Workshop on Statistics and Stochastic Analysis

任：

员：

中国科学院

山东大学

澳门大学

任：

山东大学

员：

山东大学

山东大学

 2016年7月7日，周四 时  间 主持人 报告人 报告题目 8:00-8:20 注  册 8:20-8:30 开幕式 8:30-9:10 吴  臻 蔡宗武 A New Test on Asset Return Predictability with Structural Breaks 9:10-9:50 陈振庆 Stability of Heat Kernel Estimates for Symmetric Non-local Dirichlet Forms 9:50-10:10 茶  歇 10:10-10:50 贾广岩 高付清 Cramer-type Moderate Deviations for Monotone Functions 10:50-11:30 宋  健 Hyperbolic Anderson Model with Space-time Homogeneous Gaussian Noise 11:30-12:10 冯峥晖 Nonparametric Variable Selection and Its Application to Additive Models 12:10 午  餐 14:00-14:40 栾贻会 蒋文新 Bayesian Generalized Method of Moments and Variable Selection 14:40-15:20 张希承 Stochastic Hamiltonian Flow with Singular Drifts 15:20-16:00 嵇少林 Recursive Utility Optimization with Nonsmooth Coefficients 16:00-16:30 茶  歇 16:30-17:10 林  路 钟  威 A New Test for High Dimensional Regression Coefficients 17:10-17:50 陈增敬 Limit Theorems for Capacities 17:50 晚  餐

 2016年7月8日，周五 时  间 主持人 报告人 报告题目 8:30-9:10 石玉峰 杨立坚 Oracally Efficient Estimation and Consistent Model Selection for Auto-regressive Moving Average Time Series with Trend 9:10-9:50 赵怀忠 Ergodicity of Periodic Stochastic Dynamical Systems and Spectral Characterisation 9:50-10:10 茶  歇 10:10-10:50 嵇少林 熊  捷 Leader-Follower Stochastic Differential Game with Asymmetric Information and Applications 10:50-11:30 董  昭 Stationary Measures for Stochastic Lotka-Volterra Systems with Application to Turbulent Convection 11:30-12:10 崔  霞 Identification and Estimation of Generalized Linear Models with Parametric Nonignorable Missing Data Mechanism 12:10 午  餐 14:00-14:40 高付清 宋仁明 Potential Theory of Subordinate Brownian Motions 14:40-15:20 刘  磊 Estimating and Testing High-dimensional Mediation Effects in Epigenetic Studies 15:20-16:00 郑海涛 Maximum Likelihood Method Under Equation Constraints for Case-Control Study 16:00-16:30 茶  歇 16:30-17:10 李  娟 任艳霞 Williams Decomposition for Superprocesses 17:10-17:50 林  路 Consistent Estimation for Distribution-uncertainty Regression Via Cross-sample and Semiparametric Methodologies 17:50 晚  餐

6日入住学人大厦

7日、8日午餐：学人大厦自助餐厅

A New Test onAsset Return Predictability with Structural Breaks

Thispaper considers a predictive regression with a structural break at some unknowndate. We establish a testing procedure for testing asset returns predictabilityvia empirical likelihood method based on weighted score equations. The proposedempirical likelihood method works well theoretically in the sense that theWilks’s theorem holds. It is noteworthy that theoretical results are derivedunder a unified framework that it is unneeded to distinguish whether thepredicting regressors are stationary or nonstationary. Simulation experimentsare provided to confirm theoretical results and to illustrate the finite sampleperformance. As empirical analyses, we test for predictability on the S&P500 stock returns using the dividend-price ratio and the earnings-price ratio.Our empirical likelihood-based procedure suggests a clear improvement overexisting tests and should be used in practical applications.

Limit Theorems for Capacities

Motivated by Ellsberg-type modelsand problems in mathematical finance, we investigate limit behaviors of two differentmodels: one is the very simple Bernoulli trials with ambiguity (or called Ellsberg-typemodel), and the other is sub-linear expectations arising

from mathematical finance.With a new notion of φ-convolution for randomvariables, we show that empirical averages obtained from a large number oftrials in both models have the same limit distribution. We also investigate therelation between this limit theorem and the weak law of large numbers fornonadditive probability, and show that they are equivalent under the assumptionofφ-convolution on random variables. Our results generalizewell-known laws of large numbers (LLNs), using the proofs that are completelydifferent from those in the existing literature. Finally, we discuss fourmodels which satisfy

the assumptions of our mainresults.

Stabilityof Heat KernelEstimatesfor SymmetricNon-localDirichlet Forms

In this talk, we consider symmetric jump processes of mixed-type  on metric measure spaces under general volume doublingcondition, and establish   stability of two-sided heat kernel estimatesand  heat kernel upper bounds. We obtain their stable equivalent characterizations in terms ofthe jumpingkernels, variants of cut-off Sobolev inequalities, and the Faber-Krahninequalities. In particular, we will present stability results of heatkernel estimates for $\alpha$-stable-like processes even with $\alpha\ge 2$when the underlying spaces have walk dimensions larger than $2$, which has beenone of the major open problems in this area.

Based on joint work with Takashi Kumagai and Jian Wang.

Identification and Estimation of Generalized Linear Models with Parametric Nonignorable Missing Data Mechanism

Weaddress the problem of identifying and estimating generalized linear modelswhen the response values are nonignorably missing. A Logistic/Probit/Log-logpattern is taken to specify the missing data mechanism. In this situation,likelihood based on observed data may not be identi_able. In this article, weprove the models parameters areidentifiable under very mild conditions and then construct estimators based ona likelihood-based approach. The proposed estimators are shown to be consistentand asymptotically normal. Simulation studies demonstrate that the proposedinference procedure performs well in manysettings. We apply the proposed method to a dataset from research in a realstudy.

StationaryMeasures for Stochastic Lotka-Volterra Systems with Application to Turbulent Convection

Inthis talk I will give some ergodicity and nonergodicity for a class of stochasticLotka-Volterra systems as the noise intensity vanishes. The nonergodicity casecan be illustrated the turbulent characteristics. It is a phenomenon that theturbulence in a fluid layer heated from below and rotating about a verticalaxis is robust under stochastic disturbances. This is a joint work with LifengChen, Jifa Jiang, Lei Niu and Jianliang Zhai.

NonparametricVariable Selection and Its Application to Additive Models

冯峥晖，厦门大学                zhfengwise@gmail.com

Formultivariate nonparametric regression models, existing variable selectionmethods with penalization require high-dimensional nonparametric approximationsin objective functions. When the dimension is high, none of methods withpenalization in the literature are readily available. Also, ranking andscreening approaches cannot have selection consistency when iterativealgorithms cannot be used due to inefficient nonparametric approximation. Inthis paper, a novel and easily implemented approach is proposed to makeexisting methods feasible for selection with no need of nonparametricapproximation. Selection consistency can be achieved. As an application toadditive regression models, we then suggest a two-stage procedure thatseparates selection and estimation steps. An adaptive estimation to thesmoothness of underlying components can be constructed such that theconsistency can be even at parametric rate if the underlying model is reallyparametric. Simulations are carried out to examine the performance of ourmethod, and a real data example is analyzed for illustration.

Cramer-type Moderate Deviations for Monotone Functions

We introduce Cramer-type moderate deviations fornonparametric maximum likelihood estimators of monotone functions and their applications.

Recursive Utility Optimization with Nonsmooth Coefficients

This paper study the recursive utility maximizationproblem. We assume that the coefficients of both the wealth equations and therecursive utilities may be nonlinear and nonsmooth. After given an equivalentbackward formulation of our problem, we employ the variational formulation todeal with the nonsmooth coefficients. By the convexity duality method, theprimal "sup-inf" problem is translated to a dual minimization problemand the characterization of the saddle point of this game is derived. Finally,we obtain the optimal terminal wealth. To illustrate our results, three casesare explicitly worked out under some special assumptions. (joint work withXiaomin Shi)

Bayesian Generalized Method of Moments and VariableSelection

Animportant practice in statistics is to use robust likelihood-free methods, suchas the estimating equations, which only require assumptions on the momentsinstead of specifying the full probabilistic model. We propose a Bayesianapproach for such likelihood-free methods, based on (quasi-)posteriorprobabilities from the Bayesian Generalized Method of Moments (BGMM). Thisnovel concept allows us to incorporate two important advantages of a Bayesianapproach: the expressiveness of posterior distributions and the convenientcomputational method of MCMC. Many different applications are possible,including modeling the correlated longitudinal data, the quantile regression,and the graphical models based on partial correlation. We demonstratenumerically how our method works in these applications. Under mild conditions,we show that theoretically the BGMM can achieve the

posteriorconsistency for selecting the unknown true model, and that it possesses aBayesian version of the oracle property, i.e. the posterior distribution forthe parameter of interest is asymptotically normal and is as informative as ifthe true model were known. In addition, we show that the proposedquasi-posterior is valid to be interpreted as an approximate conditionaldistribution given a data summary.

Thisis the joint work with Cheng Li, Duke University. Wenxin Jiang acknowledges the Taishan Scholar program forsupporting his adjunct appointment at Shandong University.

Consistent Estimation for Distribution-uncertaintyRegression ViaCross-sample and SemiparametricMethodologies

林路，山东大学                  linlu@sdu.edu.cn

Motivatingby the famous Ellsberg paradox, ambiguity (distribution-uncertainty) isquantitively and qualitatively important in behavior finance. We consider atype of distribution-uncertainty regressions that contains endogenous variableregression and semiparametric regression as its special cases. For such models,however, classical estimating function does involve infinitely many nuisanceparameters caused by the uncertain distributions. Consequently, the parametersof interest cannot be consistently estimated and the corresponding predictionis imprecise, even aimless. In this paper, cross-sample and semiparametrictechniques, together with a hidden-constant function, are proposed for dealingwith the infinitely many nuisance parameters. The resultant estimating functiononly contains the parameters of interest, and the estimators of them are alwaysconsistent and normally distributed with standard convergence rate. Moreover,the newly proposed methodologies can avoid the use of instrumental variable ornonparametric estimation even if actually the model under study containsendogenous variables or nonparametric components. On the other hand, themethodologies for numerical computation are simple, and the correspondingcomputation procedures are somewhat similar to those for thedistribution-certainty models. The main difference from the classicalregression analysis is that the estimation efficiency is related to the levelof distribution-uncertainty.

Estimating and Testing High-dimensional Mediation Effects in Epigenetic Studies

DNAmethylation is an important epigenetic mechanism to regulate gene expression.Genome-wide DNA methylation markers, e.g., measured by Illumina InfiniumHumanMethylation450 BeadChip, are ultra-high dimensional (around 480K). DNAmethylation markers may mediate pathways linking environmental exposures withhealth outcomes. However, there is a lack of analytical methods to identifysignificant mediators for high-dimensional mediation analysis. Based on sureindependent screening and minimax concave penalty (MCP) techniques, we havedeveloped a joint significance test for the mediation effect. We demonstrateits practical performance using Monte Carlo simulation studies and apply thismethod to investigate the extent to which DNA methylation markers mediate thecausal pathway from smoking to reduced lung function in the Normative AgingStudy.

WilliamsDecomposition for Superprocesses

Weare interested in a spinal decomposition for superprocesses involving theancestral lineage of the last individual alive (Williams' decomposition).

Forsuperprocesses with homogeneous branching mechanism, the spatial motion is independent of the genealogicalstructure. As a consequence, the law of the ancestral lineage of the lastindividual alive does not distinguish from the originalmotion. Therefore, in this setting, the description of the process at theextinction time may be deduced from Abraham and Delmas (2009) where no spatialmotion is taken into account.

Fornonhomogeneous branching mechanisms on the contrary, the law of the ancestrallineage of the last individual alive should depend on the distance to theextinction time. Using the Brownian snake, Delmas and H\'{e}nard (2013) providea description of the genealogy for superprocesses with non-homogeneousquadratic branching mechanism.

Wewould like to find conditions such that the Williams' decomposition worksfor   superprocesses with eneralnon-homogeneous branching mechanisms. The talk is based on a working paper withRenming Song and Rui Zhang.

Hyperbolic Anderson Model with Space-time HomogeneousGaussian Noise

宋健，香港大学                    txjsong@hku.hk

In this article, we study thestochastic wave equation in arbitrary spatial dimension d withmultiplicative noise, also known in the literature as the Hyperbolic AndresonModel. This equation is perturbed by a general Gaussian noise, which ishomogeneous in both space and time. We prove the existenceof a solution of this equation (in the Skorohod sense) and the Holder continuity ofits sample paths, under the same respective conditions on the spatial spectralmeasure of the noise as in the case of the white noise in time, regardless of the temporalcovariance function of the noise. This is joint work with R. M. Balan.

Potential Theory of Subordinate Brownian Motions

A subordinate Brownian motion can be obtained byreplacing the time parameter of a Brownian motion by an independent increasingLevy process (i. e., a subordinator). Subordinate Brownian motions form a largesubclass of Levy processes and they are very important in various applications.The generator of of a subordinate Brownian motion is a function of theLaplacian. In this talk, I will give a survey of some of the recent results inthe study of the potential theory of subordinate Brownian motions. Inparticular, I will present recent results on sharp two-sided estimates on thetransition densities of killed subordinate Brownian motions in smooth opensets, or equivalently, sharp two-sided estimates on the Dirichlet heat kernelsof the generators of subordinate Brownian motions.

Leader-Follower Stochastic Differential Game with Asymmetric Informationand Applications

This talk is concerned with a leader-followerstochastic differential game with asymmetric information, where the informationavailable to the follower is based on some sub-$\sigma$-algebra of thatavailable to the leader. Such kind of game problems has

wide applications in finance, economics andmanagement engineering such as newsvendor problems, cooperative advertising andpricing problems. Stochastic maximum principles and verification theorems withpartial information will be presented. As an application, a linear-quadraticleader-follower stochastic differential game with asymmetric information isstudied. It is shown that the open-loop Stackelberg equilibrium admits a statefeedback representation if some system of Riccati equations is solvable. Thistalk is based on a joint work with Shi and Wang.

Oracally Efficient Estimation and Consistent Model Selection for Auto-regressive Moving Average Time Series with Trend

杨立坚，清华大学                  yanglijian@tsinghua.edu.cn

Mosttime series that are encountered in practice contain non-zero trend, yettextbook approaches to time series analysis are typically focused on zero-meanstationary auto-regressive moving average (ARMA) processes. Trend is oftenestimated by ad hoc methods and subtracted from time series, and the residualsare used as the true ARMA noise for data analysis and inference, includingparameter estimation, lag selection and prediction. We propose a theoreticallyjustified two-step method to analyse time series consisting of a smooth trendfunction and ARMA error term, which is computationally efficient and easy forpractitioners to implement. The trend is estimated by B-spline regression, andthe maximum likelihood estimator based on residuals is shown to be oracallyefficient in the sense that it is asymptotically as efficient as if the truetrend function were known and then removed to obtain the ARMA errors. Inaddition, consistency of the Bayesian information criterion for model selectionis established for the detrended residual sequence. Finite sample performanceof the procedure is illustrated by simulation studies and real data analysis.

Stochastic Hamiltonian Flow with Singular Drifts

张希承，武汉大学                XichengZhang@gmail.com

Inthis report, I will introduce recent progress about stochastic Hamitonian flowwith singular drifts.

Ergodicityof Periodic Stochastic Dynamical Systems and Spectral Characterisation

Ergodicity ofrandom dynamical systems in the random periodic regime where a periodicmeasure exists on a Polish space is obtained. In the Markovianrandom dynamical systems case, the idea of Poincar\'e sections isintroduced. It is proved  if the $\tau$-periodic measure is PS-ergodic,then it is ergodic. Moreover, if the infinitesimal generator of the Markovsemigroup has only $\{{2m\pi\over \tilde \tau}i\}_{m\in {\mathbb Z}}$ assimple eigenvalues on the imaginary axis, where $\tilde \tau={\tau\over k}$ forsome $k\in {\mathbb N}\setminus \{0\}$, then the periodic measure isPS-ergodic. Furthermore, if the semigroup on Poincar\'e sections  hasspectral gap, then the periodic measure is PS-mixing. The distinctionbetween random periodic and stationary regimes is given by a sufficientand necessary condition in terms of the spectral structure of theinfinitesimal generators. In particular if the $\tau$-periodic measure isPS-mixing, then the infinitesimal generator of the Markovian semigroup hasonly $\{{2m\pi\over \tilde \tau}i\}_{m\in {\mathbb Z}}$ as simpleeigenvalues on the imaginary axis, where $\tilde \tau={\tau\over k}$ for some$k\in {\mathbb N}\setminus \{0\}$,  if and only if the minimal periodof the periodic measure is no less than $\tilde \tau$.

Maximum Likelihood Method Under Equation Constraints for Case-Control Study

In case-control study, the restricted maximumlikelihood method does not work and the corresponding asymptotic propertieshave yet been studied. In this article, we considered the maximumlikelihood(ML) method for case-control study under equation constraints. Westudied the  asymptotic properties of therestricted ML estimators and the related likelihood ratio test converges to achi^2 distribution asymptotically. Simulation studies were performed toevaluate the proposed method and corresponding theoretical properties.

A New Test for High Dimensional RegressionCoefficients

钟威，厦门大学                 wzhong@xmu.edu.cn

Testing a hypothesis for high dimensionalregression coefficients is of fundamental importance in the statistical theoryand applications. This paper aims to develop a new U-type test for coefficientsin high dimensional linear regression models based on an estimated U-statisticsof order two. With the aid of martingale central limit theorem, we prove thatthe asymptotic null distributions of the proposed test are normal under twodifferent distribution assumptions. The idea of refitted cross-validation (RCV)approach is utilized to reduce the bias of the sample variance in theestimation of the test statistic. We assess the finite-sample performance ofthe proposed test by examining its size and power via Monte Carlo simulationswhich show that the new test based on the RCV estimator of the varianceachieves higher powers, especially for the sparse cases. We also illustrate theapplication of the proposed test by an empirical analysis of a microarray dataset on Yorkshire gilts.

蔡宗武  University of Kansas

广州大学

中国科学院

山东大学

山东大学

Northwestern University

香港大学

山东大学

山东大学

山东大学

澳门大学

山东大学

山东大学

山东大学

厦门大学