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**1 - 3**of**3**### Inference for High-dimensional Exponential Family Graphical Models

"... Abstract Probabilistic graphical models have been widely used to model complex systems and aid scientific discoveries. Most existing work on highdimensional estimation of exponential family graphical models, including Gaussian and Ising models, is focused on consistent model selection. However, the ..."

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Abstract Probabilistic graphical models have been widely used to model complex systems and aid scientific discoveries. Most existing work on highdimensional estimation of exponential family graphical models, including Gaussian and Ising models, is focused on consistent model selection. However, these results do not characterize uncertainty in the estimated structure and are of limited value to scientists who worry whether their findings will be reproducible and if the estimated edges are present in the model due to random chance. In this paper, we propose a novel estimator for edge parameters in an exponential family graphical models. We prove that the estimator is √ n-consistent and asymptotically Normal. This result allows us to construct confidence intervals for edge parameters, as well as, hypothesis tests. We establish our results under conditions that are typically assumed in the literature for consistent estimation. However, we do not require that the estimator consistently recovers the graph structure. In particular, we prove that the asymptotic distribution of the estimator is robust to model selection mistakes and uniformly valid for a large number of data-generating processes. We illustrate validity of our estimator through extensive simulation studies.

### SLOPE is Adaptive to Unknown Sparsity and Asymptotically Minimax

, 2015

"... We consider high-dimensional sparse regression problems in which we observe y = Xβ + z, where X is an n × p design matrix and z is an n-dimensional vector of independent Gaussian errors, each with variance σ2. Our focus is on the recently introduced SLOPE estimator [15], which regularizes the least- ..."

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We consider high-dimensional sparse regression problems in which we observe y = Xβ + z, where X is an n × p design matrix and z is an n-dimensional vector of independent Gaussian errors, each with variance σ2. Our focus is on the recently introduced SLOPE estimator [15], which regularizes the least-squares estimates with the rank-dependent penalty 1≤i≤p λi|β̂|(i), where |β̂|(i) is the ith largest magnitude of the fitted coefficients. Under Gaussian designs, where the entries of X are i.i.d. N (0, 1/n), we show that SLOPE, with weights λi just about equal to σ · Φ−1(1 − iq/(2p)) (Φ−1(α) is the αth quantile of a standard normal and q is a fixed number in (0, 1)) achieves a squared error of estimation obeying sup ‖β‖0≤k P

### Incorporation of Sparsity Information in Large-scale Multiple Two-sample t Tests

, 2014

"... Large-scale multiple two-sample Student’s t testing problems often arise from the statistical analysis of scientific data. To detect components with different val-ues between two mean vectors, a well-known procedure is to apply the Benjamini and Hochberg (B-H) method and two-sample Student’s t stati ..."

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Large-scale multiple two-sample Student’s t testing problems often arise from the statistical analysis of scientific data. To detect components with different val-ues between two mean vectors, a well-known procedure is to apply the Benjamini and Hochberg (B-H) method and two-sample Student’s t statistics to control the false discovery rate (FDR). In many applications, mean vectors are expected to be sparse or asymptotically sparse. When dealing with such type of data, can we gain more power than the standard procedure such as the B-H method with Student’s t statistics while keeping the FDR under control? The answer is positive. By exploit-ing the possible sparsity information in mean vectors, we present an uncorrelated screening-based (US) FDR control procedure, which is shown to be more powerful than the B-H method. The US testing procedure depends on a novel construc-tion of screening statistics, which are asymptotically uncorrelated with two-sample Student’s t statistics. The US testing procedure is different from some existing testing following screening methods (Reiner, et al., 2007; Yekutieli, 2008) in which independence between screening and testing is crucial to control the FDR, while the independence often requires additional data or splitting of samples. An inap-propriate splitting of samples may result in a loss rather than an improvement of statistical power. Instead, the uncorrelated screening US is based on the original