Local linear regression with nonparametrically generated covariates for weakly dependent data

Double-smoothing for bias reduction in local linear regression

Local linear–additive estimation for multiple nonparametric regressions

In the case of the random design nonparametric regression, the double smoothing technique is applied to estimate the multivariate regression function. The proposed estimator has desirable properties in both the finite sample and the asymptotic cases. In the finite sample case, it has bounded conditional (and unconditional) bias and variance. On the other hand, in the asymptotic case, it has the same mean square error as the local linear estimator in Fan (Design-Adaptive Nonparametric Regression. Journal of the American Statistical Association 1992, 87, 998–1004; Local Linear Regression Smoothers and Their Minimax Efficiencies. Annals of Statistics 1993, 21, 196–216). Simulation studies demonstrate that the proposed estimator is better than the local linear estimator, because it has a smaller sample mean integrated square error and gives smoother estimates.

Nonparametric regression is a standard statistical tool with increased importance in the Big Data era. Boundary points pose additional difficulties but local polynomial regression can be used to alleviate them. Local linear regression, for example, is easy to implement and performs quite well both at interior and boundary points. Estimating the conditional distribution function and/or the quantile function at a given regressor point is immediate via standard kernel methods but problems ensue if local linear methods are to be used. In particular, the distribution function estimator is not guaranteed to be monotone increasing, and the quantile curves can “cross.” In the article at hand, a simple method of correcting the local linear distribution estimator for monotonicity is proposed, and its good performance is demonstrated via simulations and real data examples. Supplementary materials for this article are available online.

Asymptotic normality for a local composite quantile regression estimator of regression function with truncated data

The study focuses on the imputation for the longitudinal survey data which often has nonignorable nonrespondents. Local linear regression is used to impute the missing values and then the estimation of the time-dependent finite populations means. The asymptotic properties (unbiasedness and consistency) of the proposed estimator are investigated. Comparisons between different parametric and nonparametric estimators are performed based on the bootstrap standard deviation, mean square error and percentage relative bias. A simulation study is carried out to determine the best performing estimator of the time-dependent finite population means. The simulation results show that local linear regression estimator yields good properties.

This paper is about optimal estimation of the additive components of a nonparametric, additive isotone regression model. It is shown that asymp- totically up to first order, each additive component can be estimated as well as it could be by a least squares estimator if the other components were known. The algorithm for the calculation of the estimator uses backfitting. Conver- gence of the algorithm is shown. Finite sample properties are also compared through simulation experiments. In this paper we discuss nonparametric additive monotone regression models. We discuss a backfitting estimator that is based on iterative application of the pool adjacent violator algorithm to the additive components of the model. Our main result states the following oracle property. Asymptotically up to first order, each additive component is estimated as well as it would be (by a least squares estimator) if the other components were known. This goes beyond the classical finding that the estimator achieves the same rate of convergence, independently of the number of additive components. The result states that the asymptotic distribution of the estimator does not depend on the number of components. We have two motivations for considering this model. First of all we think that this is a useful model for some applications. For a discussion of isotonic additive regres- sion from a more applied point, see also Bacchetti (1), Morton-Jones et al. (32) and De Boer, Besten and Ter Braak (7). But our main motivation comes from statistical theory. We think that the study of nonparametric models with several nonparamet- ric components is not fully understood. The oracle property that is stated in this paper for additive isotone models has been shown for smoothing estimators in some other nonparametric models. This property is expected to hold if the estimation of the different nonparametric components is based on local smoothing where the lo- calization takes place in different scales. An example are additive models of smooth functions where each localization takes place with respect to another covariate. In Mammen, Linton and Nielsen (28) the oracle property has been verified for the lo- cal linear smooth backfitting estimator. As local linear estimators, also the isotonic least squares is a local smoother. The estimator is a local average of the response variable but in contrast to local linear estimators the local neighborhood is chosen � Research of this paper was supported by the Deutsche Forschungsgemeinschaft project MA

Pricing of American options in discrete time by means of regression-based Monte Carlo methods is considered in this thesis. Thereby, the options are allowed to be based on several underlyings (so-called basket options). It is assumed that the underlyings price processes satisfy the Markov property and consequently are Markov processes. In this dissertation, Monte Carlo methods are used to generate artificial sample paths of these price processes, and subsequently adaptive nonparametric least squares regression estimates are used to estimate the so-called continuation values from these data. The continuation values describe the mean values of the American options for given values of the underlyings at time t, subject to the constraint that the options are not exercised at time t, but optimally exercised in the future. Adaptive least squares neural networks as well as adaptive least squares spline estimates are used as nonparametric regression estimates. In conjunction with the pricing of American options, these estimates are theoretically analysed, and results with respect to consistency and rate of convergence are derived. Finally, the pricing of American options on simulated data by means of the estimates is illustrated.

Watershed delineation is a process to compute the drainage area for a point on the land surface, which is a critical step in hydrologic and water resources analysis. However, existing watershed delineation tools are still insufficient to support hydrologists and watershed researchers due to the lack of essential capabilities such as fully leveraging scalable and high performance computing infrastructure (public cloud), and providing predictable performance for the delineation tasks. To solve these problems, this paper reports on WDCloud, which is a system for large-scale watershed delineation on public cloud. For the design and implementation of WDCloud, we employ three main approaches: 1) an automated catchment search mechanism for a public data set, 2) three performance improvement strategies (Data-reuse, parallel-union, and MapReduce), and 3) local linear regression-based execution time estimator for watershed delineation. Moreover, WDCloud extensively utilizes several compute and storage capabilities from Amazon Web Services in order to maximize the performance, scalability, and elasticity of watershed delineation system. Our evaluations on WDCloud focus on two main aspects of WDCloud; the performance improvement for watershed delineation via three strategies and the estimation accuracy for watershed delineation time by local linear regression. The evaluation results show that WDCloud can achieve 18x–111x of speed-ups for delineating any scale of watersheds in the contiguous United States as compared to commodity laptop environments, and accurately predict execution time for watershed delineation with 85.6% of prediction accuracy, which is 23%–13% higher than other state-of-the-art approaches.

Semiparametric estimation for partially linear models with [formula omitted]-weak dependent errors

This paper provides a contribution to the growth empirics in sub-Saharan Africa with a focus on identifying the major determinants of long run economic growth among SSA countries. Being aware of the overwhelming dominance of parametric regression methodology in the extant literature and its associated numerous setbacks, we specifically employ the local linear kernel estimator which does not assume any functional form for the underlying growth model. At the end of the study, the findings suggest that there is a positive and nonlinear relationship between economic growth on one hand as well as investment in physical capital, population and democracy on the other hand. Again, while we find that human capital and inflation have no significant effect on economic growth over the study period, foreign aid was found to have negative effect on economic growth in SSA. The findings obtained in the paper have important implications for growth policy in SSA. Growth policies should thus consider population control, expanding and improving the quality of education and enrolment especially at the higher levels and strengthen democratic institutions. For research, the findings imply that researchers should be cautious in specifying the functional form of growth models when investigating the determinants of economic growth. Keywords: Bandwidth, economic growth, local linear kernel regression, nonparametric, sub-Saharan Africa DOI: 10.7176/JESD/12-16-02 Publication date: August 31 st 2021

Optimal bandwidths for local polynomial regression usually involve functionals of the derivatives of the unknown regression function. In the multivariate case, estimates of these functionals are not readily available, primarily because estimating multivariate derivatives is complicated. In this paper, an estimator of multivariate second derivative is obtained via local quadratic regression with cross terms left out. This estimator has the optimal rate of convergence but is simpler and uses a lot less computing time than the full local quadratic estimator. Using this as a pilot estimator, an estimator of the integrated squared Laplacian of a multivariate regression function is obtained which leads to a plug-in formula of the optimal bandwidth for multivariate local linear regression. This bandwidth has good theoretical properties as well as satisfactory performance in our simulation study. It is also recommended for variable selection methods.

In this paper, it is studied the asymptotic behavior of the nonparametric local linear estimation of the regression operator when the covariates are curves. Under some general conditions we give the exact expression involved in the leading terms of the quadratic error of this estimator. The obtained results affirm the superiority of the local linear modeling over the kernel method, in functional statistics framework.

We establish rates of convergences in statistical learning for time series forecasting. Using the PAC-Bayesian approach, slow rates of convergence √ d/n for the Gibbs estimator under the absolute loss were given in a previous work [7], where n is the sample size and d the dimension of the set of predictors. Under the same weak dependence conditions, we extend this result to any convex Lipschitz loss function. We also identify a condition on the parameter space that ensures similar rates for the classical penalized ERM procedure. We apply this method for quantile forecasting of the French GDP. Under additional conditions on the loss functions (satisfied by the quadratic loss function) and for uniformly mixing processes, we prove that the Gibbs estimator actually achieves fast rates of convergence d/n. We discuss the optimality of these different rates pointing out references to lower bounds when they are available. In particular, these results bring a generalization the results of [29] on sparse regression estimation to some autoregression.

We consider nonparametric estimation of the conditional qth quantile for stationary time series. We deal with stationary time series with strong time dependence and heavy tails under the setting of random design. We estimate the conditional qth quantile by local linear regression and investigate the asymptotic properties. It is shown that the asymptotic properties are affected by both the time dependence and the tail index of the errors. The results of a small simulation study are also given.