# BoundEdgeworth

This package implements the computation of the bounds described in the article Derumigny, Girard, and Guyonvarch (2021), Explicit non-asymptotic bounds for the distance to the first-order Edgeworth expansion, arxiv:2101.05780.

## How to install

You can install the release version from the CRAN:

install.packages("BoundEdgeworth")

or the development version from GitHub:

# install.packages("remotes")
remotes::install_github("AlexisDerumigny/BoundEdgeworth")

## Available bounds

Let $$X_1, \dots, X_n$$ be $$n$$ independent centered variables, and $$S_n$$ be their normalized sum, in the sense that $S_n := \sum_{i=1}^n X_i / \text{sd} \Big(\sum_{i=1}^n X_i \Big).$

The goal of this package is to compute values of $$\delta_n > 0$$ such that bounds of the form

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

or of the form

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

are valid. Here $$\lambda_{3,n}$$ denotes the average skewness of the variables $$X_1, \dots, X_n$$.

The first type of bounds is returned by the function Bound_BE() (Berry-Esseen-type bound) and the second type (Edgeworth expansion-type bound) is returned by the function Bound_EE1().

Note that these bounds depends on the assumptions made on $$(X_1, \dots, X_n)$$ and especially on $$K4$$, the average kurtosis of the variables $$X_1, \dots, X_n$$. In all cases, they need to have finite fourth moment and to be independent. To get improved bounds, several additional assumptions can be added:

• the variables $$X_1, \dots, X_n$$ are identically distributed,
• the skewness (normalized third moment) of $$X_1, \dots, X_n$$ are all $$0$$, respectively.
• the distribution of $$X_1, \dots, X_n$$ admits a continuous component.

### Example

setup = list(continuity = FALSE, iid = TRUE, no_skewness = FALSE)

Bound_EE1(setup = setup, n = 1000, K4 = 9)
#> [1] 0.1626857

This shows that

$\sup_{x \in \mathbb{R}} \left| \textrm{Prob}(S_n \leq x) - \Phi(x) - \frac{\lambda_{3,n}}{6\sqrt{n}}(1-x^2) \varphi(x) \right| \leq 0.1626857,$

as soon as the variables $$X_1, \dots, X_{1000}$$ are i.i.d. with a kurtosis smaller than $$9$$.

Adding one more regularity assumption on the distribution of the $$X_i$$ helps to achieve a better bound:

setup = list(continuity = TRUE, iid = TRUE, no_skewness = FALSE)

Bound_EE1(setup = setup, n = 1000, K4 = 9, regularity = list(kappa = 0.99))
#> [1] 0.1214038

This shows that

$\sup_{x \in \mathbb{R}} \left| \textrm{Prob}(S_n \leq x) - \Phi(x) - \frac{\lambda_{3,n}}{6\sqrt{n}}(1-x^2) \varphi(x) \right| \leq 0.1214038,$

in this case.