scholarly journals Bernstein-Type Inequality for Widely Dependent Sequence and Its Application to Nonparametric Regression Models

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Aiting Shen
Keyword(s):  
Strong Consistency ◽  
Nearest Neighbor ◽  
Type Inequality ◽  
Random Variables ◽  
Independent Random Variables ◽  
Bernstein Type ◽  
Dependent Random Variables ◽  
Bernstein Type Inequality ◽  
Dependent Sequence ◽  
Fixed Design Regression

We present the Bernstein-type inequality for widely dependent random variables. By using the Bernstein-type inequality and the truncated method, we further study the strong consistency of estimator of fixed design regression model under widely dependent random variables, which generalizes the corresponding one of independent random variables. As an application, the strong consistency for the nearest neighbor estimator is obtained.

scholarly journals The Rate of Strong Consistency of the Nearest Neighbor Density Estimator for Negatively Dependent Random Variables

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Xiang Zeng ◽  
Qunying Wu
Keyword(s):  
Strong Consistency ◽  
Nearest Neighbor ◽  
Random Variables ◽  
Density Estimator ◽  
Dependent Random Variables ◽  
Negatively Dependent Random Variables ◽  
Dependent Samples

Let {Xn:n≥1} be a sequence of negatively dependent random variables. Based on (X1,…,Xn), in this paper we investigate the rate of pointwise consistency and strong consistency of the nonparametric density estimator proposed by Yu. We extend the correspondent results under the negatively dependent samples.

scholarly journals The Strong Consistency of the Estimator of Fixed-Design Regression Model under Negatively Dependent Sequences

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xuejun Wang ◽  
Meimei Ge ◽  
Shuhe Hu ◽  
Xize Wang
Keyword(s):  
Regression Model ◽  
Strong Consistency ◽  
Nearest Neighbor ◽  
Type Inequality ◽  
Fixed Design ◽  
Rosenthal Type Inequality ◽  
Fixed Design Regression

We study the strong consistency of estimator of fixed design regression model under negatively dependent sequences by using the classical Rosenthal-type inequality and the truncated method. As an application, the strong consistency for the nearest neighbor estimator is obtained.

scholarly journals Bernstein-type inequality for a class of dependent random matrices

2016 ◽  
Vol 05 (02) ◽  
pp. 1650006 ◽  
Author(s):  
Marwa Banna ◽  
Florence Merlevède ◽  
Pierre Youssef
Keyword(s):  
Random Matrices ◽  
Type Inequality ◽  
Bernstein Inequality ◽  
Mathematical Statistics ◽  
Strong Mixing ◽  
Mixing Conditions ◽  
Bernstein Type ◽  
Bernstein Type Inequality ◽  
The Matrix ◽  
The Laplace Transform

In this paper, we obtain a Bernstein-type inequality for the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality can be viewed as an extension to the matrix setting of the Bernstein-type inequality obtained by Merlevède et al. [Bernstein inequality and moderate deviations under strong mixing conditions, in High Dimensional Probability V: The Luminy Volume, Institute of Mathematical Statistics Collection, Vol. 5 (Institute of Mathematical Statistics, Beachwood, OH, 2009), pp. 273–292.] in the context of real-valued bounded random variables that are geometrically absolutely regular. The proofs rely on decoupling the Laplace transform of a sum on a Cantor-like set of random matrices.

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