Bootstrap Confidence Intervals for Large-scale Multivariate Monotonic Regression Problems

Recently, the methods used to estimate monotonic regression (MR) models have been substantially improved, and some algorithms can now produce high-accuracy monotonic fits to multivariate datasets containing over a million observations. Nevertheless, the computational burden can be prohibitively large for resampling techniques in which numerous datasets are processed independently of each other. Here, we present efficient algorithms for estimation of confidence limits in large-scale settings that take into account the similarity of the bootstrap or jackknifed datasets to which MR models are fitted. In addition, we introduce modifications that substantially improve the accuracy of MR solutions for binary response variables. The performance of our algorithms is illustrated using data on death in coronary heart disease for a large population. This example also illustrates that MR can be a valuable complement to logistic regression.     

Monotone Nonparametric Regression and Confidence Intervals

Several variations of monotone nonparametric regression have been developed over the past 30 years. One approach is to first apply nonparametric regression to data and then monotone smooth the initial estimates to “iron out” violations to the assumed order. Here, such estimators are considered, where local polynomial regression is first used, followed by either least squares isotonic regression or a monotone method using simple averages. The primary focus of this work is to evaluate different types of confidence intervals for these monotone nonparametric regression estimators through Monte Carlo simulation. Most of the confidence intervals use bootstrap or jackknife procedures. Estimation of a response variable as a function of two continuous predictor variables is considered, where the estimation is performed at the observed values of the predictors (instead of on a grid). The methods are then applied to data involving subjects that worked at plants that use beryllium metal who have developed chronic beryllium disease.     

Testing Linear Regression Models in Non Regular Case

We propose a new statistic for testing linear hypotheses in the non parametric regression model in the case of a homoscedastic error structure and fixed design. In contrast to most models suggested in the literature, our procedure is applicable in the non parametric model case without regularity condition, and also under either the null or the alternative hypotheses. We show the asymptotic normality of the test statistic under the null hypothesis and the alternative one. A simulation study is conducted to investigate the finite sample properties of the test with application to regime switching.     

Simultaneous Two-Sided Tolerance Intervals for a Univariate Linear Regression Model

In this article we deal with simultaneous two-sided tolerance intervals for a univariate linear regression model with independent normally distributed errors. We present a method for determining the intervals derived by the general confidence-set approach (GCSA), i.e. the intervals are constructed based on a specified confidence set for unknown parameters of the model. The confidence set used in the new method is formed based on a suggested hypothesis test about all parameters of the model. The simultaneous two-sided tolerance intervals determined by the presented method are found to be efficient and fast to compute based on a preliminary numerical comparison of all the existing methods based on GCSA.     

Confidence Intervals on Regression Models with Censored Data

Stute (1993, Consistent estimation under random censorship when covariables are present. Journal of Multivariate Analysis 45, 89–103) proposed a new method to estimate regression models with a censored response variable using least squares and showed the consistency and asymptotic normality for his estimator. This article proposes a new bootstrap-based methodology that improves the performance of the asymptotic interval estimation for the small sample size case. Therefore, we compare the behavior of Stute's asymptotic confidence interval with that of several confidence intervals that are based on resampling bootstrap techniques. In order to build these confidence intervals, we propose a new bootstrap resampling method that has been adapted for the case of censored regression models. We use simulations to study the improvement the performance of the proposed bootstrap-based confidence intervals show when compared to the asymptotic proposal. Simulation results indicate that, for the new proposals, coverage percentages are closer to the nominal values and, in addition, intervals are narrower.     

Semiparametric Ridge Regression Approach in Partially Linear Models

In this article, we introduce a semiparametric ridge regression estimator for the vector-parameter in a partial linear model. It is also assumed that some additional artificial linear restrictions are imposed to the whole parameter space and the errors are dependent. This estimator is a generalization of the well-known restricted least-squares estimator and is confined to the (affine) subspace which is generated by the restrictions. Asymptotic distributional bias and risk are also derived and the comparison result is then given.     

Covariate-Adjusted Partially Linear Regression Models

Motivated by covariate-adjusted regression (CAR) proposed by Sentürk and Müller (2005 Sentürk , D. , Müller , H. G. ( 2005 ). Covariate-adjusted regression . Biometrika 92 : 75 – 89 .[Crossref], [Web of Science ®] , [Google Scholar]) and an application problem, in this article we introduce and investigate a covariate-adjusted partially linear regression model (CAPLM), in which both response and predictor vector can only be observed after being distorted by some multiplicative factors, and an additional variable such as age or period is taken into account. Although our model seems to be a special case of covariate-adjusted varying coefficient model (CAVCM) given by Sentürk (2006 Sentürk , D. ( 2006 ). Covariate-adjusted varying coefficient models . Biostatistics 7 : 235 – 251 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), the data types of CAPLM and CAVCM are basically different and then the methods for inferring the two models are different. In this article, the estimate method motivated by Cui et al. (2008 Cui , X. , Guo , W. S. , Lin , L. , Zhu , L. X. ( 2008 ). Covariate-adjusted nonlinear regression . Ann. Statist. 37 : 1839 – 1870 . [Google Scholar]) is employed to infer the new model. Furthermore, under some mild conditions, the asymptotic normality of estimator for the parametric component is obtained. Combined with the consistent estimate of asymptotic covariance, we obtain confidence intervals for the regression coefficients. Also, some simulations and a real data analysis are made to illustrate the new model and methods.     

Linear Quantile Regression Based on EM Algorithm

This article aims to put forward a new method to solve the linear quantile regression problems based on EM algorithm using a location-scale mixture of the asymmetric Laplace error distribution. A closed form of the estimator of the unknown parameter vector β based on EM algorithm, is obtained. In addition, some simulations are conducted to illustrate the performance of the proposed method. Simulation results demonstrate that the proposed algorithm performs well. Finally, the classical Engel data is fitted and the Bootstrap confidence intervals for estimators are provided.     

The Use of Polynomials to Shift Coefficients in Linear Regression Models

Polynomials are commonly used in linear regression models to capture nonlinearities in explanatory variables. It is less common, however, that polynomials are used to shift the regression coefficients, an exception being the use of polynomially distributed lag coefficients. This note recommends the technique for a wider range of applications and suggests the Lagrangian interpolation representation as the most convenient for practitioners.     

Usual and Shortest Confidence Intervals on Odds Ratios from Logistic Regression

Based on the large-sample normal distribution of the sample log odds ratio and its asymptotic variance from maximum likelihood logistic regression, shortest 95% confidence intervals for the odds ratio are developed. Although the usual confidence interval on the odds ratio is unbiased, the shortest interval is not. That is, while covering the true odds ratio with the stated probability, the shortest interval covers some values below the true odds ratio with higher probability. The upper and lower limits of the shortest interval are shifted to the left of those of the usual interval, with greater shifts in the upper limits. With the log odds model γ + Xβ, in which X is binary, simulation studies showed that the approximate average percent difference in length is 7.4% for n (sample size) = 100, and 3.8% for n = 200. Precise estimates of the covering probabilities of the two types of intervals were obtained from simulation studies, and are compared graphically. For odds ratio estimates greater (less) than one, shortest intervals are more (less) likely to include one than are the usual intervals. The usual intervals are likelihood-based and the shortest intervals are not. The usual intervals have minimum expected length among the class of unbiased intervals. Shortest intervals do not provide important advantages over the usual intervals, which we recommend for practical use.     

Bayesian Estimation of the Number of Change Points in Simple Linear Regression Models

A Bayesian approach is considered to detect the number of change points in simple linear regression models. A normal-gamma empirical prior for the regression parameters based on maximum likelihood estimator (MLE) is employed in the analysis. Under mild conditions, consistency for the number of change points and boundedness between the estimated location and the true location of the change points are established. The Bayesian approach to the detection of the number of change points is suitable whether the switching simple regression is continuous or discontinuous. Some simulation results are given to confirm the accuracy of the proposed estimator.     

Linearized Restricted Ridge Regression Estimator in Linear Regression

This article primarily aims to put forward the linearized restricted ridge regression (LRRR) estimator in linear regression models. Two types of LRRR estimators are investigated under the PRESS criterion and the optimal LRRR estimators and the optimal restricted generalized ridge regression estimator are obtained. We apply the results to the Hald data and finally make a simulation study by using the method of McDonald and Galarneau.     

Some Second-Order Asymptotics for Extreme Value Linear Regression Models

ABSTRACT

In this article we derive finite-sample corrections in matrix notation for likelihood ratio and score statistics in extreme-value linear regression models. We consider three corrected score tests that perform better than the usual score test. We also derive general formulae for second-order biases of maximum likelihood estimates of the linear parameters. Some simulations are performed to compare the likelihood ratio and score statistics with their modified versions and to illustrate the bias correction.     

Linearized Ridge Regression Estimator in Linear Regression

In this article, we aim to study the linearized ridge regression (LRR) estimator in a linear regression model motivated by the work of Liu (1993). The LRR estimator and the two types of generalized Liu estimators are investigated under the PRESS criterion. The method of obtaining the optimal generalized ridge regression (GRR) estimator is derived from the optimal LRR estimator. We apply the Hald data as a numerical example and then make a simulation study to show the main results. It is concluded that the idea of transforming the GRR estimator as a complicated function of the biasing parameters to a linearized version should be paid more attention in the future.     

Testing for Linear Regression Relationships in Randomly Right-Censored Varying-Coefficient Models

A test statistic is constructed to test linear relationships in randomly right-censored varying-coefficient models. A residual-based bootstrap procedure is employed to derive the p-value of the test. The performance of the test is examined by extensive simulations. The simulation results show that the bootstrap estimate of the null distribution of the test statistic is approximately valid and the test method with the residual-based bootstrap works satisfactorily for at least moderate censoring rates of the response. Furthermore, the proposed test is applied to the Stanford heart transplant data for exploring a linear regression relationship between the logrithm of the survival time and the age of the patients.     

On Fixed Design Regression for General Linear Processes

This article is concerned with recovering a regression function g(x) on the basis of noisy observations taken at design points x _i . The corresponding observations are corrupted by additive dependent noise induced by a general linear process. The regression function is estimated by a smoother, which is shown to have an asymptotic multivariate normal distribution at multiple points. The problem of finding confidence bands for g(x) is discussed. An illustrative example is also exhibited. The results for finite samples are evaluated by computer simulations.     

Robust Confidence Intervals for Regression Parameters

The paper considers the problem of finding accurate small sample confidence intervals for regression parameters. Its approach is to construct conditional intervals with good robustness characteristics. This robustness is obtained by the choice of the density under which the conditional interval is computed. Both bounded influence and S-estimate style intervals are given. The required tail area computations are carried out using the results of DiCiccio, Field & Fraser (1990).     

Non-Diagonal-Type Estimator in Linear Regression

This short article mainly aims to introduce the notion of the non-diagonal-type estimator (NDTE) by means of the singular value decomposition theorem in the linear regression model to improve some classical linear estimators that can be called the diagonal-type estimators. We derive the optimal NDTE under the mean squared error criterion and its iterative version through matrix techniques. A simulation study is finally conducted to illustrate the theoretical results.     

Varying-Coefficient Functional Linear Regression Models

This article considers a generalization of the functional linear regression in which an additional real variable influences smoothly the functional coefficient. We thus define a varying-coefficient regression model for functional data. We propose two estimators based, respectively, on conditional functional principal regression and on local penalized regression splines and prove their pointwise consistency. We check, with the prediction one day ahead of ozone concentration in the city of Toulouse, the ability of such nonlinear functional approaches to produce competitive estimations.