Bound_EE1 {BoundEdgeworth}  R Documentation 
Uniform bound on Edgeworth expansion
Description
This function computes a nonasymptotically uniform bound on the difference between the cdf of a normalized sum of random variables and its 1st order Edgeworth expansion. It returns a valid value \(\delta_n\) such that \[ \sup_{x \in \mathbb{R}} \left \textrm{Prob}(S_n \leq x)  \Phi(x)  \frac{\lambda_{3,n}}{6\sqrt{n}}(1x^2) \varphi(x) \right \leq \delta_n,\] where \(X_1, \dots, X_n\) be \(n\) independent centered variables, and \(S_n\) be their normalized sum, in the sense that \(S_n := \sum_{i=1}^n X_i / \textrm{sd}(\sum_{i=1}^n X_i)\). Here \(\lambda_{3,n}\) denotes the average skewness of the variables \(X_1, \dots, X_n\).
Usage
Bound_EE1(
setup = list(continuity = FALSE, iid = FALSE, no_skewness = FALSE),
n,
K4 = 9,
K3 = NULL,
lambda3 = NULL,
K3tilde = NULL,
regularity = list(C0 = 1, p = 2),
eps = 0.1,
verbose = 0
)
Arguments
setup 
logical vector of size 3 made up of the following components:

n 
sample size ( = number of random variables that appear in the sum). 
K4 
bound on the 4th normalized moment of the random variables. We advise to use K4 = 9 as a general case which covers most “usual” distributions. 
K3 
bound on the 3rd normalized moment.
If not given, an upper bound on 
lambda3 
(average) skewness of the variables.
If not given, an upper bound on \(abs(lambda3)\)
will be derived from the value of 
K3tilde 
value of
\[
K_{3,n} + \frac{1}{n}\sum_{i=1}^n
\mathbb{E}X_i \sigma_{X_i}^2 / \overline{B}_n^3\]
where \(\overline{B}_n := \sqrt{(1/n) \sum_{i=1}^n E[X_i^2]}\).
If not given, an upper bound on 
regularity 
list of length up to 3
(only used in the

eps 
a value between 0 and 1/3 on which several terms depends.
Any value of 
verbose 
if it is 
Details
Note that the variables \(X_1, \dots, X_n\) must be independent
but may have different distributions (if setup$iid = FALSE
).
Value
A vector of the same size as n
with values \(\delta_n\)
such that
\[
\sup_{x \in \mathbb{R}}
\left \textrm{Prob}(S_n \leq x)  \Phi(x)
 \frac{\lambda_{3,n}}{6\sqrt{n}}(1x^2) \varphi(x) \right
\leq \delta_n.\]
References
Derumigny A., Girard L., and Guyonvarch Y. (2021). Explicit nonasymptotic bounds for the distance to the firstorder Edgeworth expansion, ArXiv preprint arxiv:2101.05780.
See Also
Bound_BE()
for a BerryEsseen bound.
Gauss_test_powerAnalysis()
for a power analysis of the classical
Gauss test that is uniformly valid based on this bound on the Edgeworth
expansion.
Examples
setup = list(continuity = TRUE, iid = FALSE, no_skewness = TRUE)
regularity = list(C0 = 1, p = 2)
computedBound < Bound_EE1(
setup = setup, n = c(150, 2000), K4 = 9,
regularity = regularity, eps = 0.1 )
setup = list(continuity = TRUE, iid = TRUE, no_skewness = TRUE)
regularity = list(kappa = 0.99)
computedBound2 < Bound_EE1(
setup = setup, n = c(150, 2000), K4 = 9,
regularity = regularity, eps = 0.1 )
setup = list(continuity = FALSE, iid = FALSE, no_skewness = TRUE)
computedBound3 < Bound_EE1(
setup = setup, n = c(150, 2000), K4 = 9, eps = 0.1 )
setup = list(continuity = FALSE, iid = TRUE, no_skewness = TRUE)
computedBound4 < Bound_EE1(
setup = setup, n = c(150, 2000), K4 = 9, eps = 0.1 )
print(computedBound)
print(computedBound2)
print(computedBound3)
print(computedBound4)