Normal approximation for sums of weighted $U$-statistics – application to Kolmogorov bounds in random subgraph counting

Bernoulli ◽  
2020 ◽  
Vol 26 (1) ◽  
pp. 587-615
Author(s):  
Nicolas Privault ◽  
Grzegorz Serafin
Keyword(s):  
Normal Approximation ◽  
U Statistics ◽  
Random Subgraph

scholarly journals Normal Approximation of U-Statistics in Hilbert Space

1997 ◽  
Vol 41 (3) ◽  
pp. 405-424 ◽  
Author(s):  
Yu. V. Borovskikh ◽  
L. Madan Puri ◽  
V. V. Sazonov
Keyword(s):  
Hilbert Space ◽  
Normal Approximation ◽  
U Statistics

scholarly journals Random subgraph counts and U-statistics: multivariate normal approximation via exchangeable pairs and embedding

2010 ◽  
Vol 47 (2) ◽  
pp. 378-393 ◽  
Author(s):  
Gesine Reinert ◽  
Adrian Röllin
Keyword(s):  
Dimensional Space ◽  
Multivariate Normal ◽  
New Approach ◽  
U Statistics ◽  
Normal Approximations ◽  
Random Subgraph ◽  
Exchangeable Pairs ◽  
Higher Dimensional ◽  
Multivariate Normal Approximation ◽  
Linearity Condition

In Reinert and Röllin (2009) a new approach - called the ‘embedding method’ - was introduced, which allows us to make use of exchangeable pairs for normal and multivariate normal approximations with Stein's method in cases where the corresponding couplings do not satisfy a certain linearity condition. The key idea is to embed the problem into a higher-dimensional space in such a way that the linearity condition is then satisfied. Here we apply the embedding to U-statistics as well as to subgraph counts in random graphs.

scholarly journals Random subgraph counts and U-statistics: multivariate normal approximation via exchangeable pairs and embedding

2010 ◽  
Vol 47 (02) ◽  
pp. 378-393
Author(s):  
Gesine Reinert ◽  
Adrian Röllin
Keyword(s):  
Dimensional Space ◽  
Multivariate Normal ◽  
New Approach ◽  
U Statistics ◽  
Normal Approximations ◽  
Random Subgraph ◽  
Exchangeable Pairs ◽  
Higher Dimensional ◽  
Multivariate Normal Approximation ◽  
Linearity Condition

In Reinert and Röllin (2009) a new approach - called the ‘embedding method’ - was introduced, which allows us to make use of exchangeable pairs for normal and multivariate normal approximations with Stein's method in cases where the corresponding couplings do not satisfy a certain linearity condition. The key idea is to embed the problem into a higher-dimensional space in such a way that the linearity condition is then satisfied. Here we apply the embedding to U-statistics as well as to subgraph counts in random graphs.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

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