Dimension Reduction in Regressions through Weighted Variance Estimation

Because sliced inverse regression (SIR) using the conditional mean of the inverse regression fails to recover the central subspace when the inverse regression mean degenerates, sliced average variance estimation (SAVE) using the conditional variance was proposed in the sufficient dimension reduction literature. However, the efficacy of SAVE depends heavily upon the number of slices. In the present article, we introduce a class of weighted variance estimation (WVE), which, similar to SAVE and simple contour regression (SCR), uses the conditional variance of the inverse regression to recover the central subspace. The strong consistency and the asymptotic normality of the kernel estimation of WVE are established under mild regularity conditions. Finite sample studies are carried out for comparison with existing methods and an application to a real data is presented for illustration.     

An Inverse‐regression Method of Dependent Variable Transformation for Dimension Reduction with Non‐linear Confounding

Many model‐free dimension reduction methods have been developed for high‐dimensional regression data but have not paid much attention on problems with non‐linear confounding. In this paper, we propose an inverse‐regression method of dependent variable transformation for detecting the presence of non‐linear confounding. The benefit of using geometrical information from our method is highlighted. A ratio estimation strategy is incorporated in our approach to enhance the interpretation of variable selection. This approach can be implemented not only in principal Hessian directions (PHD) but also in other recently developed dimension reduction methods. Several simulation examples that are reported for illustration and comparisons are made with sliced inverse regression and PHD in ignorance of non‐linear confounding. An illustrative application to one real data is also 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.     

Bootstrap variance and bias estimation in linear models

Let θ be a nonlinear function of the regression parameters and θ be its estimator based on the least-squares method. This paper studies the bootstrap estimators of the variance and bias of θ. The bootstrap estimators are shown to be consistent and asymptotically unbiased under some conditions. Asymptotic orders of the mean squared errors of the bootstrap estimators are also obtained. The bootstrap and the classical linearization method are compared in a simulation study. Discussions about when to use the bootstrap are given.     

Random sliced inverse regression

Sliced Inverse Regression (SIR; 1991) is a dimension reduction method for reducing the dimension of the predictors without losing regression information. The implementation of SIR requires inverting the covariance matrix of the predictors—which has hindered its use to analyze high-dimensional data where the number of predictors exceed the sample size. We propose random sliced inverse regression (rSIR) by applying SIR to many bootstrap samples, each using a subset of randomly selected candidate predictors. The final rSIR estimate is obtained by aggregating these estimates. A simple variable selection procedure is also proposed using these bootstrap estimates. The performance of the proposed estimates is studied via extensive simulation. Application to a dataset concerning myocardial perfusion diagnosis from cardiac Single Proton Emission Computed Tomography (SPECT) images is presented.     

Variance Estimation under Two‐Phase Sampling

We consider the variance estimation of the weighted likelihood estimator (WLE) under two‐phase stratified sampling without replacement. Asymptotic variance of the WLE in many semiparametric models contains unknown functions or does not have a closed form. The standard method of the inverse probability weighted (IPW) sample variances of an estimated influence function is then not available in these models. To address this issue, we develop the variance estimation procedure for the WLE in a general semiparametric model. The phase I variance is estimated by taking a numerical derivative of the IPW log likelihood. The phase II variance is estimated based on the bootstrap for a stratified sample in a finite population. Despite a theoretical difficulty of dependent observations due to sampling without replacement, we establish the (bootstrap) consistency of our estimators. Finite sample properties of our method are illustrated in a simulation study.     

Confidence intervals centred on bootstrap smoothed estimators

Bootstrap smoothed (bagged) parameter estimators have been proposed as an improvement on estimators found after preliminary data‐based model selection. A result of Efron in 2014 is a very convenient and widely applicable formula for a delta method approximation to the standard deviation of the bootstrap smoothed estimator. This approximation provides an easily computed guide to the accuracy of this estimator. In addition, Efron considered a confidence interval centred on the bootstrap smoothed estimator, with width proportional to the estimate of this approximation to the standard deviation. We evaluate this confidence interval in the scenario of two nested linear regression models, the full model and a simpler model, and a preliminary test of the null hypothesis that the simpler model is correct. We derive computationally convenient expressions for the ideal bootstrap smoothed estimator and the coverage probability and expected length of this confidence interval. In terms of coverage probability, this confidence interval outperforms the post‐model‐selection confidence interval with the same nominal coverage and based on the same preliminary test. We also compare the performance of the confidence interval centred on the bootstrap smoothed estimator, in terms of expected length, to the usual confidence interval, with the same minimum coverage probability, based on the full model.     

K-medoids inverse regression

Abstract

K-means inverse regression was developed as an easy-to-use dimension reduction procedure for multivariate regression. This approach is similar to the original sliced inverse regression method, with the exception that the slices are explicitly produced by a K-means clustering of the response vectors. In this article, we propose K-medoids clustering as an alternative clustering approach for slicing and compare its performance to K-means in a simulation study. Although the two methods often produce comparable results, K-medoids tends to yield better performance in the presence of outliers. In addition to isolation of outliers, K-medoids clustering also has the advantage of accommodating a broader range of dissimilarity measures, which could prove useful in other graphical regression applications where slicing is required.     

NON-PARAMETRIC ESTIMATION OF DIRECTION IN SINGLE-INDEX MODELS WITH CATEGORICAL PREDICTORS

This paper proposes a general dimension‐reduction method targeting the partial central subspace recently introduced by Chiaromonte, Cook & Li. The dependence need not be confined to particular conditional moments, nor are restrictions placed on the predictors that are necessary for methods like partial sliced inverse regression. The paper focuses on a partially linear single‐index model. However, the underlying idea is applicable more generally. Illustrative examples are presented.     

Estimation in multiple linear regression model with doubly censored data

In this article, we propose three M-estimators for multiple regression model when response variable is subject to double censoring. The consistency of the proposed M-estimators is established. A simulation study is conducted to investigate the performance of the proposed estimators. Furthermore, the simple bootstrap methods are used to construct interval estimators.     

A generalized inverse regression estimator in multi-univariate linear calibration

A multivariate linear calibration problem, in which response variable is multivariate and explanatory variable is univariate, is considered. In this paper a class of generalized inverse regression estimators is proposed in multi-univariate linear calibration. It includes the classical estimator and the inverse regression one (or Krutchkoff estimator). For the proposed estimator we derive the expressions of bias and mean square error (MSE). Furthermore the behavior of these characteristics is investigated through an analytical method. In addition through a numerical study we confirm the existence of a generalized inverse regression estimator to improve both the classical and the inverse regression estimators on the MSE criterion.     

Estimation of Multiple Linear Regression Model with Twice-Censored Data

In this article, we propose three M-estimators for multiple regression model when response variable is subject to twice censoring. The consistency of the proposed M-estimators is established. A simulation study is conducted to investigate the performance of the proposed estimators. Furthermore, the simple bootstrap methods are used to construct interval estimators.     

Goodness‐of‐fit tests for linear regression models with missing response data

The authors show how to test the goodness‐of‐fit of a linear regression model when there are missing data in the response variable. Their statistics are based on the L₂ distance between nonparametric estimators of the regression function and a ‐consistent estimator of the same function under the parametric model. They obtain the limit distribution of the statistics and check the validity of their bootstrap version. Finally, a simulation study allows them to examine the behaviour of their tests, whether the samples are complete or not.     

Optimal quantization applied to sliced inverse regression

In this paper we consider a semiparametric regression model involving a d-dimensional quantitative explanatory variable X and including a dimension reduction of X via an index β′X. In this model, the main goal is to estimate the Euclidean parameter β and to predict the real response variable Y conditionally to X. Our approach is based on sliced inverse regression (SIR) method and optimal quantization in L^p-norm. We obtain the convergence of the proposed estimators of β and of the conditional distribution. Simulation studies show the good numerical behavior of the proposed estimators for finite sample size.     

Statistical Inference for Expectile‐based Risk Measures

Expectiles were introduced by Newey and Powell in 1987 in the context of linear regression models. Recently, Bellini et al. revealed that expectiles can also be seen as reasonable law‐invariant risk measures. In this article, we show that the corresponding statistical functionals are continuous w.r.t. the 1‐weak topology and suitably functionally differentiable. By means of these regularity results, we can derive several properties such as consistency, asymptotic normality, bootstrap consistency and qualitative robustness of the corresponding estimators in nonparametric and parametric statistical models.     

Response and predictor folding to counter symmetric dependency in dimension reduction

In the regression setting, dimension reduction allows for complicated regression structures to be detected via visualisation in a low‐dimensional framework. However, some popular dimension reduction methodologies fail to achieve this aim when faced with a problem often referred to as symmetric dependency. In this paper we show how vastly superior results can be achieved when carrying out response and predictor transformations for methods such as least squares and sliced inverse regression. These transformations are simple to implement and utilise estimates from other dimension reduction methods that are not faced with the symmetric dependency problem. We highlight the effectiveness of our approach via simulation and an example. Furthermore, we show that ordinary least squares can effectively detect multiple dimension reduction directions. Methods robust to extreme response values are also considered.     

A monte carlo comparison of regression trimmed means

Two proposals are made for constructing adaptive estimators of the parameters in a linear regression model. These estimators are based on regression trimmed means and use an idea of Jaeckel [(1971) Ann Math Statist 42, 1540-1552] and the bootstrap respectively. These adaptive trimmed means as well as some nonadaptive trimmed means are studied by Monte Carlo. A one-step biweight is also included for comparison purposes.     

BOOTSTRAP TESTS FOR THE ERROR DISTRIBUTION IN LINEAR AND NONPARAMETRIC REGRESSION MODELS

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and non‐parametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution‐free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness‐of‐fit testing of assumptions regarding the error distribution in linear and non‐parametric regression models.     

Sliced inverse regression for survival data

We apply the univariate sliced inverse regression to survival data. Our approach is different from the other papers on this subject. The right-censored observations are taken into account during the slicing of the survival times by assigning each of them with equal weight to all of the slices with longer survival. We test this method with different distributions for the two main survival data models, the accelerated lifetime model and Cox’s proportional hazards model. In both cases and under different conditions of sparsity, sample size and dimension of parameters, this non-parametric approach finds the data structure and can be viewed as a variable selector.     

Non‐parametric Copula Estimation Under Bivariate Censoring

In this paper, we consider non‐parametric copula inference under bivariate censoring. Based on an estimator of the joint cumulative distribution function, we define a discrete and two smooth estimators of the copula. The construction that we propose is valid for a large range of estimators of the distribution function and therefore for a large range of bivariate censoring frameworks. Under some conditions on the tails of the distributions, the weak convergence of the corresponding copula processes is obtained in l^∞([0,1]²). We derive the uniform convergence rates of the copula density estimators deduced from our smooth copula estimators. Investigation of the practical behaviour of these estimators is performed through a simulation study and two real data applications, corresponding to different censoring settings. We use our non‐parametric estimators to define a goodness‐of‐fit procedure for parametric copula models. A new bootstrap scheme is proposed to compute the critical values.