Bound_BE {BoundEdgeworth}

Compute a Berry-Esseen-type bound

Description

This function returns a valid value \(\delta_n\) for the bound \[\sup_{x \in \mathbb{R}} \left| \textrm{Prob}(S_n \leq x) - \Phi(x) \right| \leq \delta_n, \]

Usage

Bound_BE(
  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
)

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.

Details

where \(X_1, \dots, X_n\) be \(n\) independent centered variables, and \(S_n\) be their normalized sum, in the sense that \(S_n := \sum_{i=1}^n X_i / \textrm{sd}(\sum_{i=1}^n X_i)\). This bounds follows from the triangular inequality and the bound on the difference between a cdf and its 1st-order Edgeworth Expansion.

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 \[\sup_{x \in \mathbb{R}} \left| \textrm{Prob}(S_n \leq x) - \Phi(x) \right| \leq \delta_n. \]

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_EE1() for a bound on the distance to the first-order Edgeworth expansion.

Examples

setup = list(continuity = FALSE, iid = FALSE, no_skewness = FALSE)
regularity = list(C0 = 1, p = 2, kappa = 0.99)

computedBound_EE1 <- Bound_EE1(
  setup = setup, n = 150, K4 = 9,
  regularity = regularity, eps = 0.1 )

computedBound_BE <- Bound_BE(
  setup = setup, n = 150, K4 = 9,
  regularity = regularity, eps = 0.1 )

print(c(computedBound_EE1, computedBound_BE))