Monte Carlo Simulation and Derivation of Chi-Square Statistics

Tofik Mussa Reshid

doi:doi:10.11648/j.ajtas.20231203.13

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Monte Carlo Simulation and Derivation of Chi-Square Statistics

Tofik Mussa Reshid

Published in American Journal of Theoretical and Applied Statistics (Volume 12, Issue 3)

Received: 6 April 2023 Accepted: 22 May 2023 Published: 27 June 2023

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Abstract

Computer simulation has become an important tool in teaching statistics. Teaching using computer simulation would enhance the understanding of the concept using visual illustrations. This paper describes how to use simulation in R-programming language to perform a chi-square test. We try to show the distribution of most commonly used chi-square statistics we often found in statistical methods in both derivation and simulation. In statistical methods in such cases as test of independency, test of goodness of fit, test of significance, log likelihood ratio test, significance test and model selection we use chi-square statistic. The approach of the paper will enhance the students’ and researchers’ ability to understand simulation and sampling distribution. The paper contains an expository discussion of chi-square statistic, its derivation and distribution and its derivatives such as t-distribution and F-distribution. We consider two chi-squares, the empirical chi-square statistic and the theoretical chi-square distribution. The empirical distribution of chi-square statistic agrees closely with the theoretical chi-square distribution for large simulations, only the empirical distribution near to zero has lower density compared to the theoretical one for one degree of freedom. This is because the theoretical chi-square distribution at 1 degree of freedom has infinite density near to zero, but for any number of simulation the empirical distribution has finite density near to zero. Chi-square itself turns to normal distribution as the degree of freedom is large.

Published in	American Journal of Theoretical and Applied Statistics (Volume 12, Issue 3)
DOI	10.11648/j.ajtas.20231203.13
Page(s)	51-65
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), 2024. Published by Science Publishing Group

Keywords

Chi-Square Distribution, Chi-Square Statistic, Likelihood Ratio, T-test, F-test, Simulation

References

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[2]	Cassela George et al, 2001: Statistical Inference (2^nd edition) Duxbury. ISBN 0-534-24312-6; pp. 102.
[3]	Sorana D. Bolboala, et al, 2011: Pearson-Fisher Chi-square statistic Revisited; information journal, 528-545.
[4]	William G. Cochran, 1952, the chi-square test of goodness of fit, The Anals Of Mathematical Statistics, vol. 23 no. 3 315-345.
[5]	Teshome Hailemeskel Abebe, 2019: The derivation and choice of Approprate test Statistic (Z, T, F and Chi-square tests) in Research Methodology Mathematics letters Vol. 5 no. 3 2019, pp. 33-40. doi: 10.11648/j.ml.20190503.11.
[6]	Reshid TM, 2020: Sampling distribution and simulation in R: International journal of statistics And Mathematics, 7 (2): 154-163.
[7]	Simon J. A. Malham 2008: chi-square simulation of the ICR process and the Heston Model.
[8]	Pragyasur et al 2017: the Likelihood ratio test in High Dimensional logistic regression in asymptotically a rescaled chi-square, Maths. ST. Arxiv.
[9]	R. A Fisher, M. A. 1925: Application of “Students’ distribution, Rothamsted experimental station, Metron, 90-104.
[10]	Xuemao Zhang, et al, 2019: Using r as a simulation tool in teaching Introductory statistics, international electronic journal of mathematics education.
[11]	David M. Lane (2015): Simulations of the Sampling Distribution of the Mean Do Not Necessarily Mislead and can facilitate learning, journal of statistics education, Volume 23, Number 2.
[12]	Ana Elisa Castro Sotos et al, 2007: Students’ misconceptions of statistical inference: A review of the empirical evidence from research on statistics education, Elsevier, educational research review 98–113.
[13]	Moore, D. S. (1997): New Pedagogy and New Content: The case of Statistics. International Statistical Review 65, 123–65.
[14]	“Student” (William Seally Gosset, 1908): The probable error of a mean, Biometrika 6 (1): 1–25. doi:10.1093/biomet/6.1.1.

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Tofik Mussa Reshid. (2023). Monte Carlo Simulation and Derivation of Chi-Square Statistics. American Journal of Theoretical and Applied Statistics, 12(3), 51-65. https://doi.org/10.11648/j.ajtas.20231203.13

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ACS Style

Tofik Mussa Reshid. Monte Carlo Simulation and Derivation of Chi-Square Statistics. Am. J. Theor. Appl. Stat. 2023, 12(3), 51-65. doi: 10.11648/j.ajtas.20231203.13

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AMA Style

Tofik Mussa Reshid. Monte Carlo Simulation and Derivation of Chi-Square Statistics. Am J Theor Appl Stat. 2023;12(3):51-65. doi: 10.11648/j.ajtas.20231203.13

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@article{10.11648/j.ajtas.20231203.13,
  author = {Tofik Mussa Reshid},
  title = {Monte Carlo Simulation and Derivation of Chi-Square Statistics},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {12},
  number = {3},
  pages = {51-65},
  doi = {10.11648/j.ajtas.20231203.13},
  url = {https://doi.org/10.11648/j.ajtas.20231203.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20231203.13},
  abstract = {Computer simulation has become an important tool in teaching statistics. Teaching using computer simulation would enhance the understanding of the concept using visual illustrations. This paper describes how to use simulation in R-programming language to perform a chi-square test. We try to show the distribution of most commonly used chi-square statistics we often found in statistical methods in both derivation and simulation. In statistical methods in such cases as test of independency, test of goodness of fit, test of significance, log likelihood ratio test, significance test and model selection we use chi-square statistic. The approach of the paper will enhance the students’ and researchers’ ability to understand simulation and sampling distribution. The paper contains an expository discussion of chi-square statistic, its derivation and distribution and its derivatives such as t-distribution and F-distribution. We consider two chi-squares, the empirical chi-square statistic and the theoretical chi-square distribution. The empirical distribution of chi-square statistic agrees closely with the theoretical chi-square distribution for large simulations, only the empirical distribution near to zero has lower density compared to the theoretical one for one degree of freedom. This is because the theoretical chi-square distribution at 1 degree of freedom has infinite density near to zero, but for any number of simulation the empirical distribution has finite density near to zero. Chi-square itself turns to normal distribution as the degree of freedom is large.},
 year = {2023}
}

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TY - JOUR
T1 - Monte Carlo Simulation and Derivation of Chi-Square Statistics
AU - Tofik Mussa Reshid
Y1 - 2023/06/27
PY - 2023
N1 - https://doi.org/10.11648/j.ajtas.20231203.13
DO - 10.11648/j.ajtas.20231203.13
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 - 65
PB - Science Publishing Group
SN - 2326-9006
UR - https://doi.org/10.11648/j.ajtas.20231203.13
AB - Computer simulation has become an important tool in teaching statistics. Teaching using computer simulation would enhance the understanding of the concept using visual illustrations. This paper describes how to use simulation in R-programming language to perform a chi-square test. We try to show the distribution of most commonly used chi-square statistics we often found in statistical methods in both derivation and simulation. In statistical methods in such cases as test of independency, test of goodness of fit, test of significance, log likelihood ratio test, significance test and model selection we use chi-square statistic. The approach of the paper will enhance the students’ and researchers’ ability to understand simulation and sampling distribution. The paper contains an expository discussion of chi-square statistic, its derivation and distribution and its derivatives such as t-distribution and F-distribution. We consider two chi-squares, the empirical chi-square statistic and the theoretical chi-square distribution. The empirical distribution of chi-square statistic agrees closely with the theoretical chi-square distribution for large simulations, only the empirical distribution near to zero has lower density compared to the theoretical one for one degree of freedom. This is because the theoretical chi-square distribution at 1 degree of freedom has infinite density near to zero, but for any number of simulation the empirical distribution has finite density near to zero. Chi-square itself turns to normal distribution as the degree of freedom is large.
VL - 12
IS - 3
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Author Information

Tofik Mussa Reshid

Department of Statistics, Werabe University, Werabe, Ethiopia

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APA Style

Tofik Mussa Reshid. (2023). Monte Carlo Simulation and Derivation of Chi-Square Statistics. American Journal of Theoretical and Applied Statistics, 12(3), 51-65. https://doi.org/10.11648/j.ajtas.20231203.13

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ACS Style

Tofik Mussa Reshid. Monte Carlo Simulation and Derivation of Chi-Square Statistics. Am. J. Theor. Appl. Stat. 2023, 12(3), 51-65. doi: 10.11648/j.ajtas.20231203.13

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AMA Style

Tofik Mussa Reshid. Monte Carlo Simulation and Derivation of Chi-Square Statistics. Am J Theor Appl Stat. 2023;12(3):51-65. doi: 10.11648/j.ajtas.20231203.13

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@article{10.11648/j.ajtas.20231203.13,
  author = {Tofik Mussa Reshid},
  title = {Monte Carlo Simulation and Derivation of Chi-Square Statistics},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {12},
  number = {3},
  pages = {51-65},
  doi = {10.11648/j.ajtas.20231203.13},
  url = {https://doi.org/10.11648/j.ajtas.20231203.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20231203.13},
  abstract = {Computer simulation has become an important tool in teaching statistics. Teaching using computer simulation would enhance the understanding of the concept using visual illustrations. This paper describes how to use simulation in R-programming language to perform a chi-square test. We try to show the distribution of most commonly used chi-square statistics we often found in statistical methods in both derivation and simulation. In statistical methods in such cases as test of independency, test of goodness of fit, test of significance, log likelihood ratio test, significance test and model selection we use chi-square statistic. The approach of the paper will enhance the students’ and researchers’ ability to understand simulation and sampling distribution. The paper contains an expository discussion of chi-square statistic, its derivation and distribution and its derivatives such as t-distribution and F-distribution. We consider two chi-squares, the empirical chi-square statistic and the theoretical chi-square distribution. The empirical distribution of chi-square statistic agrees closely with the theoretical chi-square distribution for large simulations, only the empirical distribution near to zero has lower density compared to the theoretical one for one degree of freedom. This is because the theoretical chi-square distribution at 1 degree of freedom has infinite density near to zero, but for any number of simulation the empirical distribution has finite density near to zero. Chi-square itself turns to normal distribution as the degree of freedom is large.},
 year = {2023}
}

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