[HDDA] sparse subspace constrained partial least squares

ABSTRACT

In this paper, we investigate the objective function and deflation process for sparse Partial Least Squares (PLS) regression with multiple components. While many have considered variations on the objective for sparse PLS, the deflation process for sparse PLS has not received as much attention. Our work highlights a flaw in the Statistically Inspired Modification of Partial Least Squares (SIMPLS) deflation method when applied in sparse PLS regression. We also consider the Nonlinear Iterative Partial Least Squares (NIPALS) deflation in sparse PLS regression. To remedy the flaw in the SIMPLS method, we propose a new sparse PLS method wherein the direction vectors are constrained to be sparse and lie in a chosen subspace. We give insight into this new PLS procedure and show through examples and simulation studies that the proposed technique can outperform alternative sparse PLS techniques in coefficient estimation. Moreover, our analysis reveals a simple renormalization step that can be used to improve the estimation of sparse PLS direction vectors generated using any convex relaxation method.     

MORE ON THE PRE-TEST ESTIMATOR IN RIDGE REGRESSION

ABSTRACT

The problem of estimation of the regression coefficients in a multiple regression model is considered under a multicollinearity situation when it is suspected that the regression coefficients may be restricted to a subspace. The objective of this paper is to compare the usual preliminary test estimator and the preliminary test ridge regression estimator in the sense of the dispersion matrix of one dominating that of the other. In particular we proved two results giving necessary and sufficient conditions for the superiority of the preliminary test ridge regression estimator over the preliminary test estimator associated with the δ = 0 (or Δ = 0) and δ ≠ 0 (or Δ ≠ 0).     

On algorithms for ordinary least squares regression spline fitting: A comparative study

Regression spline smoothing is a popular approach for conducting nonparametric regression. An important issue associated with it is the choice of a "theoretically best" set of knots. Different statistical model selection methods, such as Akaike's information criterion and generalized cross-validation, have been applied to derive different "theoretically best" sets of knots. Typically these best knot sets are defined implicitly as the optimizers of some objective functions. Hence another equally important issue concerning regression spline smoothing is how to optimize such objective functions. In this article different numerical algorithms that are designed for carrying out such optimization problems are compared by means of a simulation study. Both the univariate and bivariate smoothing settings will be considered. Based on the simulation results, recommendations for choosing a suitable optimization algorithm under various settings will be provided.     

Robust difference-based outlier detection

Abstract

In this paper, we propose an outlier-detection approach that uses the properties of an intercept estimator in a difference-based regression model (DBRM) that we first introduce. This DBRM uses multiple linear regression, and invented it to detect outliers in a multiple linear regression. Our outlier-detection approach uses only the intercept; it does not require estimates for the other parameters in the DBRM. In this paper, we first employed a difference-based intercept estimator to study the outlier-detection problem in a multiple regression model. We compared our approach with several existing methods in a simulation study and the results suggest that our approach outperformed the others. We also demonstrated the advantage of our approach using a real data application. Our approach can extend to nonparametric regression models for outliers detection.     

Bayesian estimation for time-series regressions improved with exact likelihoods

We propose an estimation procedure for time-series regression models under the Bayesian inference framework. With the exact method of Wise [Wise, J. (1955). The autocorrelation function and spectral density function. Biometrika, 42, 151–159], an exact likelihood function can be obtained instead of the likelihood conditional on initial observations. The constraints on the parameter space arising from the stationarity conditions are handled by a reparametrization, which was not taken into consideration by Chib [Chib, S. (1993). Bayes regression with autoregressive errors: A Gibbs sampling approach. J. Econometrics, 58, 275–294] or Chib and Greenberg [Chib, S. and Greenberg, E. (1994). Bayes inference in regression model with ARMA(p, q) errors. J. Econometrics, 64, 183–206]. Simulation studies show that our method leads to better inferential results than their results.     

MOSUM monitoring for variance change in nonparametric regression models

ABSTRACT

This article considers the monitoring for variance change in nonparametric regression models. First, the local linear estimator of the regression function is given. A moving square cumulative sum procedure is proposed based on residuals of the estimator. And the asymptotic results of the statistic under the null hypothesis and the alternative hypothesis are obtained. Simulations and Application support our procedure.     

Using Liu-Type Estimator to Combat Collinearity

Abstract

Linear regression model and least squares method are widely used in many fields of natural and social sciences. In the presence of collinearity, the least squares estimator is unstable and often gives misleading information. Ridge regression is the most common method to overcome this problem. We find that when there exists severe collinearity, the shrinkage parameter selected by existing methods for ridge regression may not fully address the ill conditioning problem. To solve this problem, we propose a new two-parameter estimator. We show using both theoretic results and simulation that our new estimator has two advantages over ridge regression. First, our estimator has less mean squared error (MSE). Second, our estimator can fully address the ill conditioning problem. A numerical example from literature is used to illustrate the results.     

Duality and local sensitivity analysis in least squares,minimax, and least absolute values regressions

Abstract

This paper deals with the problem of local sensitivity analysis in regression, i.e., how sensitive the results of a regression model (objective function, parameters, and dual variables) are to changes in the data. We use a general formula for local sensitivities in optimization problems to calculate the sensitivities in three standard regression problems (least squares, minimax, and least absolute values). Closed formulas for all sensitivities are derived. Sensitivity contours are presented to help in assessing the sensitivity of each observation in the sample. The dual problems of the minimax and least absolute values are obtained and interpreted. The proposed sensitivity measures are shown to deal more effectively with the masking problem than the existing methods. The methods are illustrated by their application to some examples and graphical illustrations are given.     

Bias Reduction in Logistic Regression with Missing Responses When the Missing Data Mechanism is Nonignorable

ABSTRACT

In logistic regression with nonignorable missing responses, Ibrahim and Lipsitz proposed a method for estimating regression parameters. It is known that the regression estimates obtained by using this method are biased when the sample size is small. Also, another complexity arises when the iterative estimation process encounters separation in estimating regression coefficients. In this article, we propose a method to improve the estimation of regression coefficients. In our likelihood-based method, we penalize the likelihood by multiplying it by a noninformative Jeffreys prior as a penalty term. The proposed method reduces bias and is able to handle the issue of separation. Simulation results show substantial bias reduction for the proposed method as compared to the existing method. Analyses using real world data also support the simulation findings. An R package called brlrmr is developed implementing the proposed method and the Ibrahim and Lipsitz method.     

Principal components regression estimator of the parameters in partially linear models

ABSTRACT

As a compromise between parametric regression and non-parametric regression models, partially linear models are frequently used in statistical modelling. This paper is concerned with the estimation of partially linear regression model in the presence of multicollinearity. Based on the profile least-squares approach, we propose a novel principal components regression (PCR) estimator for the parametric component. When some additional linear restrictions on the parametric component are available, we construct a corresponding restricted PCR estimator. Some simulations are conducted to examine the performance of our proposed estimators and the results are satisfactory. Finally, a real data example is analysed.     

Efficient regression modeling for correlated and overdispersed count data

Abstract

The objective of this paper is to propose an efficient estimation procedure in a marginal mean regression model for longitudinal count data and to develop a hypothesis test for detecting the presence of overdispersion. We extend the matrix expansion idea of quadratic inference functions to the negative binomial regression framework that entails accommodating both the within-subject correlation and overdispersion issue. Theoretical and numerical results show that the proposed procedure yields a more efficient estimator asymptotically than the one ignoring either the within-subject correlation or overdispersion. When the overdispersion is absent in data, the proposed method might hinder the estimation efficiency in practice, yet the Poisson regression based regression model is fitted to the data sufficiently well. Therefore, we construct the hypothesis test that recommends an appropriate model for the analysis of the correlated count data. Extensive simulation studies indicate that the proposed test can identify the effective model consistently. The proposed procedure is also applied to a transportation safety study and recommends the proposed negative binomial regression model.     

Influence measure for the L1 regression

Because outliers and leverage observations unduly affect the least squares regression, the identification of influential observations is considered an important and integrai part of the analysis. However, very few techniques have been developed for the residual analysis and diagnostics for the minimum sum of absolute errors, L₁ regression. Although the L₁ regression is more resistant to the outliers than the least squares regression, it appears that outliers (leverage) in the predictor variables may affect it. In this paper, our objective is to develop an influence measure for the L₁ regression based on the likelihood displacement function. We illustrate the proposed influence measure with examples.     

Regression model for surrogate data in high dimensional statistics

Abstract

This paper deals with the problem of estimating the regression of a surrogated scalar response variable given a functional random one. We construct an estimator of the regression operator by using, in addition to the available (true) response data, a surrogate data. We then establish some asymptotic properties of the constructed estimator in terms of the almost-complete and the quadratic mean convergences. Notice that the obtained results generalize a part of the results obtained in the finite dimensional framework. Finally, an illustration on the applicability of our results on both simulated data and a real dataset was realized. We have thus shown the superiority of our estimator on classical estimators when we are lacking complete data.     

Regression model with elliptically contoured errors

For the regression model y=X β+ε where the errors follow the elliptically contoured distribution, we consider the least squares, restricted least squares, preliminary test, Stein-type shrinkage and positive-rule shrinkage estimators for the regression parameters, β.

We compare the quadratic risks of the estimators to determine the relative dominance properties of the five estimators.     

Exponential inequalities in linear calibration problem

ABSTRACT

Calibration, also called inverse regression, is a classical problem which appears often in a regression setup under fixed design. The aim of this article is to propose a stochastic method which gives an estimated solution for a linear calibration problem. We establish exponential inequalities of Bernstein–Frechet type for the probability of the distance between the approximate solutions and the exact one. Furthermore, we build a confidence domain for the so-mentioned exact solution. To check the validity of our results, a numerical example is proposed.     

General composite quantile regression: Theory and methods

Abstract

In this article, we propose a new regression method called general composite quantile regression (GCQR) which releases the unrealistic finite error variance assumption being imposed by the traditional least squares (LS) method. Unlike the recently proposed composite quantile regression (CQR) method, our proposed GCQR allows any continuous non-uniform density/weight function. As a result, determination of the number of uniform quantile positions is not required. Most importantly, the proposed GCQR criterion can be readily transformed to a linear programing problem, which substantially reduces the computing time. Our theoretical and empirical results show that the GCQR is generally efficient than the CQR and LS if the weight function is appropriately chosen. The oracle properties of the penalized GCQR are also provided. Our simulation results are consistent with the derived theoretical findings. A real data example is analyzed to demonstrate our methodologies.     

A beta-inflated mean regression model with mixed effects for fractional response variables

Abstract

In this article we propose a new mixed-effects regression model for fractional bounded response variables. Our model allows us to incorporate covariates directly to the expected value, so we can quantify exactly the influence of these covariates in the mean of the variable of interest rather than to the conditional mean. Estimation is carried out from a Bayesian perspective. Due to the complexity of the augmented posterior distribution, we use a Hamiltonian Monte Carlo algorithm, the No-U-Turn sampler, implemented using the Stan software. A simulation study was performed showing that our model has a better performance than other traditional longitudinal models for bounded variables. Finally, we applied our beta-inflated mean mixed-effects regression model to real data which consists of utilization of credit lines in the peruvian financial system.     

Quantile regression methods for censored gap time data

ABSTRACT

In longitudinal studies, subjects may potentially undergo a series of sequentially ordered events. The gap times, which are the times between two serial events, are often the outcome variables of interest. This study considers quantile regression models of gap times for censored serial-event data and adapts a weighted version of the estimating equation for regression coefficients. The resulting estimators are uniformly consistent and asymptotically normal. Extensive simulation studies are presented to evaluate the finite-sample performance of the proposed methods. An analysis of the tumor recurrence data for bladder cancer patients is also provided to illustrate our proposed methods.     

A model-averaging approach for smoothing spline regression

ABSTRACT

This article considers nonparametric regression problems and develops a model-averaging procedure for smoothing spline regression problems. Unlike most smoothing parameter selection studies determining an optimum smoothing parameter, our focus here is on the prediction accuracy for the true conditional mean of Y given a predictor X. Our method consists of two steps. The first step is to construct a class of smoothing spline regression models based on nonparametric bootstrap samples, each with an appropriate smoothing parameter. The second step is to average bootstrap smoothing spline estimates of different smoothness to form a final improved estimate. To minimize the prediction error, we estimate the model weights using a delete-one-out cross-validation procedure. A simulation study has been performed by using a program written in R. The simulation study provides a comparison of the most well known cross-validation (CV), generalized cross-validation (GCV), and the proposed method. This new method is straightforward to implement, and gives reliable performances in simulations.     

Formulating robust regression estimation as an optimum allocation problem

The Mallows-type estimator, one of the most reasonable bounded influence estimators, often downweights leverage points regardless of the magnitude of the corresponding residual, and this could imply a loss of efficiency. In this article, we consider whether the efficiency of this bounded influence estimator could be improved by regarding both the robust x -distance and the residual size. We develop a new robust procedure based on the ideas of the Mallows-type estimator and the general robust recipe, where data been cleaned by pulling outliers towards their fitted values. Our basic idea is to formulate the robust estimation as an allocation problem, where the objective function is a Huber-type "loss" function, but the pulling resource is restricted. Using a mathematical programming technique, the pulling resource is optimally allocated to influential points <$>({x}_i, y_i)<$> with respect to residual size and given weights, <$>w({x}_i)<$>. Three previously published approaches are compared to our proposal via simulated experiments. In the case of contaminated data by regression outliers and "good" leverage points, the proposed robust estimator is a reasonable bounded influence estimator concerning both efficiency and norm of bias. In addition, the proposed approach offers the potential to establish constraints for the regression parameters and also may potentially provide insight regarding outlier detection.