Gauss_test_powerAnalysis {BoundEdgeworth}

Computation of 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 \(H_1: \mu \leq \mu_0\). 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.

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 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

There is a relation between the sample size n, the effect size eta and the power beta. This function takes as an input two of these quantities and return the third one.

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. (2021). Explicit non-asymptotic bounds for the distance to the first-order Edgeworth expansion, ArXiv preprint 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