Bound_EE1 {BoundEdgeworth}

Uniform bound on Edgeworth expansion

Description

This function computes a non-asymptotically 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 \[ \sup_{x \in \mathbb{R}} \left| \textrm{Prob}(S_n \leq x) - \Phi(x) - \frac{\lambda_{3,n}}{6\sqrt{n}}(1-x^2) \varphi(x) \right| \leq \delta_n,\] 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:

• continuity: if TRUE, assume that the distribution is continuous.

• iid: if TRUE, assume that the random variables are i.i.d.

• no_skewness: if TRUE, assume that the distribution is unskewed.

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 K3 will be derived from the value of K4.

lambda3

(average) skewness of the variables. If not given, an upper bound on \(abs(lambda3)\) will be derived from the value of K4.

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 K3tilde will be derived from the value of K4.

regularity

list of length up to 3 (only used in the continuity=TRUE framework) with the following components:

• C0 and p: only used in the iid=FALSE case. It corresponds to the assumption of a polynomial bound on \(f_{S_n}\): \(|f_{S_n}(u)| \leq C_0 \times u^{-p}\) for every \(u > a_n\), where \(a_n := 2 t_1^* \pi \sqrt{n} / K3tilde\).

• kappa: only used in the iid=TRUE case. Corresponds to a bound on the modulus of the characteristic function of the standardized \(X_n\). More precisely, kappa is an upper bound on \(kappa :=\) sup of modulus of \(f_{X_n / \sigma_n}(t)\) over all \(t\) such that \(|t| \geq 2 t_1^* \pi / K3tilde\).

eps

a value between 0 and 1/3 on which several terms depends. Any value of eps will give a valid upper bound but some may give tighter results than others.

verbose

if it is 0 the function is silent (no printing). Higher values of verbose give more precise information about the computation. verbose = 1 prints the values of the intermediary terms that are summed to produce the final bound. This can be useful to understand which term has the largest contribution to the bound.

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 \[ \sup_{x \in \mathbb{R}} \left| \textrm{Prob}(S_n \leq x) - \Phi(x) - \frac{\lambda_{3,n}}{6\sqrt{n}}(1-x^2) \varphi(x) \right| \leq \delta_n.\]

References

Derumigny A., Girard L., and Guyonvarch Y. (2021). Explicit non-asymptotic bounds for the distance to the first-order Edgeworth expansion, ArXiv preprint arxiv:2101.05780.

See Also

Bound_BE() for a Berry-Esseen 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)