# An Exhaustive Power Comparison of Normality Tests

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## Abstract

**:**

## 1. Introduction

- it can provide insights about the observed process;
- parameters of model can be inferred from the characteristics of data distributions; and
- it can help in choosing more specific and computationally efficient methods.

## 2. Statistical Methods

#### 2.1. Chi-Square Test (CHI2)

#### 2.2. Kolmogorov–Smirnov (KS)

#### 2.3. Anderson–Darling (AD)

#### 2.4. Cramer–Von Mises (CVM)

#### 2.5. Shapiro–Wilk (SW)

#### 2.6. Lilliefors (LF)

#### 2.7. D’Agostino (DA)

#### 2.8. Shapiro–Francia (SF)

#### 2.9. D’Agostino–Pearson (DAP)

#### 2.10. Filliben (Filli)

#### 2.11. Martinez–Iglewicz (MI)

#### 2.12. Epps–Pulley (EP)

#### 2.13. Jarque–Bera (JB)

#### 2.14. Hosking (${H}_{1}-{H}_{3})$

#### 2.15. Cabaña–Cabaña (CC1-CC2)

#### 2.16. The Chen–Shapiro Test (ChenS)

#### 2.17. Modified Shapiro-Wilk (SWRG)

#### 2.18. Doornik–Hansen (DH)

#### 2.19. Zhang $Q$ (ZQ), ${Q}^{*}$(ZQstar), $Q-{Q}^{*}$ (ZQQstar)

#### 2.20. Barrio–Cuesta-Albertos–Matran–Rodriguez-Rodriguez (BCMR)

#### 2.21. Glen–Leemis–Barr (GLB)

#### 2.22. Bonett–Seier ${T}_{w}$ (BS)

#### 2.23. Bontemps–Meddahi (BM1–$B{M}_{3-4}$, BM2–$B{M}_{3-6}$)

#### 2.24. Zhang–Wu (ZW1–${Z}_{C}$, ZW2–${Z}_{A}$)

#### 2.25. Gel–Miao–Gastwirth (GMG)

#### 2.26. Robust Jarque–Bera (RJB)

#### 2.27. Coin ${\beta}_{3}^{2}$

#### 2.28. Brys–Hubert–Struyf ${T}_{MC-LR}$ (BHS)

#### 2.29. Brys–Hubert–Struyf–Bonett–Seier ${T}_{MC-LR}{T}_{w}$ (BHSBS)

#### 2.30. Desgagné–Lafaye de Micheaux–Leblanc ${R}_{n}$ (DLDMLRn), ${X}_{APD}^{a}$ (DLDMXAPD), ${Z}_{EPD}^{a}$ (DLDMZEPD)

#### 2.31. N-Metric

## 3. The Power of Test

- The distribution of the analyzed data ${x}_{1},{x}_{2},\text{}\dots ,{x}_{n}$ is formed.
- Statistics of the compatibility hypothesis test criteria are calculated. If the obtain value of statistic is greater than the corresponding critical value ($\alpha =0.05$ is used), then hypothesis ${H}_{0}$ is rejected.
- Steps 1 and 2 are repeated for $k$ (in our experiments, $k=1,000,000$) times.
- The power of a test is calculated as $count/k$, where $count$ is the number of false hypotheses rejections.

## 4. Statistical Distributions

#### 4.1. Symmetric Distributions

- three cases of the $Beta\left(a,b\right)$ distribution$\u2014Beta\left(0.5;0.5\right),Beta\left(1;1\right),$and $Beta\left(2;2\right)$, where $a$ and $b$ are the shape parameters;
- three cases of the $Cauchy\left(t,s\right)$ distribution—$Cauchy\left(0;0.5\right),Cauchy\left(0;1\right)$, and $Cauchy\left(0;2\right)$, where$t$ and$s$ are the location and scale parameters;
- one case of the $Laplace\left(t,s\right)$distribution$\u2014Laplace\left(0;1\right)$, where $t$ and$s$ are the location and scale parameters;
- one case of the $Logistic\left(t,s\right)$ distribution$\u2014Logistic\left(2;2\right)$, where $t$ and$s$ are the location and scale parameters;
- four cases of the $t-Student\left(\nu \right)$ distribution$\u2014t\left(1\right),t\left(2\right),t\left(4\right),$ and $t\left(10\right)$, where $\nu $ is the number of degrees of freedom;
- five cases of the $Tukey\left(\lambda \right)$ distribution$\u2014Tukey\left(0.14\right),Tukey\left(0.5\right),Tukey\left(2\right),Tukey\left(5\right),$ and $Tukey\left(10\right)$, where $\lambda $ is the shape parameter; and
- one case of the standard normal $N\left(0;1\right)$ distribution.

#### 4.2. Asymmetric Distributions

- four cases of the $Beta\left(a,b\right)$ distribution$\u2014Beta\left(2;1\right),\text{}Beta\left(2;5\right),\text{}Beta\left(4;0.5\right),$ and $Beta\left(5;1\right)$;
- four cases of the $Chi$-$squared\left(\nu \right)$ distribution$\u2014{\chi}^{2}\left(1\right),\text{}{\chi}^{2}\left(2\right),\text{}{\chi}^{2}\left(4\right)$, and$\text{}{\chi}^{2}\left(10\right)$, where $\nu $ is the number of degrees of freedom;
- six cases of the $Gamma\left(a,\text{}b\right)$ distribution—$Gamma\left(2;2\right),\text{}Gamma\left(3;2\right),\text{}Gamma\left(5;1\right),\text{}Gamma\left(9;1\right),\text{}Gamma\left(15;1\right)$, and $Gamma\left(100;1\right)$, where $a$ and $b$ are the shape and scale parameters;
- one case of the $Gumbel\left(t,\text{}s\right)$ distribution$\u2014Gumbel\left(1;2\right)$, where $t$ and$\text{}s$ are the location and scale parameters;
- one case of the $Lognormal\left(t,\text{}s\right)$ distribution$\u2014LN\left(0;1\right)$, where $t$ and$\text{}s$ are the location and scale parameters; and
- four cases of the $Weibull\left(a,\text{}b\right)$ distribution$\u2014Weibull\left(0.5;1\right),\text{}Weibull\left(1;2\right),\text{}Weibull\left(2;3.4\right),$ and $Weibull\left(3;4\right)$, where $a$ and $b$ are the shape and scale parameters.

#### 4.3. Modified Normal Distributions

- six cases of the standard normal distribution truncated at $a$ and $b$ $Trunc\left(a;\text{}b\right)\u2014Trunc\left(-1;1\right),\text{}Trunc\left(-2;2\right),\text{}Trunc\left(-3;3\right),\text{}Trunc\left(-2;1\right),\text{}Trunc\left(-3;1\right),\text{}$and $Trunc\left(-3;2\right),$ which are referred to as NORMAL1;
- nine cases of a location-contaminated standard normal distribution, hereon termed $LoConN\left(p;\text{}a\right)\u2014LoConN\left(0.3;1\right),\text{}LoConN\left(0.4;1\right),\text{}LoConN\left(0.5;1\right),\text{}LoConN\left(0.3;3\right),\text{}$$LoConN\left(0.4;3\right),\text{}LoConN\left(0.5;3\right),\text{}LoConN\left(0.3;5\right),\text{}LoConN\left(0.4;5\right),\text{}$and $LoConN\left(0.5;5\right)$, which are referred to as NORMAL2;
- nine cases of a scale-contaminated standard normal distribution, hereon termed $ScConN\left(p;\text{}b\right)\u2014ScConN\left(0.05;0.25\right),\text{}ScConN\left(0.10;0.25\right),\text{}ScConN\left(0.20;0.25\right),\text{}$$ScConN\left(0.05;2\right),\text{}ScConN\left(0.10;2\right),\text{}ScConN\left(0.20;2\right),\text{}ScConN\left(0.05;4\right),\text{}ScConN\left(0.10;4\right),$and $ScConN\left(0.20;4\right)$, which are referred to as NORMAL3; and
- twelve cases of a mixture of normal distributions, hereon termed $MixN\left(p;\text{}a;\text{}b\right)\u2014MixN\left(0.3;1;0.25\right),\text{}MixN\left(0.4;1;0.25\right),\text{}MixN\left(0.5;1;0.25\right),\text{}MixN\left(0.3;3;0.25\right),\text{}$$MixN\left(0.4;3;0.25\right),\text{}MixN\left(0.5;3;0.25\right),\text{}MixN\left(0.3;1;4\right),\text{}MixN\left(0.4;1;4\right),\text{}MixN\left(0.5;1;4\right),$$\text{}MixN\left(0.3;3;4\right),\text{}MixN\left(0.4;3;4\right),$and $MixN\left(0.5;3;4\right)$, which are referred to as NORMAL4.

## 5. Simulation Study and Discussion

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Average empirical power results, for all sample sizes, for the groups of symmetric distributions of five powerful goodness-of-fit tests.

**Figure A2.**Average empirical power results for the examined sample sizes for the groups of asymmetric distributions of five powerful goodness-of-fit tests.

**Figure A3.**Average empirical power results for the examined sample sizes for the groups of the modified normal distributions of five powerful goodness-of-fit tests.

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**Figure 1.**Plot of out kernel function $K\left(x\right)$ with experimentally chosen optimal parameters $a=0.95,b=0.25,\text{}\mathrm{and}c=1$.

**Figure 4.**Average empirical power results, for the examined sample sizes, for the groups of symmetric, asymmetric, and modified normal distributions of five powerful goodness-of-fit tests.

Sample Size | |||||||
---|---|---|---|---|---|---|---|

32 | 64 | 128 | 256 | 512 | 1024 | ||

Tests | AD | 0.714 | 0.799 | 0.863 | 0.909 | 0.939 | 0.955 |

BCMR | 0.718 | 0.809 | 0.875 | 0.920 | 0.947 | 0.947 | |

BHS | 0.431 | 0.551 | 0.663 | 0.752 | 0.818 | 0.868 | |

BHSBS | 0.680 | 0.778 | 0.783 | 0.903 | 0.938 | 0.959 | |

BM2 | 0.726 | 0.835 | 0.905 | 0.945 | 0.965 | 0.974 | |

BS | 0.717 | 0.810 | 0.877 | 0.920 | 0.947 | 0.961 | |

CC2 | 0.712 | 0.805 | 0.873 | 0.920 | 0.949 | 0.936 | |

CHI2 | 0.663 | 0.778 | 0.842 | 0.884 | 0.941 | 0.945 | |

CVM | 0.591 | 0.733 | 0.805 | 0.855 | 0.919 | 0.949 | |

ChenS | 0.729 | 0.806 | 0.871 | 0.915 | 0.943 | 0.960 | |

Coin | 0.735 | 0.830 | 0.891 | 0.930 | 0.952 | 0.963 | |

DA | 0.266 | 0.295 | 0.314 | 0.319 | 0.315 | 0.311 | |

DAP | 0.723 | 0.820 | 0.883 | 0.924 | 0.948 | 0.962 | |

DH | 0.709 | 0.805 | 0.877 | 0.925 | 0.950 | 0.963 | |

DLDMZEPD | 0.730 | 0.826 | 0.889 | 0.929 | 0.952 | 0.963 | |

EP | 0.706 | 0.828 | 0.974 | 0.910 | 0.946 | 0.959 | |

Filli | 0.712 | 0.805 | 0.875 | 0.922 | 0.949 | 0.962 | |

GG | 0.658 | 0.760 | 0.850 | 0.915 | 0.949 | 0.962 | |

GLB | 0.712 | 0.798 | 0.863 | 0.909 | 0.943 | 0.918 | |

GMG | 0.787 | 0.862 | 0.914 | 0.946 | 0.965 | 0.975 | |

H1 | 0.799 | 0.862 | 0.852 | 0.999 | 0.999 | 0.999 | |

JB | 0.643 | 0.762 | 0.856 | 0.918 | 0.949 | 0.963 | |

KS | 0.585 | 0.723 | 0.789 | 0.836 | 0.905 | 0.939 | |

Lillie | 0.669 | 0.758 | 0.828 | 0.883 | 0.921 | 0.947 | |

MI | 0.632 | 0.676 | 0.705 | 0.724 | 0.736 | 0.745 | |

N-metric | 0.245 | 0.585 | 0.971 | 0.999 | 0.999 | 0.999 | |

SF | 0.715 | 0.807 | 0.876 | 0.923 | 0.949 | 0.962 | |

SW | 0.718 | 0.808 | 0.874 | 0.919 | 0.946 | 0.962 | |

SWRG | 0.694 | 0.775 | 0.834 | 0.882 | 0.916 | 0.946 | |

ZQstar | 0.513 | 0.576 | 0.630 | 0.669 | 0.697 | 0.718 | |

ZW2 | 0.715 | 0.806 | 0.869 | 0.912 | 0.939 | 0.957 |

Sample Size | |||||||
---|---|---|---|---|---|---|---|

32 | 64 | 128 | 256 | 512 | 1024 | ||

Tests | AD | 0.729 | 0.835 | 0.908 | 0.949 | 0.969 | 0.984 |

BCMR | 0.749 | 0.856 | 0.924 | 0.971 | 0.995 | 0.991 | |

BHS | 0.529 | 0.664 | 0.769 | 0.855 | 0.915 | 0.950 | |

BHSBS | 0.538 | 0.652 | 0.747 | 0.914 | 0.902 | 0.944 | |

BM2 | 0.737 | 0.859 | 0.931 | 0.965 | 0.981 | 0.993 | |

BS | 0.506 | 0.588 | 0.665 | 0.738 | 0.805 | 0.859 | |

CC2 | 0.579 | 0.682 | 0.777 | 0.853 | 0.938 | 0.956 | |

CHI2 | 0.645 | 0.799 | 0.881 | 0.934 | 0.965 | 0.980 | |

CVM | 0.594 | 0.755 | 0.836 | 0.887 | 0.935 | 0.957 | |

ChenS | 0.756 | 0.862 | 0.928 | 0.961 | 0.978 | 0.991 | |

Coin | 0.480 | 0.556 | 0.630 | 0.700 | 0.769 | 0.916 | |

DA | 0.237 | 0.223 | 0.209 | 0.198 | 0.191 | 0.192 | |

DAP | 0.705 | 0.826 | 0.910 | 0.955 | 0.977 | 0.990 | |

DH | 0.724 | 0.845 | 0.921 | 0.957 | 0.977 | 0.991 | |

DLDMXAPD | 0.726 | 0.843 | 0.918 | 0.955 | 0.975 | 0.989 | |

EP | 0.753 | 0.846 | 0.913 | 0.967 | 0.975 | 0.993 | |

Filli | 0.732 | 0.842 | 0.915 | 0.953 | 0.974 | 0.991 | |

GG | 0.672 | 0.805 | 0.898 | 0.949 | 0.973 | 0.988 | |

GLB | 0.725 | 0.831 | 0.905 | 0.987 | 0.970 | 0.984 | |

GMG | 0.683 | 0.751 | 0.809 | 0.859 | 0.901 | 0.932 | |

H1 | 0.816 | 0.896 | 0.896 | 0.999 | 0.999 | 0.999 | |

JB | 0.662 | 0.808 | 0.904 | 0.953 | 0.975 | 0.989 | |

KS | 0.582 | 0.736 | 0.810 | 0.863 | 0.921 | 0.945 | |

Lillie | 0.671 | 0.786 | 0.872 | 0.929 | 0.959 | 0.976 | |

MI | 0.644 | 0.731 | 0.798 | 0.843 | 0.872 | 0.913 | |

N-metric | 0.464 | 0.761 | 0.990 | 0.999 | 0.999 | 0.999 | |

SF | 0.736 | 0.846 | 0.918 | 0.955 | 0.975 | 0.989 | |

SW | 0.753 | 0.859 | 0.925 | 0.959 | 0.977 | 0.991 | |

SWRG | 0.758 | 0.861 | 0.927 | 0.960 | 0.977 | 0.999 | |

ZQstar | 0.570 | 0.639 | 0.693 | 0.732 | 0.761 | 0.748 | |

ZW2 | 0.764 | 0.870 | 0.932 | 0.962 | 0.980 | 0.997 |

Sample Size | |||||||
---|---|---|---|---|---|---|---|

32 | 64 | 128 | 256 | 512 | 1024 | ||

Tests | AD | 0.662 | 0.756 | 0.825 | 0.872 | 0.905 | 0.931 |

BCMR | 0.652 | 0.756 | 0.831 | 0.880 | 0.913 | 0.935 | |

BHS | 0.463 | 0.585 | 0.676 | 0.744 | 0.796 | 0.834 | |

BHSBS | 0.568 | 0.701 | 0.787 | 0.847 | 0.890 | 0.918 | |

BM2 | 0.641 | 0.770 | 0.854 | 0.904 | 0.934 | 0.953 | |

BS | 0.587 | 0.688 | 0.770 | 0.833 | 0.881 | 0.916 | |

CC2 | 0.576 | 0.675 | 0.763 | 0.833 | 0.887 | 0.923 | |

CHI2 | 0.566 | 0.728 | 0.808 | 0.866 | 0.914 | 0.939 | |

CVM | 0.557 | 0.708 | 0.779 | 0.833 | 0.897 | 0.930 | |

ChenS | 0.656 | 0.759 | 0.833 | 0.882 | 0.915 | 0.937 | |

Coin | 0.579 | 0.691 | 0.781 | 0.846 | 0.889 | 0.918 | |

DA | 0.314 | 0.342 | 0.367 | 0.388 | 0.405 | 0.418 | |

DAP | 0.617 | 0.733 | 0.818 | 0.872 | 0.906 | 0.930 | |

DH | 0.617 | 0.727 | 0.815 | 0.872 | 0.907 | 0.930 | |

DLDMXAPD | 0.651 | 0.754 | 0.831 | 0.879 | 0.912 | 0.935 | |

EP | 0.640 | 0.748 | 0.819 | 0.865 | 0.906 | 0.931 | |

Filli | 0.637 | 0.743 | 0.823 | 0.877 | 0.911 | 0.933 | |

GG | 0.529 | 0.657 | 0.775 | 0.860 | 0.906 | 0.932 | |

GLB | 0.659 | 0.755 | 0.823 | 0.870 | 0.903 | 0.930 | |

GMG | 0.688 | 0.771 | 0.836 | 0.883 | 0.917 | 0.942 | |

H1 | 0.743 | 0.816 | 0.799 | 0.999 | 0.999 | 0.999 | |

JB | 0.515 | 0.662 | 0.783 | 0.861 | 0.904 | 0.930 | |

KS | 0.564 | 0.710 | 0.772 | 0.825 | 0.893 | 0.924 | |

Lillie | 0.626 | 0.724 | 0.796 | 0.850 | 0.889 | 0.917 | |

MI | 0.494 | 0.536 | 0.563 | 0.578 | 0.585 | 0.590 | |

N-metric | 0.243 | 0.582 | 0.972 | 0.999 | 0.999 | 0.999 | |

SF | 0.642 | 0.747 | 0.826 | 0.879 | 0.912 | 0.934 | |

SW | 0.654 | 0.758 | 0.832 | 0.882 | 0.915 | 0.937 | |

SWRG | 0.643 | 0.746 | 0.818 | 0.864 | 0.901 | 0.931 | |

ZQstar | 0.394 | 0.423 | 0.450 | 0.472 | 0.487 | 0.498 | |

ZW2 | 0.640 | 0.749 | 0.826 | 0.876 | 0.907 | 0.931 |

Nr. | Distribution | Groups of Distributions | $\mathbf{Minimal}\text{}\mathbf{Sample}\text{}\mathbf{Size}\text{}\left(\mathit{n}\right)$ |
---|---|---|---|

1. | Standard normal | Symmetric | 46 |

2. | Beta | Symmetric | 88 |

3. | Cauchy | Symmetric | 257 |

4. | Laplace | Symmetric | 117 |

5. | Logistic | Symmetric | 71 |

6. | Student | Symmetric | 96 |

7. | Beta | Asymmetric | 108 |

8. | Chi-square | Asymmetric | 123 |

9. | Gamma | Asymmetric | <32 |

10. | Gumbel | Asymmetric | 125 |

11. | Lognormal | Asymmetric | 255 |

12. | Weibull | Asymmetric | 65 |

13. | Normal1 | Modified normal | 70 |

14. | Normal2 | Modified normal | 93 |

15. | Normal3 | Modified normal | 72 |

16. | Normal4 | Modified normal | 117 |

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Arnastauskaitė, J.; Ruzgas, T.; Bražėnas, M.
An Exhaustive Power Comparison of Normality Tests. *Mathematics* **2021**, *9*, 788.
https://doi.org/10.3390/math9070788

**AMA Style**

Arnastauskaitė J, Ruzgas T, Bražėnas M.
An Exhaustive Power Comparison of Normality Tests. *Mathematics*. 2021; 9(7):788.
https://doi.org/10.3390/math9070788

**Chicago/Turabian Style**

Arnastauskaitė, Jurgita, Tomas Ruzgas, and Mindaugas Bražėnas.
2021. "An Exhaustive Power Comparison of Normality Tests" *Mathematics* 9, no. 7: 788.
https://doi.org/10.3390/math9070788