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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.
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")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:
setup = list(continuity = FALSE, iid = TRUE, no_skewness = FALSE)
Bound_EE1(setup = setup, n = 1000, K4 = 9)
#> [1] 0.1626857This 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] 0.1214038This 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.
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.
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
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