Generalized Method of Moments (GMM) estimators in their various forms, including the popular Maximum Likelihood (ML) estimator, are frequently applied for the evaluation of complex econometric models with not analytically computable moment or likelihood functions. As the objective functions of GMM- and ML-estimators themselves constitute the approximation of an integral, more precisely of the expected value over the real world data space, the question arises whether the approximation of the moment function and the simulation of the entire objective function can be combined. Motivated by the popular Probit and Mixed Logit models, we consider double integrals with a linking function which stems from the considered estimator, e.g. the logarithm for Maximum Likelihood, and apply a sparse tensor product quadrature to reduce the computational effort for the approximation of the combined integral. Given Hölder continuity of the linking function, we prove that this approach can improve the order of the convergence rate of the classical GMM- and ML-estimator by a factor of two, even for integrands of low regularity or high dimensionality. This result is illustrated by numerical simulations of Mixed Logit and Multinomial Probit integrals which are estimated by ML- and GMM-estimators, respectively.
Sparse PCA: A New Scalable Estimator Based On Integer Programming
Operations Research Center, Massachusetts Institute of Technology（麻省理工学院运筹学中心）；Sloan School of Management, Massachusetts Institute of Technology（麻省理工学院斯隆管理学院）
A unifying approach for doubly-robust $\ell_1$ regularized estimation of causal contrasts
We consider inference about a scalar parameter under a non-parametric model based on a one-step estimator computed as a plug in estimator plus the empirical mean of an estimator of the parameter's influence function . We focus on a class of parameters that have influence function which depends on two infinite dimensional nuisance functions and such that the bias of the one-step estimator of the parameter of interest is the expectation of the product of the estimation errors of the two nuisance functions. Our class includes many important treatment effect contrasts of interest in causal inferen…
Popularity Regression-Based Latent Space Model for Large-Scale Sparse Network Analysis
Guanghua School of Management, Peking University（北京大学光华管理学院）；Center for Applied Statistics and School of Statistics, Renmin University of China（中国人民大学统计学院与应用统计中心）；School of Statistics and Mathematics, Central University of Finance and Economics（中央财经大学统计与数学学院）
Unified Principal Component Analysis for Sparse and Dense Functional Data under Spatial Dependency
We consider spatially dependent functional data collected under a geostatistics setting, where locations are sampled from a spatial point process. The functional response is the sum of a spatially dependent functional effect and a spatially independent functional nugget effect. Observations on each function are made on discrete time points and contaminated with measurement errors . Under the assumption of spatial stationarity and isotropy, we propose a tensor product spline estimator for the spatio-temporal covariance function . When a coregionalization covariance structure is further assumed,…
Adaptive Functional Thresholding for Sparse Covariance Function Estimation in High Dimensions
Covariance function estimation is a fundamental task in multivariate functional data analysis and arises in many applications. In this paper, we consider estimating sparse covariance functions for high-dimensional functional data, where the number of random functions p is comparable to, or even larger than the sample size n. Aided by the Hilbert--Schmidt norm of functions, we introduce a new class of functional thresholding operators that combine functional versions of thresholding and shrinkage, and propose the adaptive functional thresholding estimator by incorporating the variance effects o…
Sparse Gaussian Graphical Models with Discrete Optimization: Computational and Statistical Perspectives
Operations Research Center, Massachusetts Institute of Technology（麻省理工学院运筹学研究中心）；Sloan School of Management, Massachusetts Institute of Technology（麻省理工学院斯隆管理学院）
Sparse Models and Methods for Optimal Instruments with an Application to Eminent Domain
我们开发的结果使用套索和后套索方法，形成第一阶段的预测和估计最佳的工具，在线性工具变量（IV）模型与许多工具，$p$。即使当$p$远大于样本量$n$时，我们的结果也适用。我们表明，IV估计的基础上使用Lasso或后Lasso在第一阶段是根-n一致和渐近正常时，第一阶段是近似稀疏的，即当给定的工具的内生变量的条件期望可以很好地近似一组相对较小的变量，其身份可能是未知的。我们还表明，估计是半参数有效的结构误差是同方差的。值得注意的是，我们的研究结果允许不完美的模型选择，并不依赖于不切实际的“β-min”的条件，广泛用于建立有效性的推断以下模型选择。在模拟实验中，基于拉索的IV估计与数据驱动的惩罚表现良好，最近提倡的许多仪器的鲁棒程序相比。在一个处理司法征用权决定对经济结果影响的实证例子中，基于拉索的IV估计量优于直观基准。在发展的IV结果，我们建立了一系列的新结果的Lasso和Post-Lasso估计的非参数条件期望函数，这是独立的理论和实际利益。我们构建了一个修改的Lasso设计来处理非高斯，异方差干扰，它使用数据加权$\ell_1$罚函数。利用自归一化和的中偏差理论，我们给出了Lasso和Post-Lasso估计的收敛速度，在$\log p = o(n^{1/3})$满足条件下，其收敛速度与同方差高斯情形下的相应速度一样快.
