BOOTSTRAP WAVELET IN THE NONPARAMETRIC REGRESSION MODEL WITH WEAKLY DEPENDENT PROCESSES

This paper introduces a method of bootstrap wavelet estimation in a non-parametric regression model with weakly dependent processes for both fixed and random designs. The asymptotic bounds for the bias and variance of the bootstrap wavelet estimators are given in the fixed design model. The conditional normality for a modified version of the bootstrap wavelet estimators is obtained in the fixed model. The consistency for the bootstrap wavelet estimator is also proved in the random design model. These results show that the bootstrap wavelet method is valid for the model with weakly dependent processes.

TRANSPORTATION INEQUALITIES FOR WEAKLY DEPENDENT SEQUENCES

In [3], they gave necessary and sufficient condition for T 1 C and then as applications T 1 C for weakly dependent sequences was established. In this note, based on Gozlan-L′eonard characterization for W 1 H -inequalities, we extends this result to W 1 H inequalities.

Penalized least squares estimation with weakly dependent data

In statistics and machine learning communities, the last fifteen years have witnessed a surge of high-dimensional models backed by penalized methods and other state-of-the-art variable selection techniques.The high-dimensional models we refer to differ from conventional models in that the number of all parameters p and number of significant parameters s are both allowed to grow with the sample size T. When the field-specific knowledge is preliminary and in view of recent and potential affluence of data from genetics, finance and on-line social networks, etc., such（s, T, p）-triply diverging models enjoy ultimate flexibility in terms of modeling, and they can be used as a data-guided first step of investigation. However, model selection consistency and other theoretical properties were addressed only for independent data, leaving time series largely uncovered. On a simple linear regression model endowed with a weakly dependent sequence, this paper applies a penalized least squares（PLS） approach. Under regularity conditions, we show sign consistency, derive finite sample bound with high probability for estimation error, and prove that PLS estimate is consistent in L_2 norm with rate （s log s/T）～1/2.

Law of iterated logarithm and model selection consistency for generalized linear models with independent and dependent responses

We study the law of the iterated logarithm (LIL) for the maximum likelihood estimation of the parameters (as a convex optimization problem) in the generalized linear models with independent or weakly dependent (ρ-mixing) responses under mild conditions. The LIL is useful to derive the asymptotic bounds for the discrepancy between the empirical process of the log-likelihood function and the true log-likelihood. The strong consistency of some penalized likelihood-based model selection criteria can be shown as an application of the LIL. Under some regularity conditions, the model selection criterion will be helpful to select the simplest correct model almost surely when the penalty term increases with the model dimension, and the penalty term has an order higher than O(log log n) but lower than O(n). Simulation studies are implemented to verify the selection consistency of Bayesian information criterion.

Asymptotic Behavior of Product of Two Heavy-tailed Dependent Random Variables

Let X and Y be positive weakly negatively dependent （WND） random variables with finite expectations and continuous distribution functions F and G with heavy tails, respectively. The asymptotic behavior of the tail of distribution of XY is studied and some closure properties under some suitable conditions on F（x） = 1-F（x） and G（x） = of XY when X and Y are WND random variables 1- G（x） are provided. Moreover, subexponentiality is derived.

Screening-Assisted Dynamic Multiple Testing with False Discovery Rate Control

In the era of big data,high-dimensional data always arrive in streams,making timely and accurate decision necessary.It has become particularly important to rapidly and sequentially identify individuals whose behavior deviates from the norm.Aiming at identifying as many irregular behavioral patterns as possible,the authors develop a large-scale dynamic testing system in the framework of false discovery rate(FDR)control.By fully exploiting the sequential feature of datastreams,the authors propose a screening-assisted procedure that filters streams and then only tests streams that pass the filter at each time point.A data-driven optimal screening threshold is derived,giving the new method an edge over existing methods.Under some mild conditions on the dependence structure of datastreams,the FDR is shown to be strongly controlled and the suggested approach for determining screening thresholds is asymptotically optimal.Simulation studies show that the proposed method is both accurate and powerful,and a real-data example is used for illustrative purpose.

The Almost Sure Central Limit Theorems for the Maxima of Sums under Some New Weak Dependence Assumptions

We prove the almost sure central limit theorems for the maxima of partial sums of r.v.＇s under a general condition of dependence due to Doukhan and Louhichi. We will separately consider the centered sequences and the sequences with positive expected values.