate the regression parameter and the nonlinear coefficient function, the profile least squares approach is applied with the help of local linear approximation. The asymptotic distributions of the proposed estimators are established under mild conditions. Meanwhile, the generalized likelihood ratio test is studied and the test statistics are demonstrated to follow asymptotic χ²-distribution under the null hypothesis. Furthermore, some extensions of the proposed model are discussed and several numerical examples are provided to illustrate the finite sample behavior of the proposed methods.     

Testing for constancy in varying coefficient models

We consider varying coefficient models, which are an extension of the classical linear regression models in the sense that the regression coefficients are replaced by functions in certain variables (for example, time), the covariates are also allowed to depend on other variables. Varying coefficient models are popular in longitudinal data and panel data studies, and have been applied in fields such as finance and health sciences. We consider longitudinal data and estimate the coefficient functions by the flexible B-spline technique. An important question in a varying coefficient model is whether an estimated coefficient function is statistically different from a constant (or zero). We develop testing procedures based on the estimated B-spline coefficients by making use of nice properties of a B-spline basis. Our method allows longitudinal data where repeated measurements for an individual can be correlated. We obtain the asymptotic null distribution of the test statistic. The power of the proposed testing procedures are illustrated on simulated data where we highlight the importance of including the correlation structure of the response variable and on real data.     

Quantile regression in heteroscedastic varying coefficient models

Varying coefficient models are flexible models to describe the dynamic structure in longitudinal data. Quantile regression, more than mean regression, gives partial information on the conditional distribution of the response given the covariates. In the literature, the focus has been so far mostly on homoscedastic quantile regression models, whereas there is an interest in looking into heteroscedastic modelling. This paper contributes to the area by modelling the heteroscedastic structure and estimating it from the data, together with estimating the quantile functions. The use of the proposed methods is illustrated on real-data applications. The finite-sample behaviour of the methods is investigated via a simulation study, which includes a comparison with an existing method.     

Principal varying coefficient estimator for high-dimensional models

ABSTRACT

We consider principal varying coefficient models in the high-dimensional setting, combined with variable selection, to reduce the effective number of parameters in semiparametric modelling. The estimation is based on B-splines approach. For the unpenalized estimator, we establish non-asymptotic bounds of the estimator and then establish the (asymptotic) local oracle property of the penalized estimator, as well as non-asymptotic error bounds. Monte Carlo studies reveal the favourable performance of the estimator and an application on a real dataset is presented.     

Efficient estimation for time-varying coefficient longitudinal models

For estimation of time-varying coefficient longitudinal models, the widely used local least-squares (LS) or covariance-weighted local LS smoothing uses information from the local sample average. Motivated by the fact that a combination of multiple quantiles provides a more complete picture of the distribution, we investigate quantile regression-based methods to improve efficiency by optimally combining information across quantiles. Under the working independence scenario, the asymptotic variance of the proposed estimator approaches the Cramér–Rao lower bound. In the presence of dependence among within-subject measurements, we adopt a prewhitening technique to transform regression errors into independent innovations and show that the prewhitened optimally weighted quantile average estimator asymptotically achieves the Cramér–Rao bound for the independent innovations. Fully data-driven bandwidth selection and optimal weights estimation are implemented through a two-step procedure. Monte Carlo studies show that the proposed method delivers more robust and superior overall performance than that of the existing methods.     

On two-step estimation for varying coefficient models

In this note we discuss two-step kernel estimation of varying coefficient regression models that have a common smoothing variable. The method allows one to use different bandwidths for different coefficient functions. We consider local polynomial fitting and present explicit formulas for the asymptotic biases and variances of the estimators.     

SiZer inference for generalized varying coefficient models

ABSTRACT

SiZer (significant zero crossings of derivatives) is an effective tool for exploring significant features in curves from the viewpoint of the scale space theory. In this paper, a SiZer approach is developed for generalized varying coefficient models (GVCMs) in order to achieve the task of understanding dynamic characteristics of the regression relationship at multiscales. The proposed SiZer method is based on the local-linear maximum likelihood estimation of GVCMs and the one-step estimation procedure is employed to alleviate the computational cost of estimating the coefficients and their derivatives at different scales. Simulation studies are performed to assess the performance of the SiZer inference and two real-world examples are given to demonstrate its applications.     

L 1-estimation for varying coefficient models

The varying coefficient model is a useful extension of linear models and has many advantages in practical use. To estimate the unknown functions in the model, the kernel type with local linear least-squares (L ₂) estimation methods has been proposed by several authors. When the data contain outliers or come from population with heavy-tailed distributions, L ₁-estimation should yield better estimators. In this article, we present the local linear L ₁-estimation method and derive the asymptotic distributions of the L ₁-estimators. The simulation results for two examples, with outliers and heavy-tailed distribution, respectively, show that the L ₁-estimators outperform the L ₂-estimators.     

Reducing component estimation for varying coefficient models

The estimation problem for varying coefficient models has been studied by many authors. We consider the problem in the case that the unknown functions admit different degrees of smoothness. In this paper we propose a reducing component local polynomial method to estimate the unknown functions. It is shown that all of our estimators achieve the optimal convergence rates. The asymptotic distributions of our estimators are also derived. The established asymptotic results and the simulation results show that our estimators outperform the the existing two-step estimators when the coefficient functions admit different degrees of smoothness. We also develop methods to speed up the estimation of the model and the selection of the bandwidths.     

Monitoring parameter changes for random coefficient autoregressive models

In this paper, we develop a monitoring procedure for an early detection of parameter changes in random coefficient autoregressive models. It is shown that the stopping rule signaling a parameter change satisfies the desired asymptotic property as seen in Lee, Lee, and Na (submitted for publication). Simulation results are provided for illustration.     

Selection of terms in random coefficient regression models

The selection of suitable terms in random coefficient regression models is a challenging problem to practitioners. Although many techniques, ranging from those with a theoretical flavour to those with an exploratory spirit, have been proposed for such purposes, no particular one may be considered as a paradigm. In fact, many authors advocate that they should be used in a complementary way. We consider exploratory methods based on fitting standard regression models to the individual response profiles or to the rows of the sample within-units covariance matrix (for balanced data) that may serve as additional tools in the process of selecting an appropriate model. We evaluate the performance of the proposal via a simulation study and consider applications to two examples in the field of Biostatistics.     

Estimation of semi-varying coefficient models with nonstationary regressors

We study a semivarying coefficient model where the regressors are generated by the multivariate unit root I(1) processes. The influence of the explanatory vectors on the response variable satisfies the semiparametric partially linear structure with the nonlinear component being functional coefficients. A semiparametric estimation methodology with the first-stage local polynomial smoothing is applied to estimate both the constant coefficients in the linear component and the functional coefficients in the nonlinear component. The asymptotic distribution theory for the proposed semiparametric estimators is established under some mild conditions, from which both the parametric and nonparametric estimators are shown to enjoy the well-known super-consistency property. Furthermore, a simulation study is conducted to investigate the finite sample performance of the developed methodology and results.     

Optimal designs in random coefficient cubic regression models

In this article we investigate the problem of ascertaining A- and D-optimal designs in a cubic regression model with random coefficients. Our interest lies in estimation of all the parameters or in only those except the intercept term. Assuming the variance ratios to be known, we tabulate D-optimal designs for various combinations of the variance ratios. A-optimality does not pose any new problem in the random coefficients situation.     

Small area estimation under random regression coefficient models

Statistical agencies are interested to report precise estimates of linear parameters from small areas. This goal can be achieved by using model-based inference. In this sense, random regression coefficient models provide a flexible way of modelling the relationship between the target and the auxiliary variables. Because of this, empirical best linear unbiased predictor (EBLUP) estimates based on these models are introduced. A closed-formula procedure to estimate the mean-squared error of the EBLUP estimators is also given and empirically studied. Results of several simulation studies are reported as well as an application to the estimation of household normalized net annual incomes in the Spanish Living Conditions Survey.