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.

You can install the release version from the CRAN:

`install.packages("BoundEdgeworth")`

or the development version from GitHub:

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

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.

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

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