arXiv subjects

Lixing Zhu

Publications and source records attributed to Lixing Zhu.

At least 55 records · Page 3Linked to original sources

arXiv2020

Doubly robust estimation for conditional treatment effect: a study on asymptotics

In this paper, we apply doubly robust approach to estimate, when some covariates are given, the conditional average treatment effect under parametric, semiparametric and nonparametric structure of the nuisance propensity score and outcome regression models. We then conduct a systematic study on the asymptotic distributions of nine estimators with different combinations of estimated propensity score and outcome regressions. The study covers the asymptotic properties with all models correctly specified; with either propensity score or outcome regressions locally / globally misspecified; and with all models locally / globally misspecified. The asymptotic variances are compared and the asymptotic bias correction under model-misspecification is discussed. The phenomenon that the asymptotic variance, with model-misspecification, could sometimes be even smaller than that with all models correctly specified is explored. We also conduct a numerical study to examine the theoretical results.

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arXiv2020

A Neural Generative Model for Joint Learning Topics and Topic-Specific Word Embeddings

We propose a novel generative model to explore both local and global context for joint learning topics and topic-specific word embeddings. In particular, we assume that global latent topics are shared across documents, a word is generated by a hidden semantic vector encoding its contextual semantic meaning, and its context words are generated conditional on both the hidden semantic vector and global latent topics. Topics are trained jointly with the word embeddings. The trained model maps words to topic-dependent embeddings, which naturally addresses the issue of word polysemy. Experimental results show that the proposed model outperforms the word-level embedding methods in both word similarity evaluation and word sense disambiguation. Furthermore, the model also extracts more coherent topics compared with existing neural topic models or other models for joint learning of topics and word embeddings. Finally, the model can be easily integrated with existing deep contextualized word embedding learning methods to further improve the performance of downstream tasks such as sentiment classification.

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arXiv2020

Doubly robust estimation of average treatment effect revisited

The research described herewith is to re-visit the classical doubly robust estimation of average treatment effect by conducting a systematic study on the comparisons, in the sense of asymptotic efficiency, among all possible combinations of the estimated propensity score and outcome regression. To this end, we consider all nine combinations under, respectively, parametric, nonparametric and semiparametric structures. The comparisons provide useful information on when and how to efficiently utilize the model structures in practice. Further, when there is model-misspecification, either propensity score or outcome regression, we also give the corresponding comparisons. Three phenomena are observed. Firstly, when all models are correctly specified, any combination can achieve the same semiparametric efficiency bound, which coincides with the existing results of some combinations. Secondly, when the propensity score is correctly modeled and estimated, but the outcome regression is misspecified parametrically or semiparametrically, the asymptotic variance is always larger than or equal to the semiparametric efficiency bound. Thirdly, in contrast, when the propensity score is misspecified parametrically or semiparametrically, while the outcome regression is correctly modeled and estimated, the asymptotic variance is not necessarily larger than the semiparametric efficiency bound. In some cases, the "super-efficiency" phenomenon occurs. We also conduct a small numerical study.

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arXiv2020

Neural Temporal Opinion Modelling for Opinion Prediction on Twitter

Opinion prediction on Twitter is challenging due to the transient nature of tweet content and neighbourhood context. In this paper, we model users' tweet posting behaviour as a temporal point process to jointly predict the posting time and the stance label of the next tweet given a user's historical tweet sequence and tweets posted by their neighbours. We design a topic-driven attention mechanism to capture the dynamic topic shifts in the neighbourhood context. Experimental results show that the proposed model predicts both the posting time and the stance labels of future tweets more accurately compared to a number of competitive baselines.

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arXiv2020

Integrated conditional moment test and beyond: when the number of covariates is divergent

The classic integrated conditional moment test is a promising method for testing regression model misspecification. However, it severely suffers from the curse of dimensionality. To extend it to handle the testing problem for parametric multi-index models with diverging number of covariates, we investigate three issues in inference in this paper. First, we study the consistency and asymptotically linear representation of the least squares estimator of the parameter matrix at faster rates of divergence than those in the literature for nonlinear models. Second, we propose, via sufficient dimension reduction techniques, an adaptive-to-model version of the integrated conditional moment test. We study the asymptotic properties of the new test under both the null and alternative hypothesis to examine its ability of significance level maintenance and its sensitivity to the global and local alternatives that are distinct from the null at the fastest possible rate in hypothesis testing. Third, we derive the consistency of the bootstrap approximation for the new test in the diverging dimension setting. The numerical studies show that the new test can very much enhance the performance of the original ICM test in high-dimensional scenarios. We also apply the test to a real data set for illustrations.

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arXiv2020

Model Checking for Parametric Ordinary Differential Equations System

Ordinary differential equations have been used to model dynamical systems in a broad range. Model checking for parametric ordinary differential equations is a necessary step to check whether the assumed models are plausible. In this paper we introduce three test statistics for their different purposes. We first give a trajectory matching-based test for the whole system. To further identify which component function(s) would be wrongly modelled, we introduce two test statistics that are based on integral matching and gradient matching respectively. We investigate the asymptotic properties of the three test statistics under the null, global and local alternative hypothesis. To achieve these purposes, we also investigate the asymptotic properties of nonlinear least squares estimation and two-step collocation estimation under both the null and alternatives. The results about the estimations are also new in the literature. To examine the performances of the tests, we conduct several numerical simulations. A real data example about immune cell kinetics and trafficking for influenza infection is analyzed for illustration.

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arXiv2020

Detecting multiple change points: a PULSE criterion

The research described herewith investigates detecting change points of means and of variances in a sequence of observations. The number of change points can be divergent at certain rate as the sample size goes to infinity. We define a MOSUM-based objective function for this purpose. Unlike all existing MOSUM-based methods, the novel objective function exhibits an useful ``PULSE" pattern near change points in the sense: at the population level, the value at any change point plus 2 times of the segment length of the moving average attains a local minimum tending to zero following by a local maximum going to infinity. This feature provides an efficient way to simultaneously identify all change points at the sample level. In theory, the number of change points can be consistently estimated and the locations can also be consistently estimated in a certain sense. Further, because of its visualization nature, in practice, the locations can be relatively more easily identified by plots than existing methods in the literature. The method can also handle the case in which the signals of some change points are very weak in the sense that those changes go to zero. Further, the computational cost is very inexpensive. The numerical studies we conduct validate its good performance.

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arXiv2019

Order Determination for Spiked Models

Motivated by dimension reduction in regression analysis and signal detection, we investigate the order determination for large dimension matrices including spiked models of which the numbers of covariates are proportional to the sample sizes for different models. Because the asymptotic behaviour of the estimated eigenvalues of the corresponding matrices differ completely from those in fixed dimension scenarios, we then discuss the largest possible number we can identify and introduce a "valley-cliff" criterion. We propose two versions of the criterion: one based on the original differences of eigenvalues and the other based on the transformed differences, which reduces the effects of ridge selection in the former one. This generic method is very easy to implement and computationally inexpensive, and it can be applied to various matrices. As examples, we focus on spiked population models, spiked Fisher matrices and factor models with auto-covariance matrices. Numerical studies are conducted to examine the method's finite sample performances and to compare it with existing methods.

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arXiv2019

Adaptive-to-model hybrid of tests for regressions

In model checking for regressions, nonparametric estimation-based tests usually have tractable limiting null distributions and are sensitive to oscillating alternative models, but suffer from the curse of dimensionality. In contrast, empirical process-based tests can, at the fastest possible rate, detect local alternatives distinct from the null model, but is less sensitive to oscillating alternative models and with intractable limiting null distributions. It has long been an issue on how to construct a test that can fully inherit the merits of these two types of tests and avoid the shortcomings. We in this paper propose a generic adaptive-to-model hybrid of moment and conditional moment-based test to achieve this goal. Further, a significant feature of the method is to make nonparametric estimation-based tests, under the alternatives, also share the merits of existing empirical process-based tests. This methodology can be readily applied to other kinds of data and constructing other hybrids. As a by-product in sufficient dimension reduction field, the estimation of residual-related central subspace is used to indicate the underlying models for model adaptation. A systematic study is devoted to showing when alternative models can be indicated and when cannot. This estimation is of its own interest and can be applied to the problems with other kinds of data. Numerical studies are conducted to verify the powerfulness of the proposed test.

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arXiv2018

Testing heteroscedasticity for regression models based on projections

In this paper we propose a new test of heteroscedasticity for parametric regression models and partial linear regression models in high dimensional settings. When the dimension of covariates is large, existing tests of heteroscedasticity perform badly due to the \curse of dimensionality". To attack this problem, we construct a test of heteroscedasticity by using a projection-based empirical process. We study the asymptotic properties of the test statistic under the null hypothesis and alternative hypotheses. It is shown that the test can detect local alternatives departure from the null hypothesis at the fastest possible rate in hypothesis testing. As the limiting null distribution of the test statistic is not distribution free, we propose a residual-based bootstrap. The validity of the bootstrap approximations is investigated. We present some simulation results to show the finite sample performances of the test. Two real data analyses are conducted for illustration.

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arXiv2017

Specification testing for regressions: an approach bridging between local smoothing and global smoothing methods

For regression models, most of existing specification tests can be categorized into the class of local smoothing tests and of global smoothing tests. Compared with global smoothing tests, local smoothing tests can only detect local alternatives distinct from the null hypothesis at a much slower rate when the dimension of predictor vector is high, but can be more sensitive to oscillating alternatives. In this paper, we suggest a projection-based test to bridge between the local and global smoothing-based methodologies such that the test can benefit from the advantages of these two types of tests. The test construction is based on a kernel estimation-based method and the resulting test becomes a distance-based test with a closed form. The asymptotic properties are investigated. Simulations and a real data analysis are conducted to evaluate the performance of the test in finite sample cases.

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arXiv2017

Estimation and adaptive-to-model testing for regressions with diverging number of predictors

The research described in this paper is motivated by model checking for parametric single-index models with diverging number of predictors. To construct a test statistic, we first study the asymptotic property of the estimators of involved parameters of interest under the null and alternative hypothesis when the dimension is divergent to infinity as the sample size goes to infinity. For the testing problem, we study an adaptive-to-model residual-marked empirical process as the basis for constructing a test statistic. By modifying the approach in the literature to suit the diverging dimension settings, we construct a martingale transformation. Under the null, local and global alternative hypothesis, the weak limits of the empirical process are derived and then the asymptotic properties of the test statistic are investigated. Simulation studies are carried out to examine the performance of the test.

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arXiv2016

Dimensionality determination: a thresholding double ridge ratio criterion

Popularly used eigendecomposition-based criteria such as BIC type, ratio estimation and principal component-based criterion often underdetermine model dimensionality for regressions or the number of factors for factor models. This longstanding problem is caused by the existence of one or two dominating eigenvalues compared to other nonzero eigenvalues. To alleviate this difficulty, we propose a thresholding double ridge ratio criterion such that the true dimension can be better identified and is less underdetermined. Unlike all existing eigendecomposition-based criteria, this criterion can define consistent estimate without requiring the uniqueness of minimum and can then handle possible multiple local minima scenarios. This generic strategy would be readily applied to other dimensionality or order determination problems. In this paper, we systematically investigate, for general sufficient dimension reduction theory, the dimensionality determination with fixed and divergent dimensions; for local alternative models that converge to its limiting model with fewer projected covariates, discuss when the number of projected covariates can be consistently estimated, when cannot; and for ultra-high dimensional factor models, study the estimation consistency for the number of common factors. Numerical studies are conducted to examine the finite sample performance of the method.

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arXiv2016

Penalized Maximum Likelihood Estimator for Skew Normal Mixtures

Skew normal mixture models provide a more flexible framework than the popular normal mixtures for modelling heterogeneous data with asymmetric behaviors. Due to the unboundedness of likelihood function and the divergency of shape parameters, the maximum likelihood estimators of the parameters of interest are often not well defined, leading to dissatisfactory inferential process. We put forward a proposal to deal with these issues simultaneously in the context of penalizing the likelihood function. The resulting penalized maximum likelihood estimator is proved to be strongly consistent when the putative order of mixture is equal to or larger than the true one. We also provide penalized EM-type algorithms to compute penalized estimators. Finite sample performances are examined by simulations and real data applications and the comparison to the existing methods.

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arXiv2016

An Adaptive-to-Model Test for Parametric Single-Index Errors-in-Variables Models

This paper provides some useful tests for fitting a parametric single-index regression model when covariates are measured with error and validation data is available. We propose two tests whose consistency rates do not depend on the dimension of the covariate vector when an adaptive-to-model strategy is applied. One of these tests has a bias term that becomes arbitrarily large with increasing sample size but its asymptotic variance is smaller, and the other is asymptotically unbiased with larger asymptotic variance. Compared with the existing local smoothing tests, the new tests behave like a classical local smoothing test with only one covariate, and still are omnibus against general alternatives. This avoids the difficulty associated with the curse of dimensionality. Further, a systematic study is conducted to give an insight on the effect of the values of the ratio between the sample size and the size of validation data on the asymptotic behavior of these tests. Simulations are conducted to examine the performance in several finite sample scenarios.

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arXiv2016

A projection-based adaptive-to-model test for regressions

A longstanding problem of existing empirical process-based tests for regressions is that when the number of covariates is greater than one, they either have no tractable limiting null distributions or are not omnibus. To attack this problem, we in this paper propose a projection-based adaptive-to-model approach. When the hypothetical model is parametric single-index, the method can fully utilize the dimension reduction model structure under the null hypothesis as if the covariate were one-dimensional such that the martingale transformation-based test can be asymptotically distribution-free. Further, the test can automatically adapt to the underlying model structure such that the test can be omnibus and thus detect alternative models distinct from the hypothetical model at the fastest possible rate in hypothesis testing. The method is examined through simulation studied and is illustrated by a real data analysis.

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arXiv2015

An adaptive-to-model test for partially parametric single-index models

Residual marked empirical process-based tests are commonly used in regression models. However, they suffer from data sparseness in high-dimensional space when there are many covariates. This paper has three purposes. First, we suggest a partial dimension reduction adaptive-to-model testing procedure that can be omnibus against general global alternative models although it fully use the dimension reduction structure under the null hypothesis. This feature is because that the procedure can automatically adapt to the null and alternative models, and thus greatly overcomes the dimensionality problem. Second, to achieve the above goal, we propose a ridge-type eigenvalue ratio estimate to automatically determine the number of linear combinations of the covariates under the null and alternatives. Third, a Monte-Carlo approximation to the sampling null distribution is suggested. Unlike existing bootstrap approximation methods, this gives an approximation as close to the sampling null distribution as possible by fully utilising the dimension reduction model structure under the null. Simulation studies and real data analysis are then conducted to illustrate the performance of the new test and compare it with existing tests.

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arXiv2015

Dimension reduction-based significance testing in nonparametric regression

A dimension reduction-based adaptive-to-model test is proposed for significance of a subset of covariates in the context of a nonparametric regression model. Unlike existing local smoothing significance tests, the new test behaves like a local smoothing test as if the number of covariates were just that under the null hypothesis and it can detect local alternatives distinct from the null at the rate that is only related to the number of covariates under the null hypothesis. Thus, the curse of dimensionality is largely alleviated when nonparametric estimation is inevitably required. In the cases where there are many insignificant covariates, the improvement of the new test is very significant over existing local smoothing tests on the significance level maintenance and power enhancement. Simulation studies and a real data analysis are conducted to examine the finite sample performance of the proposed test.

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