arXiv subjects

Lixing Zhu

Publications and source records attributed to Lixing Zhu.

72 records · Page 4Linked to original sources

arXiv2015

Heteroscedasticity Testing for Regression Models: A Dimension Reduction-based Model Adaptive

Heteroscedasticity testing is of importance in regression analysis. Existing local smoothing tests suffer severely from curse of dimensionality even when the number of covariates is moderate because of use of nonparametric estimation. In this paper, a dimension reduction-based model adaptive test is proposed which behaves like a local smoothing test as if the number of covariates were equal to the number of their linear combinations in the mean regression function, in particular, equal to 1 when the mean function contains a single index. The test statistic is asymptotically normal under the null hypothesis such that critical values are easily determined. The finite sample performances of the test are examined by simulations and a real data analysis.

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arXiv2015

A robust adaptive-to-model enhancement test for parametric single-index models

In the research on checking whether the underlying model is of parametric single-index structure with outliers in observations, the purpose of this paper is two-fold. First, a test that is robust against outliers is suggested. The Hampel's second-order influence function of the test statistic is proved to be bounded. Second, the test fully uses the dimension reduction structure of the hypothetical model and automatically adapts to alternative models when the null hypothesis is false. Thus, the test can greatly overcome the dimensionality problem and is still omnibus against general alternative models. The performance of the test is demonstrated by both Monte Carlo simulation studies and an application to a real dataset.

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arXiv2015

Enhancements of nonparametric generalized likelihood ratio test: Bias-correction and dimension reduction

Nonparametric generalized likelihood ratio test is popularly used for model checking for regressions. However, there are two issues that may be the barriers for its powerfulness. First, the bias term in its liming null distribution causes the test not to well control type I error and thus Monte Carlo approximation for critical value determination is required. Second, it severely suffers from the curse of dimensionality due to the use of multivariate nonparametric function estimation. The purpose of this paper is thus two-fold: a bias-correction is suggested to this test and a dimension reduction-based model-adaptive enhancement is recommended to promote the power performance. The proposed test still possesses the Wilks phenomenon, and the test statistic can converge to its limit at a much faster rate and is much more sensitive to alternative models than the original nonparametric generalized likelihood ratio test as if the dimension of covariates were one. Simulation studies are conducted to evaluate the finite sample performance and to compare with other popularly used tests. A real data analysis is conducted for illustration.

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arXiv2015

Variable selection and estimation for semi-parametric multiple-index models

In this paper, we propose a novel method to select significant variables and estimate the corresponding coefficients in multiple-index models with a group structure. All existing approaches for single-index models cannot be extended directly to handle this issue with several indices. This method integrates a popularly used shrinkage penalty such as LASSO with the group-wise minimum average variance estimation. It is capable of simultaneous dimension reduction and variable selection, while incorporating the group structure in predictors. Interestingly, the proposed estimator with the LASSO penalty then behaves like an estimator with an adaptive LASSO penalty. The estimator achieves consistency of variable selection without sacrificing the root-$n$ consistency of basis estimation. Simulation studies and a real-data example illustrate the effectiveness and efficiency of the new method.

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arXiv2014

Upper expectation parametric regression

Every observation may follow a distribution that is randomly selected in a class of distributions. It is called the distribution uncertainty. This is a fact acknowledged in some research fields such as financial risk measure. Thus, the classical expectation is not identifiable in general.In this paper, a distribution uncertainty is defined, and then an upper expectation regression is proposed, which can describe the relationship between extreme events and relevant covariates under the framework of distribution uncertainty. As there are no classical methods available to estimate the parameters in the upper expectation regression, a two-step penalized maximum least squares procedure is proposed to estimate the mean function and the upper expectation of the error. The resulting estimators are consistent and asymptotically normal in a certain sense.Simulation studies and a real data example are conducted to show that the classical least squares estimation does not work and the penalized maximum least squares performs well.

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arXiv2014

Estimation for ultra-high dimensional factor model: a pivotal variable detection based approach

For factor model, the involved covariance matrix often has no row sparse structure because the common factors may lead some variables to strongly associate with many others. Under the ultra-high dimensional paradigm, this feature causes existing methods for sparse covariance matrix in the literature not directly applicable. In this paper, for general covariance matrix, a novel approach to detect these variables that is called the pivotal variables is suggested. Then, two-stage estimation procedures are proposed to handle ultra-high dimensionality in factor model. In these procedures, pivotal variable detection is performed as a screening step and then existing approaches are applied to refine the working model. The estimation efficiency can be promoted under weaker assumptions on the model structure. Simulations are conducted to examine the performance of the new method and a real dataset is analysed for illustration.

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arXiv2014

Inference for biased models: a quasi-instrumental variable approach

For linear regression models who are not exactly sparse in the sense that the coefficients of the insignificant variables are not exactly zero, the working models obtained by a variable selection are often biased. Even in sparse cases, after a variable selection, when some significant variables are missing, the working models are biased as well. Thus, under such situations, root-n consistent estimation and accurate prediction could not be expected. In this paper, a novel remodelling method is proposed to produce an unbiased model when quasi-instrumental variables are introduced. The root-n estimation consistency and the asymptotic normality can be achieved, and the prediction accuracy can be promoted as well. The performance of the new method is examined through simulation studies.

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arXiv2014

Model checking for generalized linear models: a dimension-reduction model-adaptive approach

Local smoothing testing that is based on multivariate nonparametric regression estimation is one of the main model checking methodologies in the literature. However, relevant tests suffer from the typical curse of dimensionality resulting in slow convergence rates to their limits under the null hypotheses and less deviation from the null under alternatives. This problem leads tests to not well maintain the significance level and to be less sensitive to alternatives. In this paper, a dimension-reduction model-adaptive test is proposed for generalized linear models. The test behaves like a local smoothing test as if the model were univariate, and can be consistent against any global alternatives and can detect local alternatives distinct from the null at a fast rate that existing local smoothing tests can achieve only when the model is univariate. Simulations are carried out to examine the performance of our methodology. A real data analysis is conducted for illustration. The method can readily be extended to global smoothing methodology and other testing problems.

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arXiv2014

Transformed sufficient dimension reduction

A novel general framework is proposed in this paper for dimension reduction in regression to fill the gap between linear and fully nonlinear dimension reduction. The main idea is to transform first each of the raw predictors monotonically, and then search for a low-dimensional projection in the space defined by the transformed variables. Both user-specified and data-driven transformations are suggested. In each case, the methodology is discussed first in a general manner, and a representative method, as an example, is then proposed and evaluated by simulation. The proposed methods are applied to a real data set for illustration.

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arXiv2013

Asymptotic Composite Estimation

Composition methodologies in the current literature are mainly to promote estimation efficiency via direct composition, either, of initial estimators or of objective functions. In this paper, composite estimation is investigated for both estimation efficiency and bias reduction. To this end, a novel method is proposed by utilizing a regression relationship between initial estimators and values of model-independent parameter in an asymptotic sense. The resulting estimators could have smaller limiting variances than those of initial estimators, and for nonparametric regression estimation, could also have faster convergence rate than the classical optimal rate that the corresponding initial estimators can achieve. The simulations are carried out to examine its performance in finite sample situations.

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arXiv2012

The EFM approach for single-index models

Single-index models are natural extensions of linear models and circumvent the so-called curse of dimensionality. They are becoming increasingly popular in many scientific fields including biostatistics, medicine, economics and financial econometrics. Estimating and testing the model index coefficients $\boldsβ$ is one of the most important objectives in the statistical analysis. However, the commonly used assumption on the index coefficients, $\|\boldsβ\|=1$, represents a nonregular problem: the true index is on the boundary of the unit ball. In this paper we introduce the EFM approach, a method of estimating functions, to study the single-index model. The procedure is to first relax the equality constraint to one with (d-1) components of $\boldsβ$ lying in an open unit ball, and then to construct the associated (d-1) estimating functions by projecting the score function to the linear space spanned by the residuals with the unknown link being estimated by kernel estimating functions. The root-n consistency and asymptotic normality for the estimator obtained from solving the resulting estimating equations are achieved, and a Wilks type theorem for testing the index is demonstrated. A noticeable result we obtain is that our estimator for $\boldsβ$ has smaller or equal limiting variance than the estimator of Carroll et al. [J. Amer. Statist. Assoc. 92 (1997) 447-489]. A fixed-point iterative scheme for computing this estimator is proposed. This algorithm only involves one-dimensional nonparametric smoothers, thereby avoiding the data sparsity problem caused by high model dimensionality. Numerical studies based on simulation and on applications suggest that this new estimating system is quite powerful and easy to implement.

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arXiv2012

Robust rank correlation based screening

Independence screening is a variable selection method that uses a ranking criterion to select significant variables, particularly for statistical models with nonpolynomial dimensionality or "large p, small n" paradigms when p can be as large as an exponential of the sample size n. In this paper we propose a robust rank correlation screening (RRCS) method to deal with ultra-high dimensional data. The new procedure is based on the Kendall τcorrelation coefficient between response and predictor variables rather than the Pearson correlation of existing methods. The new method has four desirable features compared with existing independence screening methods. First, the sure independence screening property can hold only under the existence of a second order moment of predictor variables, rather than exponential tails or alikeness, even when the number of predictor variables grows as fast as exponentially of the sample size. Second, it can be used to deal with semiparametric models such as transformation regression models and single-index models under monotonic constraint to the link function without involving nonparametric estimation even when there are nonparametric functions in the models. Third, the procedure can be largely used against outliers and influence points in the observations. Last, the use of indicator functions in rank correlation screening greatly simplifies the theoretical derivation due to the boundedness of the resulting statistics, compared with previous studies on variable screening. Simulations are carried out for comparisons with existing methods and a real data example is analyzed.

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arXiv2011

Estimation and inference for high-dimensional non-sparse models

To successfully work on variable selection, sparse model structure has become a basic assumption for all existing methods. However, this assumption is questionable as it is hard to hold in most of cases and none of existing methods may provide consistent estimation and accurate model prediction in nons-parse scenarios. In this paper, we propose semiparametric re-modeling and inference when the linear regression model under study is possibly non-sparse. After an initial working model is selected by a method such as the Dantzig selector adopted in this paper, we re-construct a globally unbiased semiparametric model by use of suitable instrumental variables and nonparametric adjustment. The newly defined model is identifiable, and the estimator of parameter vector is asymptotically normal. The consistency, together with the re-built model, promotes model prediction. This method naturally works when the model is indeed sparse and thus is of robustness against non-sparseness in certain sense. Simulation studies show that the new approach has, particularly when $p$ is much larger than $n$, significant improvement of estimation and prediction accuracies over the Gaussian Dantzig selector and other classical methods. Even when the model under study is sparse, our method is also comparable to the existing methods designed for sparse models.

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arXiv2010

Component Selection in the Additive Regression Model

Similar to variable selection in the linear regression model, selecting significant components in the popular additive regression model is of great interest. However, such components are unknown smooth functions of independent variables, which are unobservable. As such, some approximation is needed. In this paper, we suggest a combination of penalized regression spline approximation and group variable selection, called the lasso-type spline method (LSM), to handle this component selection problem with a diverging number of strongly correlated variables in each group. It is shown that the proposed method can select significant components and estimate nonparametric additive function components simultaneously with an optimal convergence rate simultaneously. To make the LSM stable in computation and able to adapt its estimators to the level of smoothness of the component functions, weighted power spline bases and projected weighted power spline bases are proposed. Their performance is examined by simulation studies across two set-ups with independent predictors and correlated predictors, respectively, and appears superior to the performance of competing methods. The proposed method is extended to a partial linear regression model analysis with real data, and gives reliable results.

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arXiv2010

Adaptive post-Dantzig estimation and prediction for non-sparse "large $p$ and small $n$" models

For consistency (even oracle properties) of estimation and model prediction, almost all existing methods of variable/feature selection critically depend on sparsity of models. However, for ``large $p$ and small $n$" models sparsity assumption is hard to check and particularly, when this assumption is violated, the consistency of all existing estimations is usually impossible because working models selected by existing methods such as the LASSO and the Dantzig selector are usually biased. To attack this problem, we in this paper propose adaptive post-Dantzig estimation and model prediction. Here the adaptability means that the consistency based on the newly proposed method is adaptive to non-sparsity of model, choice of shrinkage tuning parameter and dimension of predictor vector. The idea is that after a sub-model as a working model is determined by the Dantzig selector, we construct a globally unbiased sub-model by choosing suitable instrumental variables and nonparametric adjustment. The new estimation of the parameters in the sub-model can be of the asymptotic normality. The consistent estimator, together with the selected sub-model and adjusted model, improves model predictions. Simulation studies show that the new approach has the significant improvement of estimation and prediction accuracies over the Gaussian Dantzig selector and other classical methods have.

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arXiv2010

Bounds smaller than the Fisher information for generalized linear models

In this paper, we propose a parameter space augmentation approach that is based on "intentionally" introducing a pseudo-nuisance parameter into generalized linear models for the purpose of variance reduction. We first consider the parameter whose norm is equal to one. By introducing a pseudo-nuisance parameter into models to be estimated, an extra estimation is asymptotically normal and is, more importantly, non-positively correlated to the estimation that asymptotically achieves the Fisher/quasi Fisher information. As such, the resulting estimation is asymptotically with smaller variance-covariance matrices than the Fisher/quasi Fisher information. For general cases where the norm of the parameter is not necessarily equal to one, two-stage quasi-likelihood procedures separately estimating the scalar and direction of the parameter are proposed. The traces of the limiting variance-covariance matrices are in general smaller than or equal to that of the Fisher/quasi-Fisher information. We also discuss the pros and cons of the new methodology, and possible extensions. As this methodology of parameter space augmentation is general, and then may be readily extended to handle, say, cluster data and correlated data, and other models.

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arXiv2009

Covariate-adjusted nonlinear regression

In this paper, we propose a covariate-adjusted nonlinear regression model. In this model, both the response and predictors can only be observed after being distorted by some multiplicative factors. Because of nonlinearity, existing methods for the linear setting cannot be directly employed. To attack this problem, we propose estimating the distorting functions by nonparametrically regressing the predictors and response on the distorting covariate; then, nonlinear least squares estimators for the parameters are obtained using the estimated response and predictors. Root $n$-consistency and asymptotic normality are established. However, the limiting variance has a very complex structure with several unknown components, and confidence regions based on normal approximation are not efficient. Empirical likelihood-based confidence regions are proposed, and their accuracy is also verified due to its self-scale invariance. Furthermore, unlike the common results derived from the profile methods, even when plug-in estimates are used for the infinite-dimensional nuisance parameters (distorting functions), the limit of empirical likelihood ratio is still chi-squared distributed. This property eases the construction of the empirical likelihood-based confidence regions. A simulation study is carried out to assess the finite sample performance of the proposed estimators and confidence regions. We apply our method to study the relationship between glomerular filtration rate and serum creatinine.

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arXiv2009

Estimation for a Partial-Linear Single-Index Model

In this paper, we study the estimation for a partial-linear single-index model. A two-stage estimation procedure is proposed to estimate the link function for the single index and the parameters in the single index, as well as the parameters in the linear component of the model. Asymptotic normality is established for both parametric components. For the index, a constrained estimating equation leads to an asymptotically more efficient estimator than existing estimators in the sense that it is of a smaller limiting variance. The estimator of the nonparametric link function achieves optimal convergence rates; and the structural error variance is obtained. In addition, the results facilitate the construction of confidence regions and hypothesis testing for the unknown parameters. A simulation study is performed and an application to a real dataset is illustrated. The extension to multiple indices is briefly sketched.

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