Oracle Inequality for Sparse Trace Regression Models with Exponentialβ-mixing Errors

In applications involving,e.g.,panel data,images,genomics microarrays,etc.,trace regression models are useful tools.To address the high-dimensional issue of these applications,it is common to assume some sparsity property.For the case of the parameter matrix being simultaneously low rank and elements-wise sparse,we estimate the parameter matrix through the least-squares approach with the composite penalty combining the nuclear norm and the l1norm.We extend the existing analysis of the low-rank trace regression with i.i.d.errors to exponentialβ-mixing errors.The explicit convergence rate and the asymptotic properties of the proposed estimator are established.Simulations,as well as a real data application,are also carried out for illustration.

Rates of uniform convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions

Assume that {X n } is a strictly stationary β-mixing random sequence with the β-mixing coefficient β k = O(k ?r ), 0 < r ? 1. Yu (1994) obtained convergence rates of empirical processes of strictly stationary β-mixing random sequence indexed by bounded classes of functions. Here, a new truncation method is proposed and used to study the convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions. The research results show that if the envelope of the index class of functions is in L p, p > 2 or p > 4, uniform convergence rates of empirical processes of strictly stationary β-mixing random sequence over the index classes can reach O((n r/(1+r)/log n)?1/2) or O((n r/(1+r)/log n)?3/4) and that the Central Limit Theorem does not always hold for the empirical processes.

Optimal convergence rates of nonparametric conditional quantiles in dependent cases

The ordinary quantiles for univariate data were successfully generalized to linear modelsin Koenker and Bassett. Regression quantiles provide more specific and more global in-formation on the relationship of two variables through their distributions. Mosteller andTukey argued that the use of regression quantiles helps to provide a more complete pic-

Rates of convergence of M-estimators for partly linear models involving time series

The asymptotic behaviour of M-estimalors constructed with B-spline method based on strictly stationary β-mixing observations of a partly linear model is dealt with. Under some regular conditions, it is proved that the M-estimators of the vector of parameters are asymptotically normal and the M-estimators of the nonparametric component achieve the optimal convergence rates for nonparametric regression. Our asymptotic theory includes L1-, L2-, Lp-norm, and Huber estimators as special cases.

THE RATES OF CONVERGENCE OF M-ESTIMATORS FOR PARTLY LINEAR MODELS IN DEPENDENT CASES

Consider the partly linear model K = X1& + go(Ti) + ei, where {(Ti, Xi)}T is a strictlystationary Sequence of random variable8, the ei’8 are i.i.d. random errorsl the K’s are realvalued responsest fo is a &vector of parameters, X is a &vector of explanatory variables,Ti is another explanatory variable ranging over a nondegenerate compact interval. Bnd ona segmnt of observations (T1, Xi 1 Y1 ),’’’ f (Tn, X;, Yn), this article investigates the rates ofconvrgence of the M-estimators for Po and go obtained from the minimisation problemwhere H is a space of B-spline functions of order m + 1 and p(-) is a function chosen suitablyUnder some regularity conditions, it is shown that the estimator of go achieves the optimalglobal rate of convergence of estimators for nonparametric regression, and the estdriator offo is asymptotically normal. The M-estimators here include regression quantile estimators,Li-estimators, Lp-norm estimators, Huber’s type M-estimators and usual least squares estimators. Applications of the asymptotic theory to testing the hypothesis H0: A’β0 =β are alsodiscussed, where β is a given vector and A is a known d × do matrix with rank d0.

A copula-based approximation to Markov chains

The Markov chain is well studied and widely applied in many areas.For some Markov chains,it is infeasible to obtain the explicit expressions of their corresponding finite-dimensional distributions and sometimes it is time-consuming for computation.In this paper,we propose an approximation method for Markov chains by applying the copula theory.For this purpose,we first discuss the checkerboard copula-based Markov chain,which is the Markov chain generated by the family of checkerboard copulas.This Markov chain has some appealing properties,such as self-similarity in copulas and having explicit forms of finite-dimensional distributions.Then we prove that each Markov chain can be approximated by a sequence of checkerboard copula-based Markov chains,and the error bounds of the approximate distributions are provided.Employing the checkerboard copula-based approximation method,we propose a sufficient condition for the geometric β-mixing of copula-based Markov chains.This condition allows copulas of Markov chains to be asymmetric.Finally,by applying the approximation method,analytical recurrence formulas are also derived for computing approximate distributions of both the first passage time and the occupation time of a Markov chain,and numerical results are listed to show the approximation errors.