We investigate conditions sufficient for identification of average treatment effects using instrumental variables. First we show that the existence of valid instruments is not sufficient to identify any meaningful average treatment effect. We then establish that the combination of an instrument and a condition on the relation between the instrument and the participation status is sufficient for identification of a local average treatment effect for those who can be induced to change their participation status by changing the value of the instrument. Finally we derive the probability limit of the standard IV estimator under these conditions. It is seen to be a weighted average of local average treatment effects.

Evolution and Rationality Some Recent Game-Theoretic Results. Identification and Estimation of Local Average Treatment Effects

Big Data bring new opportunities to modern society and challenges to data scientists. On the one hand, Big Data hold great promises for discovering subtle population patterns and heterogeneities that are not possible with small-scale data. On the other hand, the massive sample size and high dimensionality of Big Data introduce unique computational and statistical challenges, including scalability and storage bottleneck, noise accumulation, spurious correlation, incidental endogeneity and measurement errors. These challenges are distinguished and require new computational and statistical paradigm. This paper gives overviews on the salient features of Big Data and how these features impact on paradigm change on statistical and computational methods as well as computing architectures. We also provide various new perspectives on the Big Data analysis and computation. In particular, we emphasize on the viability of the sparsest solution in high-confidence set and point out that exogenous assumptions in most statistical methods for Big Data cannot be validated due to incidental endogeneity. They can lead to wrong statistical inferences and consequently wrong scientific conclusions.

http://www.stat.ntu.edu.tw/download/教學文件/bigdata/Challenges%20of%20Big%20Data%20analysis.pdf

Challenges of Big Data analysis

Big Data bring new opportunities to modern society and challenges to data scientists. On one hand, Big Data hold great promises for discovering subtle population patterns and heterogeneities that are not possible with small-scale data. On the other hand, the massive sample size and high dimensionality of Big Data introduce unique computational and statistical challenges, including scalability and storage bottleneck, noise accumulation, spurious correlation, incidental endogeneity, and measurement errors. These challenges are distinguished and require new computational and statistical paradigm. This article give overviews on the salient features of Big Data and how these features impact on paradigm change on statistical and computational methods as well as computing architectures. We also provide various new perspectives on the Big Data analysis and computation. In particular, we emphasis on the viability of the sparsest solution in high-confidence set and point out that exogeneous assumptions in most statistical methods for Big Data can not be validated due to incidental endogeneity. They can lead to wrong statistical inferences and consequently wrong scientific conclusions.

Challenges of Big Data Analysis

This paper deals with estimation of high-dimensional covariance with a conditional sparsity structure, which is the composition of a low-rank matrix plus a sparse matrix. By assuming sparse error covariance matrix in a multi-factor model, we allow the presence of the cross-sectional correlation even after taking out common but unobservable factors. We introduce the Principal Orthogonal complEment Thresholding (POET) method to explore such an approximate factor structure. The POET estimator includes the sample covariance matrix, the factor-based covariance matrix (Fan, Fan and Lv, 2008), the thresholding estimator (Bickel and Levina, 2008) and the adaptive thresholding estimator (Cai and Liu, 2011) as specic examples. We provide mathematical insights when the factor analysis is approximately the same as the principal component analysis for high dimensional data. The rates of convergence of the sparse residual covariance matrix and the conditional sparse covariance matrix are studied under various norms, including the spectral norm. It is shown that the impact of estimating the unknown factors vanishes as the dimensionality increases. The uniform rates of convergence for the unobserved factors and their factor loadings are derived. The asymptotic results are also veried by extensive simulation studies.

https://mpra.ub.uni-muenchen.de/38697/1/MPRA_paper_38697.pdf

Large covariance estimation by thresholding principal orthogonal complements

This paper deals with the estimation of a high-dimensional covariance with a conditional sparsity structure and fast-diverging eigenvalues. By assuming sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross-sectional correlation even after taking out common but unobservable factors. We introduce the Principal Orthogonal complEment Thresholding (POET) method to explore such an approximate factor structure with sparsity. The POET estimator includes the sample covariance matrix, the factor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresholding estimator (Bickel and Levina, 2008) and the adaptive thresholding estimator (Cai and Liu, 2011) as specific examples. We provide mathematical insights when the factor analysis is approximately the same as the principal component analysis for high-dimensional data. The rates of convergence of the sparse residual covariance matrix and the conditional sparse covariance matrix are studied under various norms. It is shown that the impact of estimating the unknown factors vanishes as the dimensionality increases. The uniform rates of convergence for the unobserved factors and their factor loadings are derived. The asymptotic results are also verified by extensive simulation studies. Finally, a real data application on portfolio allocation is presented.

Large Covariance Estimation by Thresholding Principal Orthogonal Complements

Multiple hypothesis testing is a fundamental problem in high-dimensional inference, with wide applications in many scientific fields. In genome-wide association studies, tens of thousands of tests are performed simultaneously to find if any single-nucleotide polymorphisms (SNPs) are associated with some traits and those tests are correlated. When test statistics are correlated, false discovery control becomes very challenging under arbitrary dependence. In this article, we propose a novel method—based on principal factor approximation—that successfully subtracts the common dependence and weakens significantly the correlation structure, to deal with an arbitrary dependence structure. We derive an approximate expression for false discovery proportion (FDP) in large-scale multiple testing when a common threshold is used and provide a consistent estimate of realized FDP. This result has important applications in controlling false discovery rate and FDP. Our estimate of realized FDP compares favorably with Efr...

/pdf/estimating-false-discovery-proportion-under-arbitrary-x83ktfll8b.pdf

Estimating False Discovery Proportion Under Arbitrary Covariance Dependence

Multiple hypothesis testing is a fundamental problem in high dimensional inference, with wide applications in many scientific fields. In genome-wide association studies, tens of thousands of tests are performed simultaneously to find if any SNPs are associated with some traits and those tests are correlated. When test statistics are correlated, false discovery control becomes very challenging under arbitrary dependence. In the current paper, we propose a novel method based on principal factor approximation, which successfully subtracts the common dependence and weakens significantly the correlation structure, to deal with an arbitrary dependence structure. We derive an approximate expression for false discovery proportion (FDP) in large scale multiple testing when a common threshold is used and provide a consistent estimate of realized FDP. This result has important applications in controlling FDR and FDP. Our estimate of realized FDP compares favorably with Efron (2007)'s approach, as demonstrated in the simulated examples. Our approach is further illustrated by some real data applications. We also propose a dependence-adjusted procedure, which is more powerful than the fixed threshold procedure.

Large-scale multiple testing with correlated test statistics arises frequently in many scientific research Incorporating correlation information in approximating false discovery proportion has attracted increasing attention in recent years When the covariance matrix of test statistics is known, Fan, Han & Gu (2012) provided an accurate approximation of False Discovery Proportion (FDP) under arbitrary dependence structure and some sparsity assumption However, the covariance matrix is often unknown in many applications and such dependence information has to be estimated before approximating FDP The estimation accuracy can greatly affect FDP approximation In the current paper, we aim to theoretically study the impact of unknown dependence on the testing procedure and establish a general framework such that FDP can be well approximated The impacts of unknown dependence on approximating FDP are in the following two major aspects: through estimating eigenvalues/eigenvectors and through estimating marginal variances To address the challenges in these two aspects, we firstly develop general requirements on estimates of eigenvalues and eigenvectors for a good approximation of FDP We then give conditions on the structures of covariance matrices that satisfy such requirements Such dependence structures include banded/sparse covariance matrices and (conditional) sparse precision matrices Within this framework, we also consider a special example to illustrate our method where data are sampled from an approximate factor model, which encompasses most practical situations We provide a good approximation of FDP via exploiting this specific dependence structure The results are further generalized to the situation where the multivariate normality assumption is relaxed Our results are demonstrated by simulation studies and some real data applications

Estimation of the false discovery proportion with unknown dependence

We study the causal effect of winning an Oscar Award on an actor or actress’s survival. Does the increase in social rank from a performer winning an Oscar increase the performer’s life expectancy? Previous studies of this issue have suffered from healthy performer survivor bias, that is, candidates who are healthier will be able to act in more films and have more chance to win Oscar Awards. To correct this bias, we adapt Robins’ rank preserving structural accelerated failure time model and g-estimation method. We show in simulation studies that this approach corrects the bias contained in previous studies. We estimate that the effect of winning an Oscar Award on survival is 4.2 years, with a 95% confidence interval of [−0.4, 8.4] years. There is not strong evidence that winning an Oscar increases life expectancy.

/pdf/the-effect-of-winning-an-oscar-award-on-survival-correcting-4pjm8y3m7p.pdf

The Effect of Winning an Oscar Award on Survival: Correcting for Healthy Performer Survivor Bias With a Rank Preserving Structural Accelerated Failure Time Model

Large-scale multiple testing with highly correlated test statistics arises frequently in many scientific research. Incorporating correlation information in estimating false discovery proportion has attracted increasing attention in recent years. When the covariance matrix of test statistics is known, Fan, Han & Gu (2012) provided a consistent estimate of False Discovery Proportion (FDP) under arbitrary dependence structure. However, the covariance matrix is often unknown in many applications and such dependence information has to be estimated before estimating FDP (Efron, 2010). The estimation accuracy can greatly affect the convergence result of FDP or even violate its consistency. In the current paper, we provide methodological modification and theoretical investigations for estimation of FDP with unknown covariance. First we develop requirements for estimates of eigenvalues and eigenvectors such that we can obtain a consistent estimate of FDP. Secondly we give conditions on the dependence structures such that the estimate of FDP is consistent. Such dependence structures include sparse covariance matrices, which have been popularly considered in the contemporary random matrix theory. When data are sampled from an approximate factor model, which encompasses most practical situations, we provide a consistent estimate of FDP via exploiting this specific dependence structure. The results are further demonstrated by simulation studies and some real data applications.

Xu Han

Papers

Estimating False Discovery Proportion Under Arbitrary Covariance Dependence

Estimating False Discovery Proportion Under Arbitrary Covariance Dependence

Estimation of the false discovery proportion with unknown dependence

The Effect of Winning an Oscar Award on Survival: Correcting for Healthy Performer Survivor Bias With a Rank Preserving Structural Accelerated Failure Time Model

Estimation of False Discovery Proportion with Unknown Dependence