Change points in nonparametric regresion functions

Estimators of location and size of jumps or discontinuities in a regression function and/or its derivatives are proposed. The estimators are based on the analysis of residuals obtained from the locally weighted least squares regression. The proposed estimators adapt to both fixed and random designs. The asymptotic properties of the estimators are investigated. The method is illustrated through simulation studies.     

Estimation of rational transfer function models

In the estimation of rational transfer function models, it has been recommended that starting values of a transfer function component be assumed to be zero (or a constant) in the recursive computation of the transfer function response. It is demonstrated that such algorithms may lead to serious bias in the estimation of moving average parameters. This paper discusses several other algorithms that may rectify this problem. It is found that the starting-value-free (SVF) method is a more reliable algorithm. For computer programs using the traditional algorithm, i.e., the zero-starting-value (ZSV) method, the bias problem can be easily remedied using a short-cut method that omits appropriate number of values at the beginning of the residual series.     

The Non Dominated Set in Bayesian Decision Problems with Convex Loss Functions

Bayesian decision problems require subjective elicitation of the inputs: beliefs and preferences. Sometimes, elicitation methods may not represent perfectly the judgements of the decision maker. Several foundations propose to overlay this problem using robust approaches. In these models, beliefs are modelled by a class of probability distributions and preferences by a class of loss functions. Then, we are in the conditions of a Pareto order. Hence the solution concept is the set of non dominated alternatives. In this article we focus on the computation of the efficient set when the preferences are modeled by a class of convex loss functions.     

Small-Sample Quantile Estimators in a Large Nonparametric Model

The large nonparametric model in this note is a statistical model with the family ? of all continuous and strictly increasing distribution functions. In the abundant literature of the subject, there are many proposals for nonparametric estimators that are applicable in the model. Typically the kth order statistic X _k:n is taken as a simplest estimator, with k = [nq], or k = [(n + 1)q], or k = [nq] + 1, etc. Often a linear combination of two consecutive order statistics is considered. In more sophisticated constructions, different L-statistics (e.g., Harrel–Davis, Kaigh–Lachenbruch, Bernstein, kernel estimators) are proposed. Asymptotically the estimators do not differ substantially, but if the sample size n is fixed, which is the case of our concern, differences may be serious. A unified treatment of quantile estimators in the large, nonparametric statistical model is developed.     

Shrinkage Estimators of Intercept Parameters of Two Simple Regression Models with Suspected Equal Slopes

Estimators of the intercept parameter of a simple linear regression model involves the slope estimator. In this article, we consider the estimation of the intercept parameters of two linear regression models with normal errors, when it is a priori suspected that the two regression lines are parallel, but in doubt. We also introduce a coefficient of distrust as a measure of degree of lack of trust on the uncertain prior information regarding the equality of two slopes. Three different estimators of the intercept parameters are defined by using the sample data, the non sample uncertain prior information, an appropriate test statistic, and the coefficient of distrust. The relative performances of the unrestricted, shrinkage restricted and shrinkage preliminary test estimators are investigated based on the analyses of the bias and risk functions under quadratic loss. If the prior information is precise and the coefficient of distrust is small, the shrinkage preliminary test estimator overperforms the other estimators. An example based on a medical study is used to illustrate the method.     

On Asymptotic Normality of the Local Polynomial Regression Estimator with Stochastic Bandwidths

Nonparametric density and regression estimators commonly depend on a bandwidth. The asymptotic properties of these estimators have been widely studied when bandwidths are non stochastic. In practice, however, in order to improve finite sample performance of these estimators, bandwidths are selected by data driven methods, such as cross-validation or plug-in procedures. As a result, nonparametric estimators are usually constructed using stochastic bandwidths. In this article, we establish the asymptotic equivalence in probability of local polynomial regression estimators under stochastic and nonstochastic bandwidths. Our result extends previous work by Boente and Fraiman (1995 Boente , G. , Fraiman , R. ( 1995 ). Asymptotic distribution of data-driven smoothers in density and regression estimation under dependence . Can. J. Statist. 23 : 383 – 397 .[Crossref], [Web of Science ®] , [Google Scholar]) and Ziegler (2004 Ziegler , K. ( 2004 ). Adaptive kernel estimation of the mode in nonparametric random design regression model . Probab. Mathemat. Statist. 24 : 213 – 235 . [Google Scholar]).     

Consistency and Normality of M-Estimators in Partly Linear Models with Stochastic Adapted Errors

The robust M-estimators for the partly linear model under stochastic adapted errors are considered. It is shown that the M-estimator of parameter is asymptotically normal and the M-estimator of the nonparametric function achieves the optimal rate of convergence for nonparametric regression. Some known results are improved and generalized. Some simulations and a real data example are conducted to illustrate the proposed method.     

Shrinkage Estimation Under Multivariate Elliptic Models

The estimation of the location vector of a p-variate elliptically contoured distribution (ECD) is considered using independent random samples from two multivariate elliptically contoured populations when it is apriori suspected that the location vectors of the two populations are equal. For the setting where the covariance structure of the populations is the same, we define the maximum likelihood, Stein-type shrinkage and positive-rule shrinkage estimators. The exact expressions for the bias and quadratic risk functions of the estimators are derived. The comparison of the quadratic risk functions reveals the dominance of the Stein-type estimators if p ≥ 3. A graphical illustration of the risk functions under a “typical” member of the elliptically contoured family of distributions is provided to confirm the analytical results.     

Measuring the Discrepancy of a Parametric Model via Local Polynomial Smoothing

Abstract. In the context of multivariate mean regression, we propose a new method to measure and estimate the inadequacy of a given parametric model. The measure is basically the missed fraction of variation after adjusting the best possible parametric model from a given family. The proposed approach is based on the minimum L ² ‐distance between the true but unknown regression curve and a given model. The estimation method is based on local polynomial averaging of residuals with a polynomial degree that increases with the dimension d of the covariate. For any d ≥ 1 and under some weak assumptions we give a Bahadur‐type representation of the estimator from which ‐consistency and asymptotic normality are derived for strongly mixing variables. We report the outcomes of a simulation study that aims at checking the finite sample properties of these techniques. We present the analysis of a dataset on ultrasonic calibration for illustration.