Statistical inference on asymptotic properties of two estimators for the partially linear single-index models

The outer product of gradients (OPG) estimation procedure based on least squares (LS) approach has been presented by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] to estimate the single-index parameter in partially linear single-index models (PLSIM). However, its asymptotic property has not been established yet and the efficiency of LS-based method can be significantly affected by outliers and heavy-tailed distributions. In this paper, we firstly derive the asymptotic property of OPG estimator developed by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] in theory, and a novel robust estimation procedure combining the ideas of OPG and local rank (LR) inference is further developed for PLSIM along with its theoretical property. Then, we theoretically derive the asymptotic relative efficiency (ARE) of the proposed LR-based procedure with respect to LS-based method, which is shown to possess an expression that is closely related to that of the signed-rank Wilcoxon test in comparison with the t-test. Moreover, we demonstrate that the new proposed estimator has a great efficiency gain across a wide spectrum of non-normal error distributions and almost not lose any efficiency for the normal error. Even in the worst case scenarios, the ARE owns a lower bound equalling to 0.864 for estimating the single-index parameter and a lower bound being 0.8896 for estimating the nonparametric function respectively, versus the LS-based estimators. Finally, some Monte Carlo simulations and a real data analysis are conducted to illustrate the finite sample performance of the estimators.     

On LSE in regression model for long-range dependent random fields on spheres

We study the asymptotic behaviour of least squares estimators (LSE) in regression models for long-range dependent random fields observed on spheres. The LSE can be given as a weighted functional of long-range dependent random fields. It is known that in this scenario the limits can be non-Gaussian. We derive the limit distribution and the corresponding rate of convergence for the estimators. The results were obtained under rather general assumptions on the random fields. Simulation studies were conducted to support theoretical findings.     

On the construction of optimal experimental designs: a penalty approach

Tn optimal experimental design theory there are well-known situations, in which additional constraints are implied to the design set. These constraints destroy in general the simplex structure of the set of feasible points of the design set. Thus the available iteration procedures for the unrestricted case are no longer applicable.

In this paper a penalty approach is suggested which transforms the restricted problem to the unrestricted case and allows the application of well-known algorithms such as the Fedorov-Wynn-type or the projected gradient procedure.     

Penalized Maximum Likelihood Principle for Choosing Ridge Parameter

We consider the problem of choosing the ridge parameter. Two penalized maximum likelihood (PML) criteria based on a distribution-free and a data-dependent penalty function are proposed. These PML criteria can be considered as “continuous” versions of AIC. A systematic simulation is conducted to compare the suggested criteria to several existing methods. The simulation results strongly support the use of our method. The method is also applied to two real data sets.     

Changepoint Estimation in a Segmented Linear Regression via Empirical Likelihood

For a segmented regression system with an unknown changepoint over two domains of a predictor, a new empirical likelihood ratio statistic is proposed to test the null hypothesis of no change. Under the null hypothesis of no change, the proposed test statistic is shown empirically to be Gumbel distributed with robust location and scale estimators against various parameter settings and error distributions. A power analysis is conducted to illustrate the performance of the test. Under the alternative hypothesis with a changepoint, the test statistic is utilized to estimate the changepoint between the two domains. A comparison of the frequency distributions between the proposed estimator and two parametric methods indicates that the proposed method is effective in capturing the true changepoint.     

New Prediction Interval and Band in the Nonlinear Regression Model: Application to Predictive Modeling in Foods

This article is concerned with the proposal of a new prediction interval and band for the nonlinear regression model. The construction principle of this interval and band is based on an exact (the meaning of the term “exact” will be given later) confidence region for parameters of the nonlinear regression model. This region, fully described in Vila and Gauchi (2007 Vila , J.-P. , Gauchi , J.-P. ( 2007 ). Optimal designs based on exact confidence regions for parameter estimation of a nonlinear regression model . Journal of Statistical Planning and Inference 137 ( 9 ): 2935 – 2953 .[Crossref], [Web of Science ®] , [Google Scholar]), provides a rigorous justification for the new prediction interval and band that we propose. This new band is then compared to the classical bands (which are asymptotic and thus approximate for small n), and also to the band based on the bootstrap resampling method. The comparison of these bands is undertaken with simulated and real data from predictive modeling in food science.     

An algorithm for the minimum sum of weighted absolute errors regression

We propose an algorithm to estimate the unknown constants in a multiple linear regression model under the minimum sum of weighted absolute errors (MSWAE). The proposed algorithm, a generalization of an earlier algorithm, is compared to a bounded variable algorithm. Some somputational experience is reported.     

Performing armchair roundoff analyses of statistical algorithms

Numerical stability is but one of many desirable properties which should be considered when designing statistical software. However, rigorous roundoff analysis is rarely done because it seems not worth the price; the influence of rounding error is usually of secondary importance, and the analysis is thought to be beyond the reach of all but a few specialists. This note discusses the role of roundoff analysis in the design of a statistical program and shows that new techniques sometimes make assessment of the effect of rounding errors no more difficult than the verification of other program properties.     

The Performance of Robust Two-Stage Estimator in Nonlinear Regression with Autocorrelated Error

Some statistics practitioners often ignore the underlying assumptions when analyzing a real data and employ the Nonlinear Least Squares (NLLS) method to estimate the parameters of a nonlinear model. In order to make reliable inferences about the parameters of a model, require that the underlying assumptions, especially the assumption that the errors are independent, are satisfied. However, in a real situation, we may encounter dependent error terms which prone to produce autocorrelated errors. A two-stage estimator (CTS) has been developed to remedy this problem. Nevertheless, it is now evident that the presence of outliers have an unduly effect on the least squares estimates. We expect that the CTS is also easily affected by outliers since it is based on the least squares estimator, which is not robust. In this article, we propose a Robust Two-Stage (RTS) procedure for the estimation of the nonlinear regression parameters in the situation where autocorrelated errors come together with the existence of outliers. The numerical example and simulation study signify that the RTS is more efficient than the NLLS and the CTS methods.     

An Efficient Estimation for Switching Regression Models: A Monte Carlo Study

This article investigates an efficient estimation method for a class of switching regressions based on the characteristic function (CF). We show that with the exponential weighting function, the CF-based estimator can be achieved from minimizing a closed form distance measure. Due to the availability of the analytical structure of the asymptotic covariance, an iterative estimation procedure is developed involving the minimization of a precision measure of the asymptotic covariance matrix. Numerical examples are illustrated via a set of Monte Carlo experiments examining the implementation, finite sample property and the efficiency of the proposed estimator.