Zhengyan Lin

Shrinkage-based regularization tests for high-dimensional data with application to gene set analysis

Traditional multivariate tests such as Hotellings test or Wilks test are designed for classical problems, where the number of observations is much larger than the dimension of the variables. For high-dimensional data, however, this assumption cannot be met any longer. In this article, we consider testing problems in high-dimensional MANOVA where the number of variables exceeds the sample size. To overcome the challenges with high dimensionality, we propose a new approach called a shrinkage-based regularization test, which is suitable for a variety of data structures including the one-sample problem and one-way MANOVA. Our approach uses a ridge regularization to overcome the singularity of the sample covariance matrix and applies a soft-thresholding technique to reduce random noise and improve the testing power. An appealing property of this approach is its ability to select relevant variables that provide evidence against the hypothesis. We compare the performance of our approach with some competing approaches via real microarray data and simulation studies. The results illustrate that the proposed statistics maintains relatively high power in detecting a wide family of alternatives.

Local Linear Estimation of Second-Order Diffusion Models

In this article, we present the local linear estimations for diffusion coefficient and drift coefficient in the second-order diffusion model. We show that under mild conditions, the estimators are weak consistent. We also use a Monte Carlo experiment to compare our estimators with the ones in Nicolau (2007).

Local Linear M-estimation in non-parametric spatial regression

A robust version of local linear regression smoothers augmented with variable bandwidths is investigated for dependent spatial processes. The (uniform) weak consistency as well as asymptotic normality for the local linear M-estimator (LLME) of the spatial regression function g( x ) are established under some mild conditions. Furthermore, an additive model is considered to avoid the curse of dimensionality for spatial processes and an estimation procedure based on combining the marginal integration technique with LLME is applied in this paper. Meanwhile, we present a simulated study to illustrate the proposed estimation method. Our simulation results show that the estimation method works well numerically. Copyright 2009 The Authors. Journal compilation 2009 Blackwell Publishing Ltd

Asymptotic normality of locally modelled regression estimator for functional data

We focus on the nonparametric regression of a scalar response on a functional explanatory variable. As an alternative to the well-known Nadaraya-Watson estimator for regression function in this framework, the locally modelled regression estimator performs very well [cf. [Barrientos-Marin, J., Ferraty, F., and Vieu, P. (2010), ‘Locally Modelled Regression and Functional Data’, Journal of Nonparametric Statistics, 22, 617–632]. In this paper, the asymptotic properties of locally modelled regression estimator for functional data are considered. The mean-squared convergence as well as asymptotic normality for the estimator are established. We also adapt the empirical likelihood method to construct the point-wise confidence intervals for the regression function and derive the Wilks phenomenon for the empirical likelihood inference. Furthermore, a simulation study is presented to illustrate our theoretical results.

Adaptive Lasso in high-dimensional settings

Huang et al. [J. Huang, S. Ma, and C.-H. Zhang, Adaptive Lasso for sparse high-dimensional regression models, Statist. Sinica 18 (2008), pp. 1603–1618] have studied the asymptotic properties of the adaptive Lasso estimators in sparse, high-dimensional, linear regression models when the number of covariates may increase with the sample size. They proved that the adaptive Lasso has an oracle property in the sense of Fan and Li [J. Fan and R. Li, Variable selection via nonconcave penalized likelihood and its oracle properties, J. Am. Statist. Assoc. 96 (2001), pp. 1348–1360] and Fan and Peng [J. Fan and H. Peng, Nonconcave penalized likelihood with a diverging number of parameters, Ann. Statist. 32 (2004), pp. 928–961] under appropriate conditions. Particularly, they assumed that the errors of the linear regression model have Gaussian tails. In this paper, we relax this condition and assume that the errors have the finite 2kth moment for an integer k>0. With this assumption, we prove that the adaptive Lasso also has the oracle property under some appropriate conditions. Simulations are carried out to provide understanding of our result.

Variable selection in partially time-varying coefficient models

A partially time-varying coefficient model is introduced to characterise the nonlinearity and trending phenomenon. To enhance predictability and to select significant variables in the parametric component of the model, the penalised least squares method with the help of the profile local linear technique is developed in this article. The convergence rate and the oracle property of the resulting estimator are established under mild conditions. A remarkable achievement of our results is that it does not require undersmoothing of the nonparametric component. Meanwhile, some extensions of the proposed model and method are also discussed. Furthermore, some numerical examples are provided to show that our theory and method work well in practice.

An adaptive test for the mean vector in large- p -small- n problems

The problem of testing the mean vector in a high-dimensional setting is considered. Up to date, most high-dimensional tests for the mean vector only make use of the marginal information from the variables, and do not incorporate the correlation information into the test statistics. A new testing procedure is proposed, which makes use of the covariance information between the variables. The new approach is novel in that it can select important variables that contain evidence against the null hypothesis and reduce the impact of noise accumulation. Simulations and real data analysis demonstrate that the new test has higher power than some competing methods proposed in the literature.

A Least Squares Estimator for Lévy-driven Moving Averages Based on Discrete Time Observations

This article is concerned with a least squares estimator (LSE) of the kernel function parameter θ for a Lévy-driven moving average of the form X(t) = ∫t− ∞K(θ(t − s)) dL(s), where is a Lévy process without the Brownian motion part, K is a kernel function and θ > 0 is a parameter. Let h be the time span between two consecutive observations and let n be the size of sample. As h → 0 and nh → ∞, consistency and asymptotic normality of the LSE are studied. The small-sample performance of the LSE is evaluated by means of a simulation experiment. Finally, two real-data applications show that the Lévy-driven moving average gives a good approximation to the autocorrelation of the process.

Nonparametric tests for the general multivariate multi-sample problem

Some nonparametric tests for the multivariate multi-sample problem are proposed in this paper. For the location–scale model, the univariate Kruskal–Wallis test and the bivariate Mardia test are generalised to the multivariate case. For the general multivariate multi-sample problem, a new test based on the Liu-Singh statistic is proposed and the asymptotic null distribution of this test statistic is established under some regularity conditions. The results of simulation show that these tests are more effective than the parametric tests when the assumption of multivariate normal distribution is violated, especially under the scale model or the location–scale model.

Specification testing in nonstationary time series models

In this paper, we consider a specification testing problem in nonlinear time series models with nonstationary regressors, and we propose using a nonparametric kernel‐based test statistic. The null asymptotics for the proposed nonparametric test statistic have been well developed in the existing literature. In this paper, we study the local asymptotics of the test statistic (i.e. the asymptotic properties of the test statistic under a sequence of general nonparametric local alternatives) and show that the asymptotic distribution depends on the asymptotic behaviour of the distance function, which is the local deviation from the parametrically specified model in the null hypothesis. In order to implement the proposed test in practice, we introduce a bootstrap procedure to approximate the critical values of the test statistic and establish a new Edgeworth expansion, which is used to justify the use of such an approximation. Based on the approximate critical values, we develop a bandwidth selection method, which chooses the optimal bandwidth that maximizes the local power of the test while its size is controlled at a given significance level. The local power is defined as the power of the proposed test for a given sequence of local alternatives. Such a bandwidth selection is made feasible by an approximate expression for the local power of the test as a function of the bandwidth. A Monte Carlo simulation study is provided to illustrate the finite sample performance of the proposed test.