BoundEdgeworth

This package implements the computation of the bounds described in the article Derumigny, Girard, and Guyonvarch (2023), Explicit non-asymptotic bounds for the distance to the first-order Edgeworth expansion, Sankhya A. doi:10.1007/s13171-023-00320-y 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 \(X_1, \dots, X_n\), \(\Phi\) denotes the cumulative distribution function (cdf) of the standard Gaussian distribution, and \(\varphi\) denotes its density.

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().

Such bounds are useful because they can help to control uniformly the distance between the cdf of a normalized sum \(\textrm{Prob}(S_n \leq x)\) and its limit \(\Phi(x)\) (which is known by the central limit theorem). The second type of bound is more precise, and give a control of the uniform distance between \(\textrm{Prob}(S_n \leq x)\) and its first-order Edgeworth expansion, i.e. the limit from the central limit theorem \(\Phi(x)\) plus the next term \(\frac{\lambda_{3,n}}{6\sqrt{n}}(1-x^2) \varphi(x)\).

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 \(0\).
• 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.

Applications to testing

This package also includes the function Gauss_test_powerAnalysis(), that computes a uniformly valid power for the Gauss test that is valid over a large class of non-Gaussian distribution. This uniform validity is a consequence of the above-mentioned bounds.