Likelihood-Based Confidence Intervals for the Parameters of a Simple Linear Regression Model with Cauchy Errors

Orawo Luke Akongo

doi:doi:10.11648/j.ijsd.20210704.15

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Likelihood-Based Confidence Intervals for the Parameters of a Simple Linear Regression Model with Cauchy Errors

Orawo Luke Akongo

Published in International Journal of Statistical Distributions and Applications (Volume 7, Issue 4)

Received: 21 October 2021 Accepted: 12 November 2021 Published: 24 November 2021

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Abstract

The estimators of the slope and the intercept of simple linear regression model with normal errors are normally distributed and their exact confidence intervals are constructed using the t-distribution. However, when the normality assumption is not fulfilled, it is not possible to obtain exact confidence intervals. The Wald method of interval estimation is commonly used to provide approximate confidence intervals in such cases, and since it is derived from the central limit theorem it requires large samples in order to provide reliable approximate confidence intervals. This paper considers an alternative method of constructing approximate confidence intervals for the parameters of a simple linear regression model with Cauchy errors which is based the normal approximation to the Cauchy likelihood. The normal approximation to the Cauchy likelihood is obtained by a Tailor series expansion of the Cauchy log-likelihood function about the maximum likelihood estimate of the parameters and ignoring terms of order greater two. The maximized relative log-likelihood function for each parameter is then derived from the normal Cauchy relative log-likelihood function. The approximate confidence intervals for the parameters are constructed from their respective maximized relative log-likelihood functions. These confidence intervals have closed form confidence limits, are short and have coverage probabilities close to the nominal value 0.95.

Published in	International Journal of Statistical Distributions and Applications (Volume 7, Issue 4)
DOI	10.11648/j.ijsd.20210704.15
Page(s)	108-114
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

Normal Approximations, Relative Likelihood Function, Maximized Relative Likelihood Function, Likelihood Confidence Intervals

References

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Orawo Luke Akongo. (2021). Likelihood-Based Confidence Intervals for the Parameters of a Simple Linear Regression Model with Cauchy Errors. International Journal of Statistical Distributions and Applications, 7(4), 108-114. https://doi.org/10.11648/j.ijsd.20210704.15

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ACS Style

Orawo Luke Akongo. Likelihood-Based Confidence Intervals for the Parameters of a Simple Linear Regression Model with Cauchy Errors. Int. J. Stat. Distrib. Appl. 2021, 7(4), 108-114. doi: 10.11648/j.ijsd.20210704.15

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AMA Style

Orawo Luke Akongo. Likelihood-Based Confidence Intervals for the Parameters of a Simple Linear Regression Model with Cauchy Errors. Int J Stat Distrib Appl. 2021;7(4):108-114. doi: 10.11648/j.ijsd.20210704.15

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@article{10.11648/j.ijsd.20210704.15,
  author = {Orawo Luke Akongo},
  title = {Likelihood-Based Confidence Intervals for the Parameters of a Simple Linear Regression Model with Cauchy Errors},
  journal = {International Journal of Statistical Distributions and Applications},
  volume = {7},
  number = {4},
  pages = {108-114},
  doi = {10.11648/j.ijsd.20210704.15},
  url = {https://doi.org/10.11648/j.ijsd.20210704.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsd.20210704.15},
  abstract = {The estimators of the slope and the intercept of simple linear regression model with normal errors are normally distributed and their exact confidence intervals are constructed using the t-distribution. However, when the normality assumption is not fulfilled, it is not possible to obtain exact confidence intervals. The Wald method of interval estimation is commonly used to provide approximate confidence intervals in such cases, and since it is derived from the central limit theorem it requires large samples in order to provide reliable approximate confidence intervals. This paper considers an alternative method of constructing approximate confidence intervals for the parameters of a simple linear regression model with Cauchy errors which is based the normal approximation to the Cauchy likelihood. The normal approximation to the Cauchy likelihood is obtained by a Tailor series expansion of the Cauchy log-likelihood function about the maximum likelihood estimate of the parameters and ignoring terms of order greater two. The maximized relative log-likelihood function for each parameter is then derived from the normal Cauchy relative log-likelihood function. The approximate confidence intervals for the parameters are constructed from their respective maximized relative log-likelihood functions. These confidence intervals have closed form confidence limits, are short and have coverage probabilities close to the nominal value 0.95.},
 year = {2021}
}

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TY  - JOUR
T1  - Likelihood-Based Confidence Intervals for the Parameters of a Simple Linear Regression Model with Cauchy Errors
AU  - Orawo Luke Akongo
Y1  - 2021/11/24
PY  - 2021
N1  - https://doi.org/10.11648/j.ijsd.20210704.15
DO  - 10.11648/j.ijsd.20210704.15
T2  - International Journal of Statistical Distributions and Applications
JF  - International Journal of Statistical Distributions and Applications
JO  - International Journal of Statistical Distributions and Applications
SP  - 108
EP  - 114
PB  - Science Publishing Group
SN  - 2472-3509
UR  - https://doi.org/10.11648/j.ijsd.20210704.15
AB  - The estimators of the slope and the intercept of simple linear regression model with normal errors are normally distributed and their exact confidence intervals are constructed using the t-distribution. However, when the normality assumption is not fulfilled, it is not possible to obtain exact confidence intervals. The Wald method of interval estimation is commonly used to provide approximate confidence intervals in such cases, and since it is derived from the central limit theorem it requires large samples in order to provide reliable approximate confidence intervals. This paper considers an alternative method of constructing approximate confidence intervals for the parameters of a simple linear regression model with Cauchy errors which is based the normal approximation to the Cauchy likelihood. The normal approximation to the Cauchy likelihood is obtained by a Tailor series expansion of the Cauchy log-likelihood function about the maximum likelihood estimate of the parameters and ignoring terms of order greater two. The maximized relative log-likelihood function for each parameter is then derived from the normal Cauchy relative log-likelihood function. The approximate confidence intervals for the parameters are constructed from their respective maximized relative log-likelihood functions. These confidence intervals have closed form confidence limits, are short and have coverage probabilities close to the nominal value 0.95.
VL  - 7
IS  - 4
ER  -

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Author Information

Orawo Luke Akongo

Mathematics Department, Egerton University, Njoro, Kenya

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APA Style

Orawo Luke Akongo. (2021). Likelihood-Based Confidence Intervals for the Parameters of a Simple Linear Regression Model with Cauchy Errors. International Journal of Statistical Distributions and Applications, 7(4), 108-114. https://doi.org/10.11648/j.ijsd.20210704.15

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ACS Style

Orawo Luke Akongo. Likelihood-Based Confidence Intervals for the Parameters of a Simple Linear Regression Model with Cauchy Errors. Int. J. Stat. Distrib. Appl. 2021, 7(4), 108-114. doi: 10.11648/j.ijsd.20210704.15

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AMA Style

Orawo Luke Akongo. Likelihood-Based Confidence Intervals for the Parameters of a Simple Linear Regression Model with Cauchy Errors. Int J Stat Distrib Appl. 2021;7(4):108-114. doi: 10.11648/j.ijsd.20210704.15

Copy | Download

@article{10.11648/j.ijsd.20210704.15,
  author = {Orawo Luke Akongo},
  title = {Likelihood-Based Confidence Intervals for the Parameters of a Simple Linear Regression Model with Cauchy Errors},
  journal = {International Journal of Statistical Distributions and Applications},
  volume = {7},
  number = {4},
  pages = {108-114},
  doi = {10.11648/j.ijsd.20210704.15},
  url = {https://doi.org/10.11648/j.ijsd.20210704.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsd.20210704.15},
  abstract = {The estimators of the slope and the intercept of simple linear regression model with normal errors are normally distributed and their exact confidence intervals are constructed using the t-distribution. However, when the normality assumption is not fulfilled, it is not possible to obtain exact confidence intervals. The Wald method of interval estimation is commonly used to provide approximate confidence intervals in such cases, and since it is derived from the central limit theorem it requires large samples in order to provide reliable approximate confidence intervals. This paper considers an alternative method of constructing approximate confidence intervals for the parameters of a simple linear regression model with Cauchy errors which is based the normal approximation to the Cauchy likelihood. The normal approximation to the Cauchy likelihood is obtained by a Tailor series expansion of the Cauchy log-likelihood function about the maximum likelihood estimate of the parameters and ignoring terms of order greater two. The maximized relative log-likelihood function for each parameter is then derived from the normal Cauchy relative log-likelihood function. The approximate confidence intervals for the parameters are constructed from their respective maximized relative log-likelihood functions. These confidence intervals have closed form confidence limits, are short and have coverage probabilities close to the nominal value 0.95.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Likelihood-Based Confidence Intervals for the Parameters of a Simple Linear Regression Model with Cauchy Errors
AU  - Orawo Luke Akongo
Y1  - 2021/11/24
PY  - 2021
N1  - https://doi.org/10.11648/j.ijsd.20210704.15
DO  - 10.11648/j.ijsd.20210704.15
T2  - International Journal of Statistical Distributions and Applications
JF  - International Journal of Statistical Distributions and Applications
JO  - International Journal of Statistical Distributions and Applications
SP  - 108
EP  - 114
PB  - Science Publishing Group
SN  - 2472-3509
UR  - https://doi.org/10.11648/j.ijsd.20210704.15
AB  - The estimators of the slope and the intercept of simple linear regression model with normal errors are normally distributed and their exact confidence intervals are constructed using the t-distribution. However, when the normality assumption is not fulfilled, it is not possible to obtain exact confidence intervals. The Wald method of interval estimation is commonly used to provide approximate confidence intervals in such cases, and since it is derived from the central limit theorem it requires large samples in order to provide reliable approximate confidence intervals. This paper considers an alternative method of constructing approximate confidence intervals for the parameters of a simple linear regression model with Cauchy errors which is based the normal approximation to the Cauchy likelihood. The normal approximation to the Cauchy likelihood is obtained by a Tailor series expansion of the Cauchy log-likelihood function about the maximum likelihood estimate of the parameters and ignoring terms of order greater two. The maximized relative log-likelihood function for each parameter is then derived from the normal Cauchy relative log-likelihood function. The approximate confidence intervals for the parameters are constructed from their respective maximized relative log-likelihood functions. These confidence intervals have closed form confidence limits, are short and have coverage probabilities close to the nominal value 0.95.
VL  - 7
IS  - 4
ER  -

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