Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation

Ukponmwan H. Nosakhare; Ajibade F. Bright

doi:doi:10.11648/j.ajtas.s.2017060501.18

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Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation

Ukponmwan H. Nosakhare, Ajibade F. Bright

Published in American Journal of Theoretical and Applied Statistics (Volume 6, Issue 5-1)

Received: 24 February 2017 Accepted: 1 March 2017 Published: 9 June 2017

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Abstract

This paper examines the sensitivity of nine normality test statistics; W/S, Jaque-Bera, Adjusted Jaque-Bera, D’Agostino, Shapiro-Wilk, Shapiro-Francia, Ryan-Joiner, Lilliefors’and Anderson Darlings test statistics, with a view to determining the effectiveness of the techniques to accurately determine whether a set of data is from normal distribution or not. Simulated data of sizes 5, 10, …, 100 is used for the study and each test is repeated 100 times for increased reliability. Data from normal distributions (N (2, 1) and N (0, 1)) and non-normal distributions (asymmetric and symmetric distributions: Weibull, Chi-Square, Cauchy and t-distributions) are simulated and tested for normality using the nine normality test statistics. To ensure uniformity of results, one statistical software is used in all the data computations to eliminate variations due to statistical software. The error rate of each of the test statistic is computed; the error rate for the normal distribution is the type I error and that for non-normal distribution is type II error. Power of test is computed for the non-normal distributions and use to determine the strength of the methods. The ranking of the nine normality test statistics in order of superiority for small sample sizes is; Adjusted Jarque-Bera, Lilliefor’s, D’Agostino, Ryan-Joiner, Shapiro-Francia, Shapiro-Wilk, W/S, Jarque-Bera and Anderson-Darling test statistics while for large sample sizes, we have; D’Agostino, Ryan-Joiner, Shapiro-Francia, Jarque-Bera, Anderson-Darling, Lilliefor’s, Adjusted Jarque-Bera, Shapiro-Wilk and W/S test statistics. Hence, only D’Agostino test statistic is classified as Uniformly Most Powerful since it is effective for both small and large sample sizes. Other methods are Locally Most Powerful. Shapiro-Francia, an improvement of Shapiro-Wilk is more sensitive for both small and large samples, hence should replace Shapiro-Wilk while the Adjsted Jarque-Bera and the Jarque-Bera should both be retained for small and large samples respectively.

Published in	American Journal of Theoretical and Applied Statistics (Volume 6, Issue 5-1) This article belongs to the Special Issue Statistical Distributions and Modeling in Applied Mathematics
DOI	10.11648/j.ajtas.s.2017060501.18
Page(s)	51-61
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Error Rate, Power-of-Test, Normality, Sensitivity, and Simulation

References

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APA Style

Ukponmwan H. Nosakhare, Ajibade F. Bright. (2017). Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation. American Journal of Theoretical and Applied Statistics, 6(5-1), 51-61. https://doi.org/10.11648/j.ajtas.s.2017060501.18

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ACS Style

Ukponmwan H. Nosakhare; Ajibade F. Bright. Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation. Am. J. Theor. Appl. Stat. 2017, 6(5-1), 51-61. doi: 10.11648/j.ajtas.s.2017060501.18

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AMA Style

Ukponmwan H. Nosakhare, Ajibade F. Bright. Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation. Am J Theor Appl Stat. 2017;6(5-1):51-61. doi: 10.11648/j.ajtas.s.2017060501.18

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@article{10.11648/j.ajtas.s.2017060501.18,
  author = {Ukponmwan H. Nosakhare and Ajibade F. Bright},
  title = {Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {6},
  number = {5-1},
  pages = {51-61},
  doi = {10.11648/j.ajtas.s.2017060501.18},
  url = {https://doi.org/10.11648/j.ajtas.s.2017060501.18},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.s.2017060501.18},
  abstract = {This paper examines the sensitivity of nine normality test statistics; W/S, Jaque-Bera, Adjusted Jaque-Bera, D’Agostino, Shapiro-Wilk, Shapiro-Francia, Ryan-Joiner, Lilliefors’and Anderson Darlings test statistics, with a view to determining the effectiveness of the techniques to accurately determine whether a set of data is from normal distribution or not. Simulated data of sizes 5, 10, …, 100 is used for the study and each test is repeated 100 times for increased reliability. Data from normal distributions (N (2, 1) and N (0, 1)) and non-normal distributions (asymmetric and symmetric distributions: Weibull, Chi-Square, Cauchy and t-distributions) are simulated and tested for normality using the nine normality test statistics. To ensure uniformity of results, one statistical software is used in all the data computations to eliminate variations due to statistical software. The error rate of each of the test statistic is computed; the error rate for the normal distribution is the type I error and that for non-normal distribution is type II error. Power of test is computed for the non-normal distributions and use to determine the strength of the methods. The ranking of the nine normality test statistics in order of superiority for small sample sizes is; Adjusted Jarque-Bera, Lilliefor’s, D’Agostino, Ryan-Joiner, Shapiro-Francia, Shapiro-Wilk, W/S, Jarque-Bera and Anderson-Darling test statistics while for large sample sizes, we have; D’Agostino, Ryan-Joiner, Shapiro-Francia, Jarque-Bera, Anderson-Darling, Lilliefor’s, Adjusted Jarque-Bera, Shapiro-Wilk and W/S test statistics. Hence, only D’Agostino test statistic is classified as Uniformly Most Powerful since it is effective for both small and large sample sizes. Other methods are Locally Most Powerful. Shapiro-Francia, an improvement of Shapiro-Wilk is more sensitive for both small and large samples, hence should replace Shapiro-Wilk while the Adjsted Jarque-Bera and the Jarque-Bera should both be retained for small and large samples respectively.},
 year = {2017}
}

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TY  - JOUR
T1  - Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation
AU  - Ukponmwan H. Nosakhare
AU  - Ajibade F. Bright
Y1  - 2017/06/09
PY  - 2017
N1  - https://doi.org/10.11648/j.ajtas.s.2017060501.18
DO  - 10.11648/j.ajtas.s.2017060501.18
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 51
EP  - 61
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.s.2017060501.18
AB  - This paper examines the sensitivity of nine normality test statistics; W/S, Jaque-Bera, Adjusted Jaque-Bera, D’Agostino, Shapiro-Wilk, Shapiro-Francia, Ryan-Joiner, Lilliefors’and Anderson Darlings test statistics, with a view to determining the effectiveness of the techniques to accurately determine whether a set of data is from normal distribution or not. Simulated data of sizes 5, 10, …, 100 is used for the study and each test is repeated 100 times for increased reliability. Data from normal distributions (N (2, 1) and N (0, 1)) and non-normal distributions (asymmetric and symmetric distributions: Weibull, Chi-Square, Cauchy and t-distributions) are simulated and tested for normality using the nine normality test statistics. To ensure uniformity of results, one statistical software is used in all the data computations to eliminate variations due to statistical software. The error rate of each of the test statistic is computed; the error rate for the normal distribution is the type I error and that for non-normal distribution is type II error. Power of test is computed for the non-normal distributions and use to determine the strength of the methods. The ranking of the nine normality test statistics in order of superiority for small sample sizes is; Adjusted Jarque-Bera, Lilliefor’s, D’Agostino, Ryan-Joiner, Shapiro-Francia, Shapiro-Wilk, W/S, Jarque-Bera and Anderson-Darling test statistics while for large sample sizes, we have; D’Agostino, Ryan-Joiner, Shapiro-Francia, Jarque-Bera, Anderson-Darling, Lilliefor’s, Adjusted Jarque-Bera, Shapiro-Wilk and W/S test statistics. Hence, only D’Agostino test statistic is classified as Uniformly Most Powerful since it is effective for both small and large sample sizes. Other methods are Locally Most Powerful. Shapiro-Francia, an improvement of Shapiro-Wilk is more sensitive for both small and large samples, hence should replace Shapiro-Wilk while the Adjsted Jarque-Bera and the Jarque-Bera should both be retained for small and large samples respectively.
VL  - 6
IS  - 5-1
ER  -

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Author Information

Ukponmwan H. Nosakhare

Department of General Studies, Mathematics and Computer Science Unit, Petroleum Training Institute, Warri, Nigeria
Ajibade F. Bright

Department of General Studies, Mathematics and Computer Science Unit, Petroleum Training Institute, Warri, Nigeria

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Plain Text BibTeX RIS

APA Style

Ukponmwan H. Nosakhare, Ajibade F. Bright. (2017). Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation. American Journal of Theoretical and Applied Statistics, 6(5-1), 51-61. https://doi.org/10.11648/j.ajtas.s.2017060501.18

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ACS Style

Ukponmwan H. Nosakhare; Ajibade F. Bright. Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation. Am. J. Theor. Appl. Stat. 2017, 6(5-1), 51-61. doi: 10.11648/j.ajtas.s.2017060501.18

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AMA Style

Ukponmwan H. Nosakhare, Ajibade F. Bright. Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation. Am J Theor Appl Stat. 2017;6(5-1):51-61. doi: 10.11648/j.ajtas.s.2017060501.18

Copy | Download

@article{10.11648/j.ajtas.s.2017060501.18,
  author = {Ukponmwan H. Nosakhare and Ajibade F. Bright},
  title = {Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {6},
  number = {5-1},
  pages = {51-61},
  doi = {10.11648/j.ajtas.s.2017060501.18},
  url = {https://doi.org/10.11648/j.ajtas.s.2017060501.18},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.s.2017060501.18},
  abstract = {This paper examines the sensitivity of nine normality test statistics; W/S, Jaque-Bera, Adjusted Jaque-Bera, D’Agostino, Shapiro-Wilk, Shapiro-Francia, Ryan-Joiner, Lilliefors’and Anderson Darlings test statistics, with a view to determining the effectiveness of the techniques to accurately determine whether a set of data is from normal distribution or not. Simulated data of sizes 5, 10, …, 100 is used for the study and each test is repeated 100 times for increased reliability. Data from normal distributions (N (2, 1) and N (0, 1)) and non-normal distributions (asymmetric and symmetric distributions: Weibull, Chi-Square, Cauchy and t-distributions) are simulated and tested for normality using the nine normality test statistics. To ensure uniformity of results, one statistical software is used in all the data computations to eliminate variations due to statistical software. The error rate of each of the test statistic is computed; the error rate for the normal distribution is the type I error and that for non-normal distribution is type II error. Power of test is computed for the non-normal distributions and use to determine the strength of the methods. The ranking of the nine normality test statistics in order of superiority for small sample sizes is; Adjusted Jarque-Bera, Lilliefor’s, D’Agostino, Ryan-Joiner, Shapiro-Francia, Shapiro-Wilk, W/S, Jarque-Bera and Anderson-Darling test statistics while for large sample sizes, we have; D’Agostino, Ryan-Joiner, Shapiro-Francia, Jarque-Bera, Anderson-Darling, Lilliefor’s, Adjusted Jarque-Bera, Shapiro-Wilk and W/S test statistics. Hence, only D’Agostino test statistic is classified as Uniformly Most Powerful since it is effective for both small and large sample sizes. Other methods are Locally Most Powerful. Shapiro-Francia, an improvement of Shapiro-Wilk is more sensitive for both small and large samples, hence should replace Shapiro-Wilk while the Adjsted Jarque-Bera and the Jarque-Bera should both be retained for small and large samples respectively.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation
AU  - Ukponmwan H. Nosakhare
AU  - Ajibade F. Bright
Y1  - 2017/06/09
PY  - 2017
N1  - https://doi.org/10.11648/j.ajtas.s.2017060501.18
DO  - 10.11648/j.ajtas.s.2017060501.18
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 51
EP  - 61
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.s.2017060501.18
AB  - This paper examines the sensitivity of nine normality test statistics; W/S, Jaque-Bera, Adjusted Jaque-Bera, D’Agostino, Shapiro-Wilk, Shapiro-Francia, Ryan-Joiner, Lilliefors’and Anderson Darlings test statistics, with a view to determining the effectiveness of the techniques to accurately determine whether a set of data is from normal distribution or not. Simulated data of sizes 5, 10, …, 100 is used for the study and each test is repeated 100 times for increased reliability. Data from normal distributions (N (2, 1) and N (0, 1)) and non-normal distributions (asymmetric and symmetric distributions: Weibull, Chi-Square, Cauchy and t-distributions) are simulated and tested for normality using the nine normality test statistics. To ensure uniformity of results, one statistical software is used in all the data computations to eliminate variations due to statistical software. The error rate of each of the test statistic is computed; the error rate for the normal distribution is the type I error and that for non-normal distribution is type II error. Power of test is computed for the non-normal distributions and use to determine the strength of the methods. The ranking of the nine normality test statistics in order of superiority for small sample sizes is; Adjusted Jarque-Bera, Lilliefor’s, D’Agostino, Ryan-Joiner, Shapiro-Francia, Shapiro-Wilk, W/S, Jarque-Bera and Anderson-Darling test statistics while for large sample sizes, we have; D’Agostino, Ryan-Joiner, Shapiro-Francia, Jarque-Bera, Anderson-Darling, Lilliefor’s, Adjusted Jarque-Bera, Shapiro-Wilk and W/S test statistics. Hence, only D’Agostino test statistic is classified as Uniformly Most Powerful since it is effective for both small and large sample sizes. Other methods are Locally Most Powerful. Shapiro-Francia, an improvement of Shapiro-Wilk is more sensitive for both small and large samples, hence should replace Shapiro-Wilk while the Adjsted Jarque-Bera and the Jarque-Bera should both be retained for small and large samples respectively.
VL  - 6
IS  - 5-1
ER  -

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