Gauss_test_powerAnalysis {BoundEdgeworth} | R Documentation |

## 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 |

`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, |

### 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
```

*BoundEdgeworth*version 0.1.2.1 Index]