# Ranking of Normality Tests: An Appraisal through Skewed Alternative Space

## Abstract

**:**

_{w}) turns out to represent the best statistics for slightly skewed alternatives and the Anderson–Darling (AD); Chen–Shapiro (CS); Shapiro–Wilk (W); and Bispo, Marques, and Pestana (BCMR) statistics are the best choices for moderately skewed alternative distributions. The maximum loss of Jarque–Bera (JB) and its robust form (RJB), in terms of deviations from the power envelope, is greater than 50%, even for large sample sizes, which makes them less attractive in testing the hypothesis of normality against the moderately skewed alternatives. On balance, all selected normality tests except T

_{w}and Daniele Coin’s COIN-test performed exceptionally well against the highly skewed alternative space.

## 1. Introduction

## 2. Stringency Framework

## 3. Tests and Alternative Distributions

## 4. Discussion of Results

#### 4.1. Slightly Skewed Alternatives

#### 4.1.1. Performance of the Moments-Based Tests

#### 4.1.2. Performance of the Regression and Correlation Tests

#### 4.1.3. Performance of the ECDF Tests

#### 4.1.4. Performance of the Special Test

#### 4.2. Moderately Skewed Alternatives

#### 4.2.1. Performance of the Moments-Based Tests

#### 4.2.2. Performance of the Regression and Correlation Tests

#### 4.2.3. Performance of the ECDF Tests

#### 4.2.4. Performance of the Other Tests

#### 4.3. Highly Skewed Alternatives

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Sr. No | Student t Distribution | Mixture Distribution | |||||||
---|---|---|---|---|---|---|---|---|---|

t1 | t2 | ||||||||

d.f | Mean | d.f | Mean | Alpha | Mean | SD | β_{1} | β_{2} | |

1 | 8 | 2.0 | 12 | 5.0 | 0.50 | 3.50 | 1.88 | −0.05 | 2.33 |

2 | 100 | 4.0 | 75 | 6.0 | 0.50 | 5.00 | 1.42 | 0.00 | 2.53 |

3 | 10 | 0.0 | .. | .. | 1.00 | 0.00 | 1.12 | 0.00 | 4.00 |

4 | 100 | −1.5 | 75 | 1.5 | 0.50 | 0.00 | 1.81 | 0.00 | 2.06 |

5 | 10 | 3.0 | 5 | 50.0 | 0.50 | 26.50 | 23.53 | 0.00 | 1.01 |

6 | 100 | −4.0 | 75 | 4.0 | 0.50 | 0.00 | 4.13 | 0.00 | 1.23 |

7 | 50 | −1.2 | 25 | 1.2 | 0.50 | 0.00 | 1.58 | 0.02 | 2.38 |

8 | 8 | 5.0 | 10 | 3.0 | 0.50 | 4.00 | 1.51 | 0.04 | 3.02 |

9 | 5 | 2.0 | 7 | 4.0 | 0.70 | 2.60 | 1.56 | 0.09 | 4.95 |

10 | 5 | 10.0 | 6 | 12.0 | 0.95 | 10.10 | 1.36 | 0.12 | 7.84 |

11 | 5 | 10.0 | 7 | 12.0 | 0.90 | 10.20 | 1.41 | 0.15 | 6.90 |

12 | 10 | 5.0 | 5 | 7.0 | 0.50 | 6.00 | 1.57 | 0.16 | 4.20 |

13 | 100 | 4.0 | 75 | 6.0 | 0.70 | 4.60 | 1.36 | 0.27 | 2.77 |

14 | 8 | 5.0 | 10 | 3.0 | 0.10 | 3.20 | 1.27 | 0.30 | 3.95 |

15 | 100 | −1.0 | 75 | 1.0 | 0.75 | −0.50 | 1.33 | 0.32 | 2.91 |

16 | 8 | 5.0 | 10 | 3.0 | 0.20 | 3.40 | 1.38 | 0.32 | 3.57 |

17 | 10 | 5.0 | 5 | 7.0 | 0.90 | 5.20 | 1.29 | 0.38 | 4.65 |

18 | 100 | −1.2 | 75 | 1.2 | 0.75 | −0.60 | 1.45 | 0.43 | 2.85 |

19 | 8 | −1.0 | 10 | 2.0 | 0.95 | −0.85 | 1.33 | 0.48 | 4.68 |

20 | 8 | −1.0 | 12 | 2.0 | 0.85 | −0.55 | 1.57 | 0.59 | 3.70 |

21 | 100 | −1.5 | 75 | 1.5 | 0.77 | −0.81 | 1.62 | 0.61 | 2.88 |

22 | 100 | −4.0 | 75 | 4.0 | 0.70 | −1.60 | 3.80 | 0.78 | 1.93 |

23 | 5 | 10.0 | 7 | 25.0 | 0.70 | 14.50 | 6.99 | 0.82 | 1.83 |

24 | 10 | 3.0 | 5 | 50.0 | 0.70 | 17.10 | 21.57 | 0.87 | 1.77 |

25 | 100 | −4.0 | 75 | 4.0 | 0.75 | −2.00 | 3.61 | 1.02 | 2.44 |

26 | 8 | −10.0 | 12 | 5.0 | 0.78 | −6.70 | 6.32 | 1.28 | 2.83 |

27 | 8 | 0.0 | 12 | 5.0 | 0.90 | 0.50 | 1.89 | 1.31 | 5.11 |

28 | 8 | 0.0 | 12 | 5.0 | 0.95 | 0.25 | 1.59 | 1.32 | 6.63 |

29 | 8 | −10.0 | 12 | 5.0 | 0.80 | −7.00 | 6.11 | 1.42 | 3.22 |

30 | 8 | −10.0 | 12 | 5.0 | 0.82 | −7.30 | 5.88 | 1.57 | 3.71 |

31 | 8 | −1.0 | 12 | 5.0 | 0.90 | −0.40 | 2.14 | 1.58 | 5.60 |

32 | 5 | 5.0 | 7 | 15.0 | 0.85 | 6.50 | 3.79 | 1.62 | 4.45 |

33 | 5 | 5.0 | 6 | 15.0 | 0.90 | 6.00 | 3.26 | 2.06 | 6.73 |

34 | 100 | −4.0 | 75 | 4.0 | 0.90 | −3.20 | 2.60 | 2.09 | 6.69 |

35 | 5 | 10.0 | 7 | 25.0 | 0.90 | 11.50 | 4.68 | 2.36 | 7.35 |

36 | 8 | −10.0 | 12 | 5.0 | 0.90 | −8.50 | 4.64 | 2.42 | 7.48 |

37 | 10 | 3.0 | 5 | 50.0 | 0.90 | 7.70 | 14.15 | 2.64 | 8.06 |

_{1}and β

_{2}represent the skewness and kurtosis of the distribution.

Distribution | Skew | Kurt | JB | RJB | Best Test |
---|---|---|---|---|---|

D(5,0.00,1.01) | 0.00 | 1.01 | 0.27 | 0.04 | 1.00 |

D(6,0.00,1.23) | 0.00 | 1.23 | 0.03 | 0.02 | 1.00 |

Beta(0.5,0.5) | 0.00 | 1.50 | 0.00 | 0.00 | 0.91 |

Beta(1,1) | 0.00 | 1.80 | 0.00 | 0.00 | 0.44 |

Tukey(2) | 0.00 | 1.80 | 0.00 | 0.00 | 0.44 |

D(4,0.00,2.06) | 0.00 | 2.06 | 0.01 | 0.00 | 0.54 |

Tukey(0.5) | 0.00 | 2.08 | 0.00 | 0.00 | 0.14 |

Beta (2,2) | 0.00 | 2.14 | 0.00 | 0.00 | 0.11 |

D(2,0.00,2.53) | 0.00 | 2.53 | 0.02 | 0.01 | 0.16 |

Tukey(5) | 0.00 | 2.90 | 0.03 | 0.07 | 0.14 |

## Appendix B

## References

- Delong, J.B.; Summers, L.H. Are Business Cycle Symmetrical in American Business Cycle: Continuity and Change; University of Chicago Press: Chicago, IL, USA, 1985; pp. 166–178. [Google Scholar]
- Henderson, A.R. Testing experimental data for univariate normality. Clin. Chim. Acta
**2006**, 366, 112–129. [Google Scholar] [CrossRef] - Blanca, M.J.; Arnau, J.; López-Montiel, D.; Bono, R.; Bendayan, R. Skewness and Kurtosis in Real Data Samples. Methodology
**2013**, 9, 78–84. [Google Scholar] [CrossRef] - Bonett, D.G.; Seier, E. A test of normality with high uniform power. Comput. Stat. Data Anal.
**2002**, 40, 435–445. [Google Scholar] [CrossRef] - D’Agostino, R.; Pearson, E.S. Tests for departure from normality. Empirical results for the distributions of b2 and b1. Biometrika
**1973**, 60, 613–622. [Google Scholar] [CrossRef] - Gel, Y.R.; Gastwirth, J.L. A robust modification of the Jarque–Bera test of normality. Econ. Lett.
**2008**, 99, 30–32. [Google Scholar] [CrossRef] - Jarque, C.M.; Bera, A.K. A Test for Normality of Observations and Regression Residuals. Int. Stat. Rev.
**1987**, 55, 163–172. [Google Scholar] [CrossRef] - Bispo, R.; Marques, T.A.; Pestana, D. Statistical power of goodness-of-fit tests based on the empirical distribution function for type_I right-censored data. J. Stat. Comput. Simul.
**2012**, 82, 21–38. [Google Scholar] [CrossRef] - Shapiro, S.S.; Francia, R.S. An approximate analysis of variance test for normality. J. Am. Stat. Assoc.
**1972**, 67, 215–216. [Google Scholar] [CrossRef] - Shapiro, S.S.; Wilk, M.B. An analysis of variance test for the exponential distribution (Complete samples). Biometrika
**1965**, 54, 591–611. [Google Scholar] [CrossRef] - Anderson, T.W.; Darling, D.A. A test of goodness of fit. J. Am. Stat. Assoc.
**1954**, 49, 765–769. [Google Scholar] [CrossRef] - Yazici, B.; Yolacan, S. A comparison of various tests of normality. J. Stat. Comput. Simul.
**2007**, 77, 175–183. [Google Scholar] [CrossRef] - Zhang, J.; Wu, Y. Likelihood-ration tests for normality. Comput. Stat. Data Anal.
**2005**, 49, 709–721. [Google Scholar] [CrossRef] - Gel, Y.R.; Miao, W.; Gastwirth, J.L. Robust directed tests of normality against heavy-tailed alternatives. Comput. Stat. Data Anal.
**2007**, 51, 2734–2746. [Google Scholar] [CrossRef] - Önder, A.Ö.; Zaman, A. Robust tests for normality of errors in regression models. Econ. Lett.
**2005**, 86, 63–68. [Google Scholar] [CrossRef] - Islam, T.U. Stringency-based ranking of normality tests. Commun. Stat. Simul. Comput.
**2017**, 46, 655–668. [Google Scholar] [CrossRef] - Lehmann, E.L.; Stein, C. On the Theory of Some Non-Parametric Hypotheses. Ann. Math. Stat.
**1949**, 20, 28–45. [Google Scholar] [CrossRef] - Shapiro, S.S.; Wilk, M.B.; Chen, H.J. A comparative study of various tests for normality. J. Am. Stat. Assoc.
**1968**, 63, 1343–1372. [Google Scholar] [CrossRef] - Thorsten, T.; Buning, H. Jarque-Bera Test and its Competitors for Testing Normality-A Power Comparison. J. Appl. Stat.
**2007**, 34, 87–105. [Google Scholar] - Yap, B.W.; Sim, C.H. Comparisons of various types of normality tests. J. Stat. Comput. Simul.
**2011**, 81, 1–15. [Google Scholar] [CrossRef] - Romao, X.; Delgado, R.; Costa, A. An empirical power comparison of univariate goodness-of-fit tests for normality. J. Stat. Comput. Simul.
**2010**, 80, 1–47. [Google Scholar] [CrossRef] - Bowman, K.O.; Shenton, L.R. Omnibus test contours for departures from normality based on b1 and b2. Biometrika
**1975**, 62, 243–250. [Google Scholar] [CrossRef] - Coin, D. A goodness-of-fit test for normality based on polynomial regression. Comput. Stat. Data Anal.
**2008**, 52, 2185–2198. [Google Scholar] [CrossRef]

Test | Class of Test |
---|---|

Za, Zc, AD, and KS | ECDF |

JB, RJB, K, and Tw | Moments |

W, Wsf, D, CS, BCMR, and COIN | Correlation and Regression |

Rsj | Special |

n = 25 | n = 50 | n = 75 | ||||||
---|---|---|---|---|---|---|---|---|

Test | Rank | Gap | Test | Rank | Gap | Test | Rank | Gap |

Tw | 1 | 24.0% | Tw | 1 | 22.9% | Tw | 1 | 31.8% |

COIN | 2 | 34.6% | Rsj | 2 | 26.4% | Rsj | 1 | 32.4% |

AD | 2 | 34.7% | AD | 3 | 38.0% | AD | 1 | 32.6% |

CS | 2 | 34.8% | CS | 3 | 39.8% | CS | 2 | 38.6% |

Rsj | 3 | 36.1% | COIN | 4 | 42.5% | W | 3 | 43.3% |

W | 3 | 37.5% | K2 | 4 | 44.8% | K2 | 3 | 44.7% |

KS | 3 | 38.1% | W | 4 | 45.5% | KS | 3 | 45.1% |

Zc | 3 | 39.0% | Zc | 5 | 48.0% | COIN | 3 | 45.2% |

BCMR | 3 | 39.9% | BCMR | 5 | 48.3% | BCMR | 3 | 46.1% |

K2 | 4 | 42.6% | KS | 5 | 49.9% | Zc | 4 | 50.5% |

Za | 4 | 43.1% | Za | 6 | 51.9% | Za | 4 | 51.4% |

Wsf | 4 | 46.5% | Wsf | 7 | 61.3% | Wsf | 5 | 56.2% |

D | 5 | 91.6% | JB | 8 | 80.5% | D | 6 | 85.3% |

JB | 6 | 97.2% | D | 9 | 90.9% | JB | 7 | 88.0% |

RJB | 6 | 98.2% | RJB | 10 | 99.5% | RJB | 8 | 92.9% |

n = 25 | n = 50 | n = 75 | ||||||
---|---|---|---|---|---|---|---|---|

Tests | Rank | Gap | Test | Rank | Gap | Test | Rank | Gap |

CS | 1 | 28.5% | AD | 1 | 25.0% | AD | 1 | 26.7% |

W | 1 | 29.0% | W | 2 | 28.3% | CS | 1 | 28.9% |

AD | 1 | 29.5% | BCMR | 2 | 28.7% | W | 1 | 29.5% |

BCMR | 1 | 29.8% | CS | 2 | 29.8% | BCMR | 1 | 31.4% |

Za | 2 | 32.8% | Wsf | 3 | 34.9% | Wsf | 2 | 35.8% |

Wsf | 2 | 33.5% | KS | 3 | 35.2% | Za | 2 | 36.2% |

Zc | 2 | 33.5% | Za | 3 | 36.5% | Zc | 2 | 38.2% |

KS | 3 | 42.2% | Zc | 3 | 38.3% | KS | 2 | 40.4% |

K2 | 4 | 46.7% | JB | 4 | 59.8% | JB | 3 | 50.6% |

D | 5 | 49.8% | RJB | 4 | 61.9% | K2 | 4 | 57.9% |

Rsj | 6 | 55.5% | K2 | 5 | 64.6% | RJB | 4 | 58.0% |

Tw | 6 | 55.7% | D | 6 | 74.6% | D | 5 | 81.3% |

JB | 7 | 59.0% | Rsj | 6 | 75.6% | Rsj | 5 | 83.9% |

RJB | 8 | 64.4% | Tw | 7 | 78.4% | Tw | 6 | 88.0% |

COIN | 9 | 68.8% | COIN | 7 | 79.8% | COIN | 6 | 88.7% |

n = 25 | n = 50 | n = 75 | ||||||
---|---|---|---|---|---|---|---|---|

Test | Rank | Gap | Test | Rank | Gap | Test | Rank | Gap |

Wsf | 1 | 8.8% | Wsf | 1 | 0.6% | RJB | 1 | 0.0% |

BCMR | 1 | 9.3% | BCMR | 1 | 0.7% | Zc | 1 | 0.0% |

W | 1 | 10.1% | Zc | 1 | 0.7% | JB | 1 | 0.0% |

CS | 1 | 10.4% | W | 1 | 0.7% | Wsf | 1 | 0.0% |

Za | 1 | 10.9% | JB | 1 | 0.7% | W | 1 | 0.1% |

Zc | 1 | 11.0% | RJB | 1 | 0.7% | CS | 1 | 0.1% |

AD | 1 | 11.9% | CS | 1 | 0.8% | D | 1 | 0.1% |

RJB | 1 | 12.5% | Za | 1 | 0.9% | BCMR | 1 | 0.1% |

JB | 1 | 13.9% | D | 1 | 1.0% | K2 | 1 | 0.1% |

D | 2 | 16.1% | K2 | 1 | 1.2% | Za | 1 | 0.1% |

K2 | 3 | 20.4% | AD | 1 | 1.3% | AD | 1 | 0.2% |

KS | 3 | 21.2% | Rsj | 1 | 2.1% | Rsj | 1 | 0.2% |

Rsj | 3 | 21.5% | KS | 1 | 3.6% | KS | 1 | 0.5% |

Tw | 4 | 46.9% | Tw | 2 | 45.3% | Tw | 2 | 42.5% |

COIN | 5 | 61.4% | COIN | 3 | 69.1% | COIN | 3 | 72.0% |

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Islam, T.U.
Ranking of Normality Tests: An Appraisal through Skewed Alternative Space. *Symmetry* **2019**, *11*, 872.
https://doi.org/10.3390/sym11070872

**AMA Style**

Islam TU.
Ranking of Normality Tests: An Appraisal through Skewed Alternative Space. *Symmetry*. 2019; 11(7):872.
https://doi.org/10.3390/sym11070872

**Chicago/Turabian Style**

Islam, Tanweer Ul.
2019. "Ranking of Normality Tests: An Appraisal through Skewed Alternative Space" *Symmetry* 11, no. 7: 872.
https://doi.org/10.3390/sym11070872