Inference in Multiple Linear Regression Model with Generalized Secant Hyperbolic Distribution Errors
Main Article Content
Keywords
Maximum likelihood, Modified maximum likelihood, Least square, Generalized Secant Hyperbolic distribution, Robustness, Hypothesis testing
Abstract
We study multiple linear regression model under non-normally distributed random error by considering the family of generalized secant hyperbolic distributions. We derive the estimators of model parameters by using
modified maximum likelihood methodology and explore the properties of the modified maximum likelihood estimators so obtained. We show that the proposed estimators are more efficient and robust than the commonly used least square estimators. We also develop the relevant test of hypothesis procedures and compared the performance of such tests vis-a-vis the classical tests that are based upon the least square approach.
Downloads
Download data is not yet available.
References
[1] E. Pearson, “The analysis of variance in cases of nonnormal variation,” Biometrika., vol. 22, no. 1/2, pp. 231–235, 1986. https://doi.org/10.2307/ 2333631
[2] P. Huber, Robust Statistics, 2nd ed. Jonh Wiley: New York, 1981.
[3] J. Tukey, A survey of sampling from contaminated distributions. Stanford University Press, Stanford: Contributions to Probability and Statistics, 1960.
[4] V. D. Barnett, “Order statistic estimators of the location of the cauchy distribution,” Journal of American Statistical Association., vol. 61, no. 316, pp. 1205–1218, 1966. https://doi.org/10.2307/2283210
[5] D. C. Vaughan, “On the tiku-suresh method of estimation,” Communications in Statistics - Theory and Methods, vol. 21, no. 2, pp. 451–469, 1992. https://doi.org/10.1080/03610929208830788
[6] M. L. Tiku and R. P. Suresh, “A new method of estimation for location and scale parameters,” Journal of Statistical Planning and Inference., vol. 30, no. 2, pp. 281–292, 1992. https://doi.org/10.1016/0378-3758(92)90088-A
[7] D. C. Vaughan and M. L. Tiku, “Estimation and hypothesis testing for a nonnormal bivariate distribution with applications,” Mathematical and Computer Modelling, vol. 32, no. 1/2, pp. 53–67, 2000. https: //doi.org/10.1016/S0895-7177(00)00119-9
[8] M. L. Tiku and A. D. Akkaya, Robust Estimation and Hypothesis Testing, 2nd ed. New York: New Age, 2004.
[9] M. Q. Islam and M. L. Tiku, “Multiple linear regression model under nonnormality,” Communications in Statistics - Theory and Methods., vol. 33, no. 10, pp. 2443–2467, 2004. https://doi.org/10.1081/STA-200031519
[10] D. C. Vaughan, “The generalized secant hyperbolic distribution and its properties,” Communications in Statistics - Theory and Methods., vol. 31, no. 2, pp. 219–238, 2002. https://doi.org/10.1081/STA-120002647
[11] Y. E. Yilmaz and A. D. Akkaya, “Analysis of variance and linear contrasts in experimental design with generalized secant hyperbolic distribution,” Journal of Computational and Applied Mathematics., vol. 216, no. 2, pp. 545–553, 2008. https://doi.org/10.1016/j.cam.2007.06.001
[12] V. D. Barnett, “Evaluation of the maximum likelihood estimator when the likelihood equation has multiple roots,” Biometrika., vol. 53, no. 1/2, pp. 151–165, 1996. https://doi.org/10.2307/2334061
[13] S. Puthenpura and N. K. Sinha, “Modified maximum likelihood method for the robust estimation of system parametrs from very noisy data,” Automatica., vol. 22, pp. 231–235, 1986. https://doi.org/10.1016/ 0005-1098(86)90085-3
[14] B. Senoglu and M. L. Tiku, “Analysis of variance in experimental design with non-normal error distributions,” Communications in Statistics - Theory and Methods., vol. 30, pp. 1335–1352, 2001. https://doi.org/10.1081/ STA-100104748
[15] M. Q. Islam, M. L. Tiku, and F. Yildirim, “Nonnormal regression. I. skew distributions,” Communications in Statistics - Theory and Methods., vol. 30, no. 6, pp. 993–1020., 2001. https://doi.org/10.1081/STA-100104347
[16] M. L. Tiku, W. K. Wong, and G. Bian, “Estimating parameters in autoregressive models in non-normal situations: symmetric innovations,” Communications in Statistics - Theory and Methods., vol. 2, no. 28, pp. 315–341, 1999. https://doi.org/10.1080/03610929908832300
[17] M. L. Tiku, M. Q. Islam, and A. S. Selcuk, “Nonnormal regression. II. symmetric distributions,” Communications in Statistics - Theory and Methods., vol. 30, no. 6, pp. 1021–1045, 2001. https://doi.org/10.1081/ STA-100104348
[18] B. L. Joiner and J. R. Rosenblatt, “Some properties of the range in samples from tukey’s symmetric lambda distributions,” Journal of American Statistical Association, vol. 66, pp. 394–399, 1971. https://doi.org/10.2307/2283943
[19] G. Box, “Non-normality and test of variances,” Biometrika., vol. 40, pp. 336–346, 1953. https://doi.org/10.2307/2333350
[20] M. L. Tiku, W. Y. Tan, and N. Balakrishnan, Robust Inference. New York: Marcel Dekker, 1986.
[21] M. Barrow, Statistics for Economics, Accounting and Business Studies, 5th ed. Pearson Education, 2009.
[22] D. C. Montgomery, E. A. Peck, and G. G. Vining, Introduction to linear regression analysis, 5th ed. John wiley & Sons, 2015
[2] P. Huber, Robust Statistics, 2nd ed. Jonh Wiley: New York, 1981.
[3] J. Tukey, A survey of sampling from contaminated distributions. Stanford University Press, Stanford: Contributions to Probability and Statistics, 1960.
[4] V. D. Barnett, “Order statistic estimators of the location of the cauchy distribution,” Journal of American Statistical Association., vol. 61, no. 316, pp. 1205–1218, 1966. https://doi.org/10.2307/2283210
[5] D. C. Vaughan, “On the tiku-suresh method of estimation,” Communications in Statistics - Theory and Methods, vol. 21, no. 2, pp. 451–469, 1992. https://doi.org/10.1080/03610929208830788
[6] M. L. Tiku and R. P. Suresh, “A new method of estimation for location and scale parameters,” Journal of Statistical Planning and Inference., vol. 30, no. 2, pp. 281–292, 1992. https://doi.org/10.1016/0378-3758(92)90088-A
[7] D. C. Vaughan and M. L. Tiku, “Estimation and hypothesis testing for a nonnormal bivariate distribution with applications,” Mathematical and Computer Modelling, vol. 32, no. 1/2, pp. 53–67, 2000. https: //doi.org/10.1016/S0895-7177(00)00119-9
[8] M. L. Tiku and A. D. Akkaya, Robust Estimation and Hypothesis Testing, 2nd ed. New York: New Age, 2004.
[9] M. Q. Islam and M. L. Tiku, “Multiple linear regression model under nonnormality,” Communications in Statistics - Theory and Methods., vol. 33, no. 10, pp. 2443–2467, 2004. https://doi.org/10.1081/STA-200031519
[10] D. C. Vaughan, “The generalized secant hyperbolic distribution and its properties,” Communications in Statistics - Theory and Methods., vol. 31, no. 2, pp. 219–238, 2002. https://doi.org/10.1081/STA-120002647
[11] Y. E. Yilmaz and A. D. Akkaya, “Analysis of variance and linear contrasts in experimental design with generalized secant hyperbolic distribution,” Journal of Computational and Applied Mathematics., vol. 216, no. 2, pp. 545–553, 2008. https://doi.org/10.1016/j.cam.2007.06.001
[12] V. D. Barnett, “Evaluation of the maximum likelihood estimator when the likelihood equation has multiple roots,” Biometrika., vol. 53, no. 1/2, pp. 151–165, 1996. https://doi.org/10.2307/2334061
[13] S. Puthenpura and N. K. Sinha, “Modified maximum likelihood method for the robust estimation of system parametrs from very noisy data,” Automatica., vol. 22, pp. 231–235, 1986. https://doi.org/10.1016/ 0005-1098(86)90085-3
[14] B. Senoglu and M. L. Tiku, “Analysis of variance in experimental design with non-normal error distributions,” Communications in Statistics - Theory and Methods., vol. 30, pp. 1335–1352, 2001. https://doi.org/10.1081/ STA-100104748
[15] M. Q. Islam, M. L. Tiku, and F. Yildirim, “Nonnormal regression. I. skew distributions,” Communications in Statistics - Theory and Methods., vol. 30, no. 6, pp. 993–1020., 2001. https://doi.org/10.1081/STA-100104347
[16] M. L. Tiku, W. K. Wong, and G. Bian, “Estimating parameters in autoregressive models in non-normal situations: symmetric innovations,” Communications in Statistics - Theory and Methods., vol. 2, no. 28, pp. 315–341, 1999. https://doi.org/10.1080/03610929908832300
[17] M. L. Tiku, M. Q. Islam, and A. S. Selcuk, “Nonnormal regression. II. symmetric distributions,” Communications in Statistics - Theory and Methods., vol. 30, no. 6, pp. 1021–1045, 2001. https://doi.org/10.1081/ STA-100104348
[18] B. L. Joiner and J. R. Rosenblatt, “Some properties of the range in samples from tukey’s symmetric lambda distributions,” Journal of American Statistical Association, vol. 66, pp. 394–399, 1971. https://doi.org/10.2307/2283943
[19] G. Box, “Non-normality and test of variances,” Biometrika., vol. 40, pp. 336–346, 1953. https://doi.org/10.2307/2333350
[20] M. L. Tiku, W. Y. Tan, and N. Balakrishnan, Robust Inference. New York: Marcel Dekker, 1986.
[21] M. Barrow, Statistics for Economics, Accounting and Business Studies, 5th ed. Pearson Education, 2009.
[22] D. C. Montgomery, E. A. Peck, and G. G. Vining, Introduction to linear regression analysis, 5th ed. John wiley & Sons, 2015
Article Sidebar
Published
May 12, 2021
Article Details
How to Cite
Burbano Moreno, Álvaro A., Melo-Martinez, O. O., & Qamarul Islam, M. (2021). Inference in Multiple Linear Regression Model with Generalized Secant Hyperbolic Distribution Errors. Ingeniería Y Ciencia, 17(33), 45–70. https://doi.org/10.17230/ingciencia.17.33.3
Issue
Section
Articles
Supporting Agencies
Ninguna
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).