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.     

Convergence rates for uniform confidence intervals based on local polynomial regression estimators

We investigate the convergence rates of uniform bias-corrected confidence intervals for a smooth curve using local polynomial regression for both the interior and boundary region. We discuss the cases when the degree of the polynomial is odd and even. The uniform confidence intervals are based on the volume-of-tube formula modified for biased estimators. We empirically show that the proposed uniform confidence intervals attain, at least approximately, nominal coverage. Finally, we investigate the performance of the volume-of-tube based confidence intervals for independent non-Gaussian errors.     

Local Likelihood Estimation in Generalized Additive Models

ABSTRACT.  Generalized additive models are a popular class of multivariate non-parametric regression models, due in large part to the ease of use of the local scoring estimation algorithm. However, the theoretical properties of the local scoring estimator are poorly understood. In this article, we propose a local likelihood estimator for generalized additive models that is closely related to the local scoring estimator fitted by local polynomial regression. We derive the statistical properties of the estimator and show that it achieves the same asymptotic convergence rate as a one-dimensional local polynomial regression estimator. We also propose a wild bootstrap estimator for calculating point-wise confidence intervals for the additive component functions. The practical behaviour of the proposed estimator is illustrated through a simulation experiment.     

Nonstationary regression models with a lagged dependent variable

This paper studies regression models with a lagged dependent variable when both the dependent and independent variables are nonstationary, and the regression model is misspecified in some dimension. In particular, we discuss the limiting properties of leastsquares estimates of the parameters in such regression models, and the limiting distributions of their test statistics. We show that the estimate of the lagged dependent variable tends to unity asymptotically independent of its true value, while the estimates of the independent variables tend to zero. The limiting distributions of their test statistics are shown to diverge with sample size.     

Robust Inference in Conditionally Linear Nonlinear Regression Models

Abstract.  We consider robust methods of likelihood and frequentist inference for the nonlinear parameter, say α , in conditionally linear nonlinear regression models. We derive closed-form expressions for robust conditional, marginal, profile and modified profile likelihood functions for α under elliptically contoured data distributions. Next, we develop robust exact-F confidence intervals for α and consider robust Fieller intervals for ratios of regression parameters in linear models. Several well-known examples are considered and Monte Carlo simulation results are presented.     

Fast and robust bootstrap

In this paper we review recent developments on a bootstrap method for robust estimators which is computationally faster and more resistant to outliers than the classical bootstrap. This fast and robust bootstrap method is, under reasonable regularity conditions, asymptotically consistent. We describe the method in general and then consider its application to perform inference based on robust estimators for the linear regression and multivariate location-scatter models. In particular, we study confidence and prediction intervals and tests of hypotheses for linear regression models, inference for location-scatter parameters and principal components, and classification error estimation for discriminant analysis.     

Variance Reduction in Smoothing Splines

Abstract.  We develop a variance reduction method for smoothing splines. For a given point of estimation, we define a variance-reduced spline estimate as a linear combination of classical spline estimates at three nearby points. We first develop a variance reduction method for spline estimators in univariate regression models. We then develop an analogous variance reduction method for spline estimators in clustered/longitudinal models. Simulation studies are performed which demonstrate the efficacy of our variance reduction methods in finite sample settings. Finally, a real data analysis with the motorcycle data set is performed. Here we consider variance estimation and generate 95% pointwise confidence intervals for the unknown regression function.     

On partial linear additive isotonic regression

In this paper we discuss semiparametric additive isotonic regression models. We discuss the efficiency bound of the model and the least squares estimator under this model. We show that the ordinary least square estimator studied by Huang (2002) and Cheng (2009) for the semiparametric isotonic regression achieves the efficiency bound for the regular estimator when the true parameter belongs to the interior of the parameter space. We also show that the result by Cheng (2009) can be generalized to the case that the covariates are dependent on each other.     

Simultaneous tolerance intervals in the random one-way model with covariates

Simultaneous tolerance intervals developed by Limam and Thomas (19881, for the normal regression model, are generalized to the random one-way model with covariates. Simultaneous tolerance intervals for unit means are developed for the balanced model. A simulation study is used to estimate the exact confidence of the tolerance intervals for models with one covariate.     

A New Regression Model: Modal Linear Regression

The mode of a distribution provides an important summary of data and is often estimated on the basis of some non‐parametric kernel density estimator. This article develops a new data analysis tool called modal linear regression in order to explore high‐dimensional data. Modal linear regression models the conditional mode of a response Y given a set of predictors x as a linear function of x . Modal linear regression differs from standard linear regression in that standard linear regression models the conditional mean (as opposed to mode) of Y as a linear function of x . We propose an expectation–maximization algorithm in order to estimate the regression coefficients of modal linear regression. We also provide asymptotic properties for the proposed estimator without the symmetric assumption of the error density. Our empirical studies with simulated data and real data demonstrate that the proposed modal regression gives shorter predictive intervals than mean linear regression, median linear regression and MM‐estimators.     

Optimal designs for estimating the slope of a regression

In the common linear model with quantitative predictors we consider the problem of designing experiments for estimating the slope of the expected response in a regression. We discuss locally optimal designs, where the experimenter is only interested in the slope at a particular point, and standardized minimax optimal designs, which could be used if precise estimation of the slope over a given region is required. General results on the number of support points of locally optimal designs are derived if the regression functions form a Chebyshev system. For polynomial regression and Fourier regression models of arbitrary degree the optimal designs for estimating the slope of the regression are determined explicitly for many cases of practical interest.     

On two-step estimation for varying coefficient models

In this note we discuss two-step kernel estimation of varying coefficient regression models that have a common smoothing variable. The method allows one to use different bandwidths for different coefficient functions. We consider local polynomial fitting and present explicit formulas for the asymptotic biases and variances of the estimators.     

Non‐parametric Regression for Circular Responses

Regression with a circular response is a topic of current interest. We introduce non‐parametric smoothing for this problem. Simple adaptations of a weight function enable a unified formulation for both real‐line and circular predictors, whereas these cases have been tackled by quite distinct parametric methods. Additionally, we discuss various methodological extensions, obtaining a number of promising techniques – totally new in circular statistics – such as confidence intervals for the value of a circular regression and non‐parametric autoregression in circular time series. The findings are also illustrated through real data examples.     

Graphics for studying logistic regression models

In this article we focus on logistic regression models for binary responses. An existing result shows that the log-odds can be modelled depending on the log of the ratio between the conditional densities of the predictors given the response variable. This suggests that relevant statistical information could be extracted investigating the inverse problem. Thus, we present different methods for studying the log-density ratio through graphs, which allow us to select which predictors are needed, and how they should be included in a logistic regression model. We also discuss data analysis examples based on real datasets available in literature in order to provide further insights into the methodology proposed.     

Model‐Averaged Confidence Intervals

We develop an approach to evaluating frequentist model averaging procedures by considering them in a simple situation in which there are two‐nested linear regression models over which we average. We introduce a general class of model averaged confidence intervals, obtain exact expressions for the coverage and the scaled expected length of the intervals, and use these to compute these quantities for the model averaged profile likelihood (MPI) and model‐averaged tail area confidence intervals proposed by D. Fletcher and D. Turek. We show that the MPI confidence intervals can perform more poorly than the standard confidence interval used after model selection but ignoring the model selection process. The model‐averaged tail area confidence intervals perform better than the MPI and postmodel‐selection confidence intervals but, for the examples that we consider, offer little over simply using the standard confidence interval for θ under the full model, with the same nominal coverage.     

Quantile regression via iterative least squares computations

We present an estimating framework for quantile regression where the usual L ₁-norm objective function is replaced by its smooth parametric approximation. An exact path-following algorithm is derived, leading to the well-known ‘basic’ solutions interpolating exactly a number of observations equal to the number of parameters being estimated. We discuss briefly possible practical implications of the proposed approach, such as early stopping for large data sets, confidence intervals, and additional topics for future research.     

Tail Functions and Iterative Weights in Binary Regression

In interpreting the binary regression models often used in the analysis of dose-response data, it is common to introduce the idea of an underlying continuous tolerance distribution. Different choices of link function lead to different tolerance distributions. A useful way of comparing these alternatives is to compare the hazard functions or tail functions associated with each tolerance distribution. Tail functions can also be applied to give numerically preferable formulas for the iterative weights and the adjusted dependent variable in the fitting of binary regression models by the iteratively reweighted least-squares algorithm.     

Financial data modeling by Poisson mixture regression

In many financial applications, Poisson mixture regression models are commonly used to analyze heterogeneous count data. When fitting these models, the observed counts are supposed to come from two or more subpopulations and parameter estimation is typically performed by means of maximum likelihood via the Expectation–Maximization algorithm. In this study, we discuss briefly the procedure for fitting Poisson mixture regression models by means of maximum likelihood, the model selection and goodness-of-fit tests. These models are applied to a real data set for credit-scoring purposes. We aim to reveal the impact of demographic and financial variables in creating different groups of clients and to predict the group to which each client belongs, as well as his expected number of defaulted payments. The model's conclusions are very interesting, revealing that the population consists of three groups, contrasting with the traditional good versus bad categorization approach of the credit-scoring systems.     

A recursive approach to parameter estimation in regression and time series models

In this paper we discuss the recursive (or on line) estimation in (i) regression and (ii) autoregressive integrated moving average (ARIMA) time series models. The adopted approach uses Kalman filtering techniques to calculate estimates recursively. This approach is used for the estimation of constant as well as time varying parameters. In the first section of the paper we consider the linear regression model. We discuss recursive estimation both for constant and time varying parameters. For constant parameters, Kalman filtering specializes to recursive least squares. In general, we allow the parameters to vary according to an autoregressive integrated moving average process and update the parameter estimates recursively. Since the stochastic model for the parameter changes will "be rarely known, simplifying assumptions have to be made. In particular we assume a random walk model for the time varying parameters and show how to determine whether the parameters are changing over time. This is illustrated with an example.     

Weighted composite quantile regression method via empirical likelihood for non linear models

In this paper, we investigate empirical likelihood (EL) inferences via weighted composite quantile regression for non linear models. Under regularity conditions, we establish that the proposed empirical log-likelihood ratio is asymptotically chi-squared, and then the confidence intervals for the regression coefficients are constructed. The proposed method avoids estimating the unknown error density function involved in the asymptotic covariance matrix of the estimators. Simulations suggest that the proposed EL procedure is more efficient and robust, and a real data analysis is used to illustrate the performance.