Package 'BoundEdgeworth'

Title: Bound on the Error of the First-Order Edgeworth Expansion
Description: Computes uniform bounds on the distance between the cumulative distribution function of a standardized sum of random variables and its first-order Edgeworth expansion, following the article Derumigny, Girard, Guyonvarch (2023) <doi:10.1007/s13171-023-00320-y>.
Authors: Alexis Derumigny [aut, cre] , Lucas Girard [aut], Yannick Guyonvarch [aut]
Maintainer: Alexis Derumigny <[email protected]>
License: GPL-3
Version: 0.1.2.2
Built: 2024-11-03 05:36:39 UTC
Source: https://github.com/alexisderumigny/boundedgeworth

Help Index

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. (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.

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))

Uniform bound on Edgeworth expansion

Description

This function computes a non-asymptotically uniform bound on the difference between the cdf of a normalized sum of random variables and its 1st order Edgeworth expansion. It returns a valid value \(\delta_n\) such that \[ \sup_{x \in \mathbb{R}} \left| \textrm{Prob}(S_n \leq x) - \Phi(x) - \frac{\lambda_{3,n}}{6\sqrt{n}}(1-x^2) \varphi(x) \right| \leq \delta_n,\] where \(X_1, \dots, X_n\) be \(n\) independent centered variables, and \(S_n\) be their normalized sum, in the sense that \(S_n := \sum_{i=1}^n X_i / \textrm{sd}(\sum_{i=1}^n X_i)\). Here \(\lambda_{3,n}\) denotes the average skewness of the variables \(X_1, \dots, X_n\).

Usage

Bound_EE1(
  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,
  verbose = 0
)

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.
verbose

if it is 0 the function is silent (no printing). Higher values of verbose give more precise information about the computation. verbose = 1 prints the values of the intermediary terms that are summed to produce the final bound. This can be useful to understand which term has the largest contribution to the bound.

Details

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

References

Derumigny A., Girard L., and Guyonvarch Y. (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.

See Also

Bound_BE() for a Berry-Esseen bound.

Gauss_test_powerAnalysis() for a power analysis of the classical Gauss test that is uniformly valid based on this bound on the Edgeworth expansion.

Examples

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

computedBound <- Bound_EE1(
  setup = setup, n = c(150, 2000), K4 = 9,
  regularity = regularity, eps = 0.1 )

setup = list(continuity = TRUE, iid = TRUE, no_skewness = TRUE)
regularity = list(kappa = 0.99)

computedBound2 <- Bound_EE1(
  setup = setup, n = c(150, 2000), K4 = 9,
  regularity = regularity, eps = 0.1 )

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

computedBound3 <- Bound_EE1(
  setup = setup, n = c(150, 2000), K4 = 9, eps = 0.1 )

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

computedBound4 <- Bound_EE1(
  setup = setup, n = c(150, 2000), K4 = 9, eps = 0.1 )

print(computedBound)
print(computedBound2)
print(computedBound3)
print(computedBound4)

Computation of uniformly valid power and sufficient sample size for the one-sided Gauss test

Description

Let \(X_1, \dots, X_n\) be \(n\) i.i.d. variables with mean \(\mu\), variance \(\sigma^2\). Assume that we want to test the hypothesis \(H_0: \mu \leq \mu_0\) against the alternative \(H_1: \mu \leq \mu_0\). For this, we want to use the classical Gauss test, which rejects the null hypothesis if \(\sqrt{n}(\bar{X}_n - \mu)\) is larger than the quantile of the Gaussian distribution at level \(1 - \alpha\). Let \(\eta := (\mu - \mu_0) / \sigma\) be the effect size, i.e. the distance between the null and the alternative hypotheses, measured in terms of standard deviations. Let beta be the uniform power of this test: \[ beta = \inf_{H_1} \textrm{Prob}(\textrm{Rejection}),\] where the infimum is taken over all distributions under the alternative hypothesis, i.e. that have mean \(\mu = \mu_0 + \eta \sigma\), bounded kurtosis K4, and that satisfy the regularity condition kappa described below. This means that this power beta is uniformly valid over a large (infinite-dimensional) class of alternative distributions, much beyond the Gaussian family even though the test is based on the Gaussian quantile. There is a relation between the sample size n, the effect size eta and the uniform power beta of this test. This function takes as an input two of the three quantities (the sample size n, the effect size eta, and the uniform power beta) and return the other one.

Usage

Gauss_test_powerAnalysis(
  eta = NULL,
  n = NULL,
  beta = NULL,
  alpha = 0.05,
  K4 = 9,
  kappa = 0.99
)

Arguments

eta

the effect size \(\eta\) that characterizes the alternative hypothesis
n

sample size
beta

the power of detecting the effect eta using the sample size n
alpha

the level of the test
K4

the kurtosis of the \(X_i\)
kappa

Regularity parameter of the distribution of the \(X_i\) It corresponds to a bound on the modulus of the characteristic function \(f_{X_n / \sigma_n}(t)\) 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 \(|t| \geq 2 t_1^* \pi / K3tilde\).

Details

This function can be used to plan experiments, for example to know what would be a sufficient sample size to attain a fixed power against a given effect size that the researcher would like to detect.

Note that the results given by this function are formally valid only for the Gauss test (i.e., when the variance of the distribution is assumed to be known).

Value

The computed value of either the sufficient sample size n, or the minimum effect size eta, or the power beta.

References

Derumigny A., Girard L., and Guyonvarch Y. (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.

Examples

# Sufficient sample size to detect an effect of 0.5 standard deviation with probability 80%
Gauss_test_powerAnalysis(eta = 0.5, beta = 0.8)
# We can detect an effect of 0.5 standard deviations with probability 80% for n >= 548

# Power of an experiment to detect an effect of 0.5 with a sample size of n = 800
Gauss_test_powerAnalysis(eta = 0.5, n = 800)
# We can detect an effect of 0.5 standard deviations with probability 85.1% for n = 800

# Smallest effect size that can be detected with a probability of 80% for a sample size of n = 800
Gauss_test_powerAnalysis(n = 800, beta = 0.8)
# We can detect an effect of 0.114 standard deviations with probability 80% for n = 800