Weighted Wilcoxon Estimators in Nonlinear Regression

In this paper, we consider the estimation of parameters of a general near regression model. An estimator that minimises the weighted Wilcoxon dispersion function is considered and its asymptotic properties established under mild regularity conditions similar to those used in least squares and least absolute deviations estimation. As in linear models, the procedure provides estimators that are robust and highly efficient. The estimates depend on the choice of a weight function and diagnostics which differentiate between nonlinear fits are provided along with appropriate benchmarks. The behavior of these estimates is discussed on a real data set. A simulation study verifies the robustness, efficiency and validity of these estimates over several error distributions including the normal and a family of contaminated normal distributions.     

Composite Estimation for Single‐Index Models with Responses Subject to Detection Limits

We propose a semiparametric estimator for single‐index models with censored responses due to detection limits. In the presence of left censoring, the mean function cannot be identified without any parametric distributional assumptions, but the quantile function is still identifiable at upper quantile levels. To avoid parametric distributional assumption, we propose to fit censored quantile regression and combine information across quantile levels to estimate the unknown smooth link function and the index parameter. Under some regularity conditions, we show that the estimated link function achieves the non‐parametric optimal convergence rate, and the estimated index parameter is asymptotically normal. The simulation study shows that the proposed estimator is competitive with the omniscient least squares estimator based on the latent uncensored responses for data with normal errors but much more efficient for heavy‐tailed data under light and moderate censoring. The practical value of the proposed method is demonstrated through the analysis of a human immunodeficiency virus antibody data set.     

Partially adaptive estimation of nonlinear models via a normal mixture

This paper extends the partially adaptive method Phillips (1994) provided for linear models to nonlinear models. Asymptotic results are established under conditions general enough they cover both cross-sectional and time series applications. The sampling efficiency of the new estimator is illustrated in a small Monte Carlo study in which the parameters of an autoregressive moving average are estimated. The study indicates that, for non-normal distributions, the new estimator improves on the nonlinear least squares estimator in terms of efficiency.     

Partially adaptive estimation of nonlinear models via a normal mixture

This paper extends the partially adaptive method Phillips (1994) provided for linear models to nonlinear models. Asymptotic results are established under conditions general enough they cover both cross-sectional and time series applications. The sampling efficiency of the new estimator is illustrated in a small Monte Carlo study in which the parameters of an autoregressive moving average are estimated. The study indicates that, for non-normal distributions, the new estimator improves on the nonlinear least squares estimator in terms of efficiency.     

Least squares variogram fitting by spatial subsampling

Summary. Least squares methods are popular for fitting valid variogram models to spatial data. The paper proposes a new least squares method based on spatial subsampling for variogram model fitting. We show that the method proposed is statistically efficient among a class of least squares methods, including the generalized least squares method. Further, it is computationally much simpler than the generalized least squares method. The method produces valid variogram estimators under very mild regularity conditions on the underlying random field and may be applied with different choices of the generic variogram estimator without analytical calculation. An extension of the method proposed to a class of spatial regression models is illustrated with a real data example. Results from a simulation study on finite sample properties of the method are also reported.     

On weak consistency in linear models with equi-correlated random errors

A new estimator in linear models with equi-correlated random errors is postulated. Consistency properties of the proposed estimator and the ordinary least squares estimator are studied. It is shown that the new estimator has smaller variance than the usual least squares estimator under some mild conditions. In addition, it is observed that the new estimator tends to be weakly consistent in many cases where the usual least squares estimator is not.     

Multivariate regression models for discrete and continuous repeated measurements

A general class of multivariate regression models is considered for repeated measurements with discrete and continuous outcome variables. The proposed model is based on the seemingly unrelated regression model (Zellner, 1962) and an extension of the model of Park and Woolson(1992). The regression parameters of the model are consistently estimated using the two-stage least squares method. When the out come variables are multivariate normal, the two-stage estimator reduces to Zellner’s two-stage estimator. As a special case, we consider the marginal distribution described by Liang and Zeger (1986). Under this this distributional assumption, we show that the two-stage estimator has similar asymptotic properties and comparable small sample properties to Liang and Zeger's estimator. Since the proposed approach is based on the least squares method, however, any distributional assumption is not required for variables outcome variables. As a result, the proposed estimator is more robust to the marginal distribution of outcomes.     

Two Kinds of Restricted Estimators in Singular Linear Regression

In this article, the parameter estimators in singular linear model with linear equality restrictions are considered. The restricted root estimator and the generalized restricted root estimator are proposed and some properties of the estimators are also studied. Furthermore, we compare them with the restricted unified least squares estimator and show their sufficient conditions under which their superior over the restricted unified least squares estimator in terms of mean squares error, and discuss the choice of the unknown parameters of the generalized restricted root estimator.     

Semiparametric Sieve-Type Generalized Least Squares Inference

This article considers the problem of statistical inference in linear regression models with dependent errors. A sieve-type generalized least squares (GLS) procedure is proposed based on an autoregressive approximation to the generating mechanism of the errors. The asymptotic properties of the sieve-type GLS estimator are established under general conditions, including mixingale-type conditions as well as conditions which allow for long-range dependence in the stochastic regressors and/or the errors. A Monte Carlo study examines the finite-sample properties of the method for testing regression hypotheses.     

ON ESTIMATION OF LONG-MEMORY TIME SERIES MODELS

This paper discusses estimation associated with the long-memory time series models proposed by Granger & Joyeux (1980) and Hosking (1981). We consider the maximum likelihood estimator and the least squares estimator. Certain regularity conditions introduced by several authors to develop the asymptotic theory of these estimators do not hold in this model. However we can show that these estimators are strongly consistent, and we derive the limiting distribution and the rate of convergence.     

A unified approach to estimation of nonlinear mixed effects and Berkson measurement error models

Mixed effects models and Berkson measurement error models are widely used. They share features which the author uses to develop a unified estimation framework. He deals with models in which the random effects (or measurement errors) have a general parametric distribution, whereas the random regression coefficients (or unobserved predictor variables) and error terms have nonparametric distributions. He proposes a second-order least squares estimator and a simulation-based estimator based on the first two moments of the conditional response variable given the observed covariates. He shows that both estimators are consistent and asymptotically normally distributed under fairly general conditions. The author also reports Monte Carlo simulation studies showing that the proposed estimators perform satisfactorily for relatively small sample sizes. Compared to the likelihood approach, the proposed methods are computationally feasible and do not rely on the normality assumption for random effects or other variables in the model.     

Robust estimation for zero-inflated poisson autoregressive models based on density power divergence

In this study, we consider a robust estimation for zero-inflated Poisson autoregressive models using the minimum density power divergence estimator designed by Basu et al. [Robust and efficient estimation by minimising a density power divergence. Biometrika. 1998;85:549–559]. We show that under some regularity conditions, the proposed estimator is strongly consistent and asymptotically normal. The performance of the estimator is evaluated through Monte Carlo simulations. A real data analysis using New South Wales crime data is also provided for illustration.     

A consistent simulation-based estimator in generalized linear mixed models

We propose a strongly root-n consistent simulation-based estimator for the generalized linear mixed models. This estimator is constructed based on the first two marginal moments of the response variables, and it allows the random effects to have any parametric distribution (not necessarily normal). Consistency and asymptotic normality for the proposed estimator are derived under fairly general regularity conditions. We also demonstrate that this estimator has a bounded influence function and that it is robust against data outliers. A bias correction technique is proposed to reduce the finite sample bias in the estimation of variance components. The methodology is illustrated through an application to the famed seizure count data and some simulation studies.     

On the equality of usual and amemiya's partially generalized least squares estimator

Equivalent conditions are derived for the equality of GLSE (generalized least squares estimator) and partially GLSE (PGLSE), the latter introduced by Amemiya (1983). By adopting a more general approach the ordinary least squares estimator (OLSE) can shown to be a special PGLSE. Furthcrmore, linearly restricted estimators proposed by Balestra (1983) are investigated in this context. To facilitate the comparison of estimators extensive use of oblique and orthogonal projectors is made.     

Superiority of the r–k Class Estimator Over Some Estimators In A Linear Model

In regression analysis, to overcome the problem of multicollinearity, the r ? k class estimator is proposed as an alternative to the ordinary least squares estimator which is a general estimator including the ordinary ridge regression estimator, the principal components regression estimator and the ordinary least squares estimator. In this article, we derive the necessary and sufficient conditions for the superiority of the r ? k class estimator over each of these estimators under the Mahalanobis loss function by the average loss criterion. Then, we compare these estimators with each other using the same criterion. Also, we suggest to test to verify if these conditions are indeed satisfied. Finally, a numerical example and a Monte Carlo simulation are done to illustrate the theoretical results.     

SHRINKAGE, PRETEST AND ABSOLUTE PENALTY ESTIMATORS IN PARTIALLY LINEAR MODELS

We consider a partially linear model in which the vector of coefficients β in the linear part can be partitioned as ( β ₁, β ₂) , where β ₁ is the coefficient vector for main effects (e.g. treatment effect, genetic effects) and β ₂ is a vector for ‘nuisance’ effects (e.g. age, laboratory). In this situation, inference about β ₁ may benefit from moving the least squares estimate for the full model in the direction of the least squares estimate without the nuisance variables (Steinian shrinkage), or from dropping the nuisance variables if there is evidence that they do not provide useful information (pretesting). We investigate the asymptotic properties of Stein‐type and pretest semiparametric estimators under quadratic loss and show that, under general conditions, a Stein‐type semiparametric estimator improves on the full model conventional semiparametric least squares estimator. The relative performance of the estimators is examined using asymptotic analysis of quadratic risk functions and it is found that the Stein‐type estimator outperforms the full model estimator uniformly. By contrast, the pretest estimator dominates the least squares estimator only in a small part of the parameter space, which is consistent with the theory. We also consider an absolute penalty‐type estimator for partially linear models and give a Monte Carlo simulation comparison of shrinkage, pretest and the absolute penalty‐type estimators. The comparison shows that the shrinkage method performs better than the absolute penalty‐type estimation method when the dimension of the β ₂ parameter space is large.     

Kernel Estimators for Additive Models with Dependent Observations from Random Fields

Nonparametric estimation of the regression function for additive models is investigated in cases where the observed data are dependent. An additive kernel estimator for the regression function under some general mixing conditions is proposed. Under the mixing conditions, the additive kernel estimator is shown to be asymptotically normal.     

Sieve Empirical Likelihood and Extensions of the Generalized Least Squares

The empirical likelihood cannot be used directly sometimes when an infinite dimensional parameter of interest is involved. To overcome this difficulty, the sieve empirical likelihoods are introduced in this paper. Based on the sieve empirical likelihoods, a unified procedure is developed for estimation of constrained parametric or non-parametric regression models with unspecified error distributions. It shows some interesting connections with certain extensions of the generalized least squares approach. A general asymptotic theory is provided. In the parametric regression setting it is shown that under certain regularity conditions the proposed estimators are asymptotically efficient even if the restriction functions are discontinuous. In the non-parametric regression setting the convergence rate of the maximum estimator based on the sieve empirical likelihood is given. In both settings, it is shown that the estimator is adaptive for the inhomogeneity of conditional error distributions with respect to predictor, especially for heteroscedasticity.     

A robust and efficient estimation method for nonparametric models with jump points

Nonparametric models with jump points have been considered by many researchers. However, most existing methods based on least squares or likelihood are sensitive when there are outliers or the error distribution is heavy tailed. In this article, a local piecewise-modal method is proposed to estimate the regression function with jump points in nonparametric models, and a piecewise-modal EM algorithm is introduced to estimate the proposed estimator. Under some regular conditions, the large-sample theory is established for the proposed estimators. Several simulations are presented to evaluate the performances of the proposed method, which shows that the proposed estimator is more efficient than the local piecewise-polynomial regression estimator in the presence of outliers or heavy tail error distribution. What is more, the proposed procedure is asymptotically equivalent to the local piecewise-polynomial regression estimator under the assumption that the error distribution is a Gaussian distribution. The proposed method is further illustrated via the sea-level pressures.     

Instrumental variable estimation in nonlinear measurement error models

This paper presents an extension of instrumental variable estimation to nonlinear regression models. For the linear model, the extended estimator is equivalent to the two-stage least squares estimator. The extended estimator is consistent for an important class of nonlinear models, including the logistic model, under relatively weak assumptions on the distribution of the measurement error. An example and simulation study are presented for the logistic regression model. The simulations suggest the estimator is reasonably efficient.