We investigate the extension of the nonparametric regression technique of local polynomial fitting with a kernel weight to generalized linear models and quasi-likelihood contexts. In the ordinary regression case, local polynomial fitting has been seen to have several appealing features in terms of intuitive and mathematical simplicity. One noteworthy feature is the better performance near the boundaries compared to the traditional kernel regression estimators. These properties are shown to carry over to generalized linear model and quasi-likelihood settings. We also derive the asymptotic distributions of the proposed class of estimators that allow for straightforward interpretation and extensions of state-of-the-art bandwidth selection methods.

We propose a novel technique to boost the power of testing a highdimensional vector <i>H</i>:<i><b></b></i>=<b>0</b> against sparse alternatives where the null hypothesis is violated by only a few components. Existing tests based on quadratic forms such as the Wald statistic often suffer from low powers due to the accumulation of errors in estimating highdimensional parameters. More powerful tests for sparse alternatives such as thresholding and extreme value tests, on the other hand, require either stringent conditions or bootstrap to derive the null distribution and often suffer from size distortions due to the slow convergence. Based on a screening technique, we introduce a power enhancement component, which is zero under the null hypothesis with high probability, but diverges quickly under sparse alternatives. The proposed test statistic combines the power enhancement component with an asymptotically pivotal statistic, and

Big data is transforming our world, revolutionizing operations and analytics everywhere, from financial engineering to biomedical sciences. The complexity of big data often makes dimension reduction techniques necessary before conducting statistical inference. Principal component analysis, commonly referred to as PCA, has become an essential tool for multivariate data analysis and unsupervised dimension reduction, the goal of which is to find a lower dimensional subspace that captures most of the variation in the dataset. This article provides an overview of methodological and theoretical developments of PCA over the past decade, with focus on its applications to big data analytics. We first review the mathematical formulation of PCA and its theoretical development from the view point of perturbation analysis. We then briefly discuss the relationship between PCA and factor analysis as well as its applications to

Prediction error is critical to assessing the performance of statistical methods and selecting statistical models. We propose the cross-validation and approximated cross-validation methods for estimating prediction error under a broad q-class of Bregman divergence for error measures which embeds nearly all of the commonly used loss functions in regression, classification procedures and machine learning literature. The approximated cross-validation formulas are analytically derived, which facilitate fast estimation of prediction error under the Bregman divergence. We then study a data-driven optimal bandwidth selector for the local-likelihood estimation that minimizes the overall prediction error or equivalently the covariance penalty. It is shown that the covariance penalty and cross-validation methods converge to the same mean-prediction-error-criterion. We also propose a lower-bound scheme for computing the local logistic regression estimates and demonstrate that it is as simple and stable as the local least-squares regression estimation. The algorithm monotonically enhances the target local-likelihood and converges. The idea and methods are extended to the generalized varying-coefficient models and semiparametric models.

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