Backfitting Procedure Research Articles

Cure Rate Model With Mismeasured Covariates Under Transformation

Cure rate models explicitly account for the survival fraction in failure time data. When the covariates are measured with errors, naively treating mismeasured covariates as error-free would cause estimation bias and thus lead to incorrect inference. Under the proportional hazards cure model, we propose a corrected score approach as well as its generalization, and implement a transformation on the mismeasured covariates toward error additivity and/or normality. The corrected score equations can be easily solved through the backfitting procedure, and the biases in the parameter estimates are successfully eliminated. We show that the proposed estimators for the regression coefficients are consistent and asymptotically normal. We conduct simulation studies to examine the finite-sample properties of the new method and apply it to a real data set for illustration.

Exploring spatial nonlinearity using additive approximation

We propose to approximate the conditional expectation of a spatial random variable given its nearest-neighbour observations by an additive function. The setting is meaningful in practice and requires no unilateral ordering. It is capable of catching nonlinear features in spatial data and exploring local dependence structures. Our approach is different from both Markov field methods and disjunctive kriging. The asymptotic properties of the additive estimators have been established for $\alpha$-mixing spatial processes by extending the theory of the backfitting procedure to the spatial case. This facilitates the confidence intervals for the component functions, although the asymptotic biases have to be estimated via (wild) bootstrap. Simulation results are reported. Applications to real data illustrate that the improvement in describing the data over the auto-normal scheme is significant when nonlinearity or non-Gaussianity is pronounced.

Estimation of generalized partially linear models with measurement error using sufficiency scores

We study the partially linear model in logistic and other types of canonical exponential family regression when the explanatory variable is measured with independent normal error. We develop a backfitting estimation procedure to this model based upon the parametric idea of sufficiency scores so that no assumptions are made about the latent variable measured with error. We derive the method's asymptotic properties and present a numerical example and a simulation study.

Estimation of hedonic price functions via additive nonparametric regression

We model a hedonic price function for housing as an additive nonparametric regression. Estimation is done via a backfitting procedure in combination with a local polynomial estimator. It avoids the pitfalls of an unrestricted nonparametric estimator, such as slow convergence rates and the curse of dimensionality. Bandwidths are chosen using a novel plug in method that minimizes the asymptotic mean average squared error (AMASE) of the regression. We compare our results to alternative parametric models and find evidence of the superiority of our nonparametric model. From an empirical perspective our study is interesting in that the effects on housing prices of a series of environmental characteristics are modeled in the regression. We find these characteristics to be important in the determination of housing prices.

Smoothing Spline Nonlinear Nonparametric Regression Models

Almost all of the current nonparametric regression methods, such as smoothing splines, generalized additive models, and varyingcoefficients models, assume a linear relationship when nonparametric functions are regarded as parameters. In this article we propose a general class of smoothing spline nonlinear nonparametric models that allow nonparametric functions to act nonlinearly. They arise in many fields as either theoretical or empirical models. Our new estimation methods are based on an extension of the Gauss–Newton method to infinite-dimensional spaces and the backfitting procedure. We extend the generalized cross-validation and generalized maximum likelihood methods to estimate smoothing parameters. We establish connections between some nonlinear nonparametric models and nonlinear mixed-effects models. We derive approximate Bayesian confidence intervals for inference. We illustrate the methods with an application to term structure of interest rates and conduct simulations to evaluate the finite-sample performance of our methods.

Semi- and Nonparametric Modeling of Ordinal Data

Parametric models for categorical ordinal response variables, like the proportional odds model or the continuation ratio model, assume that the predictor is given by a linear form of covariates. In this article the parametric models are extended to include smooth components in a semiparametric or partially parametric fashion. Parts of the covariates are thereby modeled linearly while other covariates are modeled as unspecified but smooth functions. Estimation is based on a combination of local likelihood and profile likelihood and asymptotic properties of the estimates are derived. In a simulation study it is demonstrated that the profile likelihood approach is to be preferred over a backfitting procedure. Two data examples demonstrate the applicability of the models.

Integration and backfitting methods in additive models-finite sample properties and comparison

We examine and compare the finite sample performance of the competing back-fitting and integration methods for estimating additive nonparametric regression using simulated data. Although, the asymptotic properties of the integration estimator, and to some extent the backfitting, method too, are well understood, its small sample properties are not well investigated. Apart from some small experiments in the above cited papers, there is little hard evidence concerning the exact distribution of the estimates. It is our purpose to provide an extensive finite sample comparison between the backfitting procedure and the integration procedure using simulated data.

The existence and asymptotic properties of a backfitting projection algorithm under weak conditions

We derive the asymptotic distribution of a new backfitting procedure for estimating the closest additive approximation to a nonparametric regression function. The procedure employs a recent projection interpretation of popular kernel estimators provided by Mammen, Marron, Turlach and Wand and the asymptotic theory of our estimators is derived using the theory of additive projections reviewed in Bickel, Klaassen, Ritov and Wellner. Our procedure achieves the same bias and variance as the oracle estimator based on knowing the other components, and in this sense improves on the method analyzed in Opsomer and Ruppert. We provide ‘‘high level’’ conditions independent of the sampling scheme. We then verify that these conditions are satisfied in a regression and a time series autoregression under weak conditions.

Estimation of Derivatives for Additive Separable Models

Additive regression models have a long history in nonparametric regression. It is well known that these models can be estimated at the one dimensional rate. Until recently, however, these models have been estimated by a backfitting procedure. Although the procedure converges quickly, its iterative nature makes analyzing its statistical properties difficult. Recently, an integration approach has been studied that allows for the derivation of a closed form for the estimator. Although they seem to be competing procedures for the same problem, their interpretation is in fact different. For none of them the quite important question in economics of derivative estimation has been investigated so far. This paper extends the approach of marginal integration to the simultaneous estimation of both the function and its derivatives by combining the integration procedure with a local polynomial approach. Thus, we additionally get a design adaptive estimator. Finally the merits of this procedure with respect to the estimation of a production function subject to separability conditions are discussed. The procedure is applied to livestock production data from Wisconsin, showing performance and handling of these methods in practice. We demonstrate e.g., that there is some evidence of increasing returns to scale for larger farms.This work was first revised in 1995. The research was supported by the Deutsche Forschungsgemeinschaft, SFB 373. The first author, E.Severance-Lossing died in 1996; the mentioned address refers to the second author.

Miscellanea. Efficient estimation of additive nonparametric regression models

SUMMARY We define a new procedure for estimating additive nonparametric regression models. We use the integration method of Linton & Nielsen (1995) to provide starting values that are then used in a one-step backfitting procedure. We show that our new method is efficient in a certain sense and dominates the straight integration method according to mean squared error.

The seismic analysis of sandy sites

This paper presents an approach for performing one dimensional effective stress site response analyses for sandy sites, including the evaluation of liquefaction potential. This type of analysis differs from the more common total stress response analyses in that induced pore pressures in saturated sandy soils are accounted for, including the resulting influence on soil properties. This analytical method has been refined to the point where the need for complex and expensive laboratory soil testing is no longer required, a factor which has traditionally held back developments in the effective stress area. The effective stress analysis requires the determination of five soil specific parameters. A trial and error backfitting procedure was developed to successfully determine these parameters from traditional site investigation data rather than detailed laboratory testing. This procedure was investigated using two case studies, the Edgecumbe earthquake of 1987 and the Loma Prieta event of 1989, which both exhibited significant liquefaction damage. The Edgecumbe analysis produced useful results. The predicted ground acceleration required to initiate liquefaction was 2.8 m/s2 (0.29g) which is close to the estimated value of 3 m/s2 (0.31g). This was a good result as a reasonable amount of estimated and correlated data had to be used due to a lack of specific site data. The case study of Treasure Island, in the San Francisco Bay area, also produced encouraging results with both the prediction of liquefaction and surface response spectra in good agreement with recorded data. Both case studies used liquefaction resistance curves determined empirically from SPT blow count data. While this data proved acceptable it was discovered that care must be taken in the use of such overseas derived empirical data, particularly if no corroborating site specific information is available.