Adaptation of the jackknifed ridge methods to the linear mixed models

The purpose of this article is to obtain the jackknifed ridge predictors in the linear mixed models and to examine the superiorities, the linear combinations of the jackknifed ridge predictors over the ridge, principal components regression, r?k class and Henderson's predictors in terms of bias, covariance matrix and mean square error criteria. Numerical analyses are considered to illustrate the findings and a simulation study is conducted to see the performance of the jackknifed ridge predictors.     

Pruning a sufficient dimension reduction with a p-value guided hard-thresholding

Principal fitted component (PFC) models are a class of likelihood-based inverse regression methods that yield a so-called sufficient reduction of the random p-vector of predictors X given the response Y. Assuming that a large number of the predictors has no information about Y, we aimed to obtain an estimate of the sufficient reduction that ‘purges’ these irrelevant predictors, and thus, select the most useful ones. We devised a procedure using observed significance values from the univariate fittings to yield a sparse PFC, a purged estimate of the sufficient reduction. The performance of the method is compared to that of penalized forward linear regression models for variable selection in high-dimensional settings.     

Admissible Linear Predictors in the Superpopulation Model with Respect to Inequality Constraints

Admissibility of linear predictors of the linear quantity Qy is investigated under a general linear regression superpopulation model with some inequality constraints. The relation between admissible homogeneous and inhomogeneous linear predictors is characterized. Further, necessary and sufficient conditions for a linear predictor to be admissible in two cases of inequality constraints in the classes of homogeneous and inhomogeneous linear predictors are given, respectively.     

Linear predictors of future order statistics

In this paper we establish an optimal asymptotic linear predictor which does not involve the finite-sample variance-covariance structure. Extensions to the problem of finding the best linear unbiased and simple linear unbiased predictors for k samples are given. Moreover, we obtain alternative linear predictors by modifying the covariance matrix by either an identity matrix or a diagonal matrix. For normal, logistic and Rayleigh samples of size 10, the alternative linear predictors with these modifications have high efficiency when compared with the best linear unbiased predictor.     

Robust group-Lasso for functional regression model

In this article, we consider the problem of selecting functional variables using the L1 regularization in a functional linear regression model with a scalar response and functional predictors, in the presence of outliers. Since the LASSO is a special case of the penalized least-square regression with L1 penalty function, it suffers from the heavy-tailed errors and/or outliers in data. Recently, Least Absolute Deviation (LAD) and the LASSO methods have been combined (the LAD-LASSO regression method) to carry out robust parameter estimation and variable selection simultaneously for a multiple linear regression model. However, variable selection of the functional predictors based on LASSO fails since multiple parameters exist for a functional predictor. Therefore, group LASSO is used for selecting functional predictors since group LASSO selects grouped variables rather than individual variables. In this study, we propose a robust functional predictor selection method, the LAD-group LASSO, for a functional linear regression model with a scalar response and functional predictors. We illustrate the performance of the LAD-group LASSO on both simulated and real data.     

Best linear predictors for factor scores

From the literature three types of predictors for factor scores are available. These are characterized by the constraints: linear, linear conditionally unbiased, and linear correlation preserving. Each of these constraints generates a class of predictors. Best predictors are defined in terms of Lowner's partial matrix order applied to matrices of mean square error of prediction. It is shown that within the first two classes a best predictor exists and that it does not exist in the third.     

Spatial generalized linear mixed models in small area estimation

In survey sampling, policy decisions regarding the allocation of resources to sub‐groups of a population depend on reliable predictors of their underlying parameters. However, in some sub‐groups, called small areas due to small sample sizes relative to the population, the information needed for reliable estimation is typically not available. Consequently, data on a coarser scale are used to predict the characteristics of small areas. Mixed models are the primary tools in small area estimation (SAE) and also borrow information from alternative sources (e.g., previous surveys and administrative and census data sets). In many circumstances, small area predictors are associated with location. For instance, in the case of chronic disease or cancer, it is important for policy makers to understand spatial patterns of disease in order to determine small areas with high risk of disease and establish prevention strategies. The literature considering SAE with spatial random effects is sparse and mostly in the context of spatial linear mixed models. In this article, small area models are proposed for the class of spatial generalized linear mixed models to obtain small area predictors and corresponding second‐order unbiased mean squared prediction errors via Taylor expansion and a parametric bootstrap approach. The performance of the proposed approach is evaluated through simulation studies and application of the models to a real esophageal cancer data set from Minnesota, U.S.A. The Canadian Journal of Statistics 47: 426–437; 2019 © 2019 Statistical Society of Canada     

Pitman closeness for hierarchical bayes predictors in mixed linear models

The present article considers the Pitman Closeness (PC) criterion of certain hierarchical Bayes (HB) predictors derived under a normal mixed linear models for known ratios of variance components using a uniform prior for the vector of fixed effects and some proper or improper prior on the error variance. For a generalized Euclidean error, simultaneous HB predictors of several linear combinations of vector of effects are shown to be the Pitman-closest in the frequentist sense in the class of equivariant predictors for location group of transformations. The normality assumption can be relaxed to show that these HB predictors are the Pitman-closest for location-scale group of transformations for a wider family of elliptically symmetric distributions. Also for this family, the HB predictors turn out to be Pitman-closest in the class of all linear unbiased predictors (LUPs). All these results are extended for the HB predictor of finite population mean vector in the context of finite population sampling.     

Penalised spline estimation for generalised partially linear single-index models

Generalised linear models are frequently used in modeling the relationship of the response variable from the general exponential family with a set of predictor variables, where a linear combination of predictors is linked to the mean of the response variable. We propose a penalised spline (P-spline) estimation for generalised partially linear single-index models, which extend the generalised linear models to include nonlinear effect for some predictors. The proposed models can allow flexible dependence on some predictors while overcome the “curse of dimensionality”. We investigate the P-spline profile likelihood estimation using the readily available R package mgcv, leading to straightforward computation. Simulation studies are considered under various link functions. In addition, we examine different choices of smoothing parameters. Simulation results and real data applications show effectiveness of the proposed approach. Finally, some large sample properties are established.     

Minimax prediction in the linear model with a relative squared error

Arnold and Stahlecker (Stat Pap 44:107–115, 2003) considered the prediction of future values of the dependent variable in the linear regression model with a relative squared error and deterministic disturbances. They found an explicit form for a minimax linear affine solution d* of that problem. In the paper we generalize this result proving that the decision rule d* is also minimax when the class D{\mathcal{D}} of possible predictors of the dependent variable is unrestricted. Then we show that d* remains minimax in D{\mathcal{D}} when the disturbances are random with the mean vector zero and the known positive definite covariance matrix.     

Testing the Goodness of Fit of Parametric Regression Models with Random Toeplitz Forms

ABSTRACT: We introduce a class of Toeplitz‐band matrices for simple goodness of fit tests for parametric regression models. For a given length r of the band matrix the asymptotic optimal solution is derived. Asymptotic normality of the corresponding test statistic is established under a fixed and random design assumption as well as for linear and non‐linear models, respectively. This allows testing at any parametric assumption as well as the computation of confidence intervals for a quadratic measure of discrepancy between the parametric model and the true signal g;. Furthermore, the connection between testing the parametric goodness of fit and estimating the error variance is highlighted. As a by‐product we obtain a much simpler proof of a result of 34 ) concerning the optimality of an estimator for the variance. Our results unify and generalize recent results by 9 ) and 15 , 16 ) in several directions. Extensions to multivariate predictors and unbounded signals are discussed. A simulation study shows that a simple jacknife correction of the proposed test statistics leads to reasonable finite sample approximations.     

Parallelism,uniqueness, and large‐sample asymptotics for the Dantzig selector

The Dantzig selector (Candès & Tao, 2007) is a popular $\ell^{1}$ ‐regularization method for variable selection and estimation in linear regression. We present a very weak geometric condition on the observed predictors which is related to parallelism and, when satisfied, ensures the uniqueness of Dantzig selector estimators. The condition holds with probability 1, if the predictors are drawn from a continuous distribution. We discuss the necessity of this condition for uniqueness and also provide a closely related condition which ensures the uniqueness of lasso estimators (Tibshirani, 1996). Large sample asymptotics for the Dantzig selector, that is, almost sure convergence and the asymptotic distribution, follow directly from our uniqueness results and a continuity argument. The limiting distribution of the Dantzig selector is generally non‐normal. Though our asymptotic results require that the number of predictors is fixed (similar to Knight & Fu, 2000), our uniqueness results are valid for an arbitrary number of predictors and observations. The Canadian Journal of Statistics 41: 23–35; 2013 © 2012 Statistical Society of Canada     

Test for high dimensional regression coefficients of partially linear models

Abstract

Partially linear models attract much attention to investigate the association between predictors and the response variable when the dependency on some predictors may be nonlinear. However, the hypothesis test for significance of predictors is still challenging, especially when the number of predictors is larger than sample size. In this paper, we reconsider the test procedure of Zhong and Chen (2011 Zhong, P., and S. Chen. 2011. Tests for high-dimensional regression coefficients with factorial designs. Journal of the American Statistical Association 106 (493):260–74. doi:10.1198/jasa.2011.tm10284.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) when regression models have nonlinear components, and propose a generalized U-statistic for testing the linear components of the high dimensional partially linear models. The asymptotic properties of test statistic are obtained under null and alternative hypotheses, where the effect of nonlinear components should be considered and thus is different from that in linear models. Through simulation studies, we demonstrate good finite-sample performance of the proposed test in comparison with the existing methods. The practical utility of our proposed method is illustrated by a real data example.     

Bayesian variable selection with related predictors

In data sets with many predictors, algorithms for identifying a good subset of predictors are often used. Most such algorithms do not allow for any relationships between predictors. For example, stepwise regression might select a model containing an interaction AB but neither main effect A or B. This paper develops mathematical representations of this and other relations between predictors, which may then be incorporated in a model selection procedure. A Bayesian approach that goes beyond the standard independence prior for variable selection is adopted, and preference for certain models is interpreted as prior information. Priors relevant to arbitrary interactions and polynomials, dummy variables for categorical factors, competing predictors, and restrictions on the size of the models are developed. Since the relations developed are for priors, they may be incorporated in any Bayesian variable selection algorithm for any type of linear model. The application of the methods here is illustrated via the stochastic search variable selection algorithm of George and McCulloch (1993), which is modified to utilize the new priors. The performance of the approach is illustrated with two constructed examples and a computer performance dataset.     

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.     

Transformation approaches of linear random-effects models

Assume that a linear random-effects model \(\mathbf{y}= \mathbf{X}\varvec{\beta }+ \varvec{\varepsilon }= \mathbf{X}(\mathbf{A}\varvec{\alpha }+ \varvec{\gamma }) + \varvec{\varepsilon }\) is transformed as \(\mathbf{T}\mathbf{y}= \mathbf{T}\mathbf{X}\varvec{\beta }+ \mathbf{T}\varvec{\varepsilon }= \mathbf{T}\mathbf{X}(\mathbf{A}\varvec{\alpha }+ \varvec{\gamma }) + \mathbf{T}\varvec{\varepsilon }\) by pre-multiplying a given matrix \(\mathbf{T}\) of arbitrary rank. These two models are not necessarily equivalent unless \(\mathbf{T}\) is of full column rank, and we have to work with this derived model in many situations. Because predictors/estimators of the parameter spaces under the two models are not necessarily the same, it is primary work to compare predictors/estimators in the two models and to establish possible links between the inference results obtained from two models. This paper presents a general algebraic approach to the problem of comparing best linear unbiased predictors (BLUPs) of parameter spaces in an original linear random-effects model and its transformations, and provides a group of fundamental and comprehensive results on mathematical and statistical properties of the BLUPs. In particular, we construct many equalities for the BLUPs under an original linear random-effects model and its transformations, and obtain necessary and sufficient conditions for the equalities to hold.     

Hierarchical Variable Selection in Polynomial Regression Models

Significance tests on coefficients of lower-order terms in polynomial regression models are affected by linear transformations. For this reason, a polynomial regression model that excludes hierarchically inferior predictors (i.e., lower-order terms) is considered to be not well formulated. Existing variable-selection algorithms do not take into account the hierarchy of predictors and often select as “best” a model that is not hierarchically well formulated. This article proposes a theory of the hierarchical ordering of the predictors of an arbitrary polynomial regression model in m variables, where m is any arbitrary positive integer. Ways of modifying existing algorithms to restrict their search to well-formulated models are suggested. An algorithm that generates all possible well-formulated models is presented.     

Canonical Correlation Analysis Through Linear Modeling

In this paper, we introduce linear modeling of canonical correlation analysis, which estimates canonical direction matrices by minimising a quadratic objective function. The linear modeling results in a class of estimators of canonical direction matrices, and an optimal class is derived in the sense described herein. The optimal class guarantees several of the following desirable advantages: first, its estimates of canonical direction matrices are asymptotically efficient; second, its test statistic for determining the number of canonical covariates always has a chi‐squared distribution asymptotically; third, it is straight forward to construct tests for variable selection. The standard canonical correlation analysis and other existing methods turn out to be suboptimal members of the class. Finally, we study the role of canonical variates as a means of dimension reduction for predictors and responses in multivariate regression. Numerical studies and data analysis are presented.     

Mean square error behavior for prediction in linear regression models

For the problem of individual prediction in linear regression models, that is, estimation of a linear combination of regression coefficients, mean square error behavior of a general class of adaptive predictors is examined.     

Best quadratic predictions in finite populations

For a finite population and the resulting linear model Y=Xβ+e, the problem of the optimal invariant quadratic predictors including optimal invariant quadratic unbiased predictor and optimal invariant quadratic (potentially) biased predictor for the population quadratic quantities, f(H)=Y′HY , is of interest and has been previously considered in the literature for the case of HX=0. However, the special case does not contain all of situations at all. So, predicting f(H) in general situations may be of particular interest. In this paper, we make an effort to investigate how to offer a good predictor for f(H), not restricted yet to the mentioned case. Permutation matrix techniques play an important role in handling the process. The expected predictors are finally derived. In addition, we mention that the resulting predictors can be viewed as acceptable in all situations.