Unknown Coefficient Functions Research Articles

Quantile Regression for Longitudinal Functional Data with Application to Feed Intake of Lactating Sows

AbstractThis article focuses on the study of lactating sows, where the main interest is the influence of temperature, measured throughout the day, on the lower quantiles of the daily feed intake. We outline a model framework and estimation methodology for quantile regression in scenarios with longitudinal data and functional covariates. The quantile regression model uses a time-varying regression coefficient function to quantify the association between covariates and the quantile level of interest, and it includes subject-specific intercepts to incorporate within-subject dependence. Estimation relies on spline representations of the unknown coefficient functions and can be carried out with existing software. We introduce bootstrap procedures for bias adjustment and computation of standard errors. Analysis of the lactation data indicates, among others, that the influence of temperature increases during the lactation period.Supplementary materials accompanying this paper appear on-line.

Generalized Structural Equation Model with Survival Outcomes and Time-Varying Coefficients

The conventional Cox proportional hazards (PH) model typically assumes fully observed predictors and constant regression coefficients. However, some predictors are latent variables, each of which must be characterized by multiple observed indicators from various perspectives. Moreover, the predictor effects may vary with time in practice. Accommodating such latent variables and identifying temporal covariate effects are frequently of primary interest. This study proposes a generalized structural equation model to investigate the temporal effects of observed and latent risk factors on the hazards of interest. The proposed model comprises a confirmatory factor analysis model as the measurement equation and a varying-coefficient PH model with observed and latent predictors as the structural equation. A hybrid procedure that combines the expectation-maximization (EM) algorithm and the corrected estimating equation approach is developed to estimate unknown parameters and coefficient functions. Simulation studies demonstrate the satisfactory performance of the proposed method. An application to a health survey study reveals insights into risk factors for elders’ life expectancy.

Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming

Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse problem, for which unique reconstructability results, stability estimates and global convergence of numerical methods are very hard to achieve. The aim of this note is to point out a new connection between inverse coefficient problems and semidefinite programming that may help addressing these challenges. We show that an inverse elliptic Robin transmission problem with finitely many measurements can be equivalently rewritten as a uniquely solvable convex non-linear semidefinite optimization problem. This allows to explicitly estimate the number of measurements that is required to achieve a desired resolution, to derive an error estimate for noisy data, and to overcome the problem of local minima that usually appears in optimization-based approaches for inverse coefficient problems.

A Dynamic Interaction Semiparametric Function-on-Scalar Model

Motivated by recent work studying massive functional data, such as the COVID-19 data, we propose a new dynamic interaction semiparametric function-on-scalar (DISeF) model. The proposed model is useful to explore the dynamic interaction among a set of covariates and their effects on the functional response. The proposed model includes many important models investigated recently as special cases. By tensor product B-spline approximating the unknown bivariate coefficient functions, a three-step efficient estimation procedure is developed to iteratively estimate bivariate varying-coefficient functions, the vector of index parameters, and the covariance functions of random effects. We also establish the asymptotic properties of the estimators including the convergence rate and their asymptotic distributions. In addition, we develop a test statistic to check whether the dynamic interaction varies with time/spatial locations, and we prove the asymptotic normality of the test statistic. The finite sample performance of our proposed method and of the test statistic are investigated with several simulation studies. Our proposed DISeF model is also used to analyze the COVID-19 data and the ADNI data. In both applications, hypothesis testing shows that the bivariate varying-coefficient functions significantly vary with the index and the time/spatial locations. For instance, we find that the interaction effect of the population aging and the socio-economic covariates, such as the number of hospital beds, physicians, nurses per 1000 people and GDP per capita, on the COVID-19 mortality rate varies in different periods of the COVID-19 pandemic. The healthcare infrastructure index related to the COVID-19 mortality rate is also obtained for 141 countries estimated based on the proposed DISeF model.

Functional‐coefficient regression models with GARCH errors

AbstractThe GARCH models are widely used to model various financial data with nonlinearity and heteroscedasticity structures. In this article, we propose a functional‐coefficient regression model with GARCH(r, s) errors to model these kinds of data. To deal with the effect of heteroscedasticity, we introduce a two‐step approach to estimating the unknown coefficient functions and the volatility, which results in unweighted and weighted local linear estimators. Asymptotic properties of the proposed estimators are established. Our results demonstrate that the weighted estimator is more efficient than the unweighted one, and the functional coefficients can be estimated by the weighted estimator as if the volatility was known. Both simulations and real data examples support our theoretical results. In particular, when there are GARCH effects, our two‐step estimator mimics the oracle estimator, with the true volatility being known in advance.

Estimation for Varying Coefficient Models with Hierarchical Structure

The varying coefficient (VC) model is a generalization of ordinary linear model, which can not only retain strong interpretability but also has the flexibility of the nonparametric model. In this paper, we investigate a VC model with hierarchical structure. A unified variable selection method for VC model is proposed, which can simultaneously select the nonzero effects and estimate the unknown coefficient functions. Meanwhile, the selected model enforces the hierarchical structure, that is, interaction terms can be selected into the model only if the corresponding main effects are in the model. The kernel method is employed to estimate the varying coefficient functions, and a combined overlapped group Lasso regularization is introduced to implement variable selection to keep the hierarchical structure. It is proved that the proposed penalty estimators have oracle properties, that is, the coefficients are estimated as well as if the true model were known in advance. Simulation studies and a real data analysis are carried out to examine the performance of the proposed method in finite sample case.

Network-Based Clustering for Varying Coefficient Panel Data Models

In this article, we introduce a novel varying-coefficient panel-data model with locally stationary regressors and unknown group structure, in which the number of groups and the group membership are left unspecified. We develop a triple-localization approach to estimate the unknown subject-specific coefficient functions and then identify the latent group structure via community detection. To improve the efficiency of the first-stage estimator, we further propose a two-stage estimation method that enables the estimator to achieve optimal rates of convergence. In the theoretical part of the article, we derive the asymptotic theory of the resultant estimators. In the empirical part, we present several simulated examples together with an analysis of real data to illustrate the finite-sample performance of the proposed method.

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.

Inverse problem for higher order ultraparabolic equation with unknown minor coefficient and right-hand side function

Inverse problem for higher order ultraparabolic equation with unknown minor coefficient and right-hand side function

Convergence Rates for Penalized Least Squares Estimators in PDE Constrained Regression Problems

We consider PDE constrained nonparametric regression problems in which the parameter $f$ is the unknown coefficient function of a second order elliptic partial differential operator $L_f$, and the ...

Time-Varying Panel Data Models with an Additive Factor Structure

Motivated by many key features of real data from economics and finance, we study a semiparametric panel data model with time-varying regression coefficients associated with an additive factor structure. In our model, factor loadings are unknown functions of observable variables which can capture time-variant and heterogeneous covariate information. A profile marginal integration (PMI) method is proposed to estimate unknown coefficient functions, factors and their loadings jointly in a single step, which can result in estimators with closed forms. Asymptotic distributions for the proposed profile estimators are established. Two empirical applications on US stock returns and OECD health care expenditure are provided. Thorough numerical results demonstrate the finite sample performance of our estimation and its advantage over traditional models in the relevant literature.

Composite Quantile Regression for Varying Coefficient Models with Response Data Missing at Random

Composite quantile regression (CQR) estimation and inference are studied for varying coefficient models with response data missing at random. Three estimators including the weighted local linear CQR (WLLCQR) estimator, the nonparametric WLLCQR (NWLLCQR) estimator, and the imputed WLLCQR (IWLLCQR) estimator are proposed for unknown coefficient functions. Under some mild conditions, the proposed estimators are asymptotic normal. Simulation studies demonstrate that the unknown coefficient estimators with IWLLCQR are superior to the other two with WLLCQR and NWLLCQR. Moreover, bootstrap test procedures based on the IWLLCQR fittings is developed to test whether the coefficient functions are actually varying. Finally, a type of investigated real-life data is analyzed to illustrated the applications of the proposed method.

Estimation for varying coefficient panel data model with cross-sectional dependence

This paper describes a method for estimation and inference with a nonparametric varying coefficients panel data model that allows for cross-sectional dependence and heteroscedasticity, wherein the time series length T is larger than the cross-sectional size N. We first eliminate fixed effects by taking the cross-sectional average, and then use a local linear approach to obtain the initial estimator of the unknown coefficient functions. However, the initial estimator ignores the cross-sectional dependence and heteroscedasticity, which will lead to a loss of efficiency. Thus, we propose a weighted local linear method to obtain a more efficient estimator. In the theoretical part of the paper, we derive the asymptotic theory of the resulting estimator. Simulation results and a real data analysis are provided to illustrate the finite sample performance of the proposed method.

Local composite partial likelihood estimation for length-biased and right-censored data

ABSTRACTLength-biased data, which are often encountered in engineering, economics and epidemiology studies, are generally subject to right censoring caused by the research ending or the follow-up loss. The structure of length-biased data is distinct from conventional survival data, since the independent censoring assumption is often violated due to the biased sampling. In this paper, a proportional hazard model with varying coefficients is considered for the length-biased and right-censored data. A local composite likelihood procedure is put forward for the estimation of unknown coefficient functions in the model, and large sample properties of the proposed estimators are also obtained. Additionally, an extensive simulation studies are conducted to assess the finite sample performance of the proposed method and a data set from the Academy Awards is analyzed.

Event-triggered optimal consensus tracking control for multi-agent systems with unknown internal states and disturbances

In this paper, we propose an event-triggered consensus tracking control for unknown nonlinear multi-agent systems (NMASs) with disturbances by using adaptive dynamic programming (ADP) technique. Considering the disturbances as control inputs, the optimal consensus tracking control problem can be transformed into multi-player zero-sum differential games, where the controllers are designed to minimize the cost functions under the worst-case disturbances. To recover the unknown internal states and control coefficient functions, neural network (NN)-based observers are established. Then, NN-based critics are applied to approximate the value functions and help calculate the optimal control policies and disturbance policies. In order to save the computation resource and reduce the transmission load, the designed observers and controllers are updated only when the designed events are triggered. Stability of the proposed method is demonstrated by Lyapunov analysis for both the continuous and jump dynamics. Meanwhile, we can obtain that the weight estimated errors of the observer NNs, the critic NNs and the local neighbor consensus tracking errors are uniformly ultimately bounded (UUB). Finally, a simulation example is given to illustrate the effectiveness of the proposed approach.

Statistical inference for varying-coefficient partially linear errors-in-variables models with missing data

The purpose of this paper is twofold. First, we investigate estimations in varying-coefficient partially linear errors-in-variables models with covariates missing at random. However, the estimators are often biased due to the existence of measurement errors, the bias-corrected profile least-squares estimator and local liner estimators for unknown parametric and coefficient functions are obtained based on inverse probability weighted method. The asymptotic properties of the proposed estimators both for the parameter and nonparametric parts are established. Second, we study asymptotic distributions of an empirical log-likelihood ratio statistic and maximum empirical likelihood estimator for the unknown parameter. Based on this, more accurate confidence regions of the unknown parameter can be constructed. The methods are examined through simulation studies and illustrated by a real data analysis.

Risk-adjusted monitoring of surgical performance.

We propose a nonparametric risk-adjusted cumulative sum chart to monitor surgical outcomes for patients with different risks of post-operative mortality due to risk factors that exist before the surgery. Using varying-coefficient logistic regression models, we accomplish the risk adjustment. Unknown coefficient functions are estimated by global polynomial spline approximation based on the maximum likelihood principle. We suggest a bisection minimization approach and a bootstrap method to determine the chart testing limit value. Compared with the previous (parametric) risk-adjusted cumulative sum chart, a major advantage of our method is that the morality rate can be modeled more flexibly by related covariates, which significantly enhances the monitoring efficiency. Simulations demonstrate nice performance of our proposed procedure. An application to a UK cardiac surgery dataset illustrates the use of our methodology.

Tail index varying coefficient model

ABSTRACTThis paper deals with a new class of tail index varying coefficient models with the random covariate under Pareto-type distributions. To estimate the unknown coefficient functions, we develop an estimation procedure via a local polynomial maximum likelihood techniques. The asymptotic normality of the estimated coefficient functions under some mild regularity conditions are established. Two numerical examples and one application are used to illustrate the performance of the proposed procedure.

Functional coefficient time series models with trending regressors

ABSTRACTThis paper studies a functional coefficient time series model with trending regressors, where the coefficients are unknown functions of time and random variables. We propose a local linear estimation method to estimate the unknown coefficient functions, and establish the corresponding asymptotic theory under mild conditions. We also develop a test procedure to see if the functional coefficients take particular parametric forms. For practical use, we further propose a Bayesian approach to select the bandwidths, and conduct several numerical experiments to examine the finite sample performance of our proposed local linear estimator and the test procedure. The results show that the local linear estimator works well and the proposed test has satisfactory size and power. In addition, our simulation studies show that the Bayesian bandwidth selection method performs better than the cross-validation method. Furthermore, we use the functional coefficient model to study the relationship between consumption per capita and income per capita in United States, and it was shown that the functional coefficient model with our proposed local linear estimator and Bayesian bandwidth selection method performs well in both in-sample fitting and out-of-sample forecasting.

Solving inverse coefficient problems of non-uniform fractionally diffusive reactive material by a boundary functional method

The inverse coefficient problems for estimating the unknown space-time dependent conductivity function and reaction coefficient function of a time-fractional diffusion reaction equation for non-uniform material are solved in the paper, without needing of initial condition, final time condition and internal measurement. We tackle these inverse coefficient problems supplementing with boundary data. After a homogenization technique by using these boundary data, a sequence of boundary functions are derived, which together with the zero element constitute a linear space. As an approximation, a boundary functional is proved in the linear space, of which the time-dependent energy is preserved for the energetic boundary function at each time step. The linear systems and iterative algorithms used to recover the unknown conductivity function and reaction coefficient with energetic boundary functions as bases are developed, which are convergent fast at each time step. The data required are merely the boundary temperatures and fluxes, and the boundary values and slopes of unknown functions to be recovered. The accuracy and robustness of present methods are confirmed by comparing the estimated results under large noise with exact solutions.