Linear Regression with Errors not Normal: Generalized Hyperbolic Secant

Main Article Content

Álvaro Alexander Burbano Moreno https://orcid.org/0000-0001-8293-9705
Oscar Orlando Melo Martinez https://orcid.org/0000-0002-0296-4511

Keywords

distribution, classical linear model, modified maximum likelihood, least squares

Abstract

This paper presents a study of the model of linear regression of the type y = Θx + e, where the error has generalized hyperbolic secant distribution (GHS). The method to estimate the parameters are obtained by setting maximum likelihood expressing the non-linear equations in linear form (modified likelihood). The resulting estimators are analytical expressions in terms of values of the sample and, therefore, are easily calculables. Through the application of various types of data, the methodology described above is shown, and plausible models against the true underlying distributions of data are.

MSC: 60E05, 62E10

 

Downloads

Download data is not yet available.
Abstract 1883 | PDF (Español) Downloads 1854 HTML (Español) Downloads 646

References

D. C. Vaughan, "The Generalized Secant Hyperbolic Distribution And ItsProperties," Communications in statistics, vol. 31, no. 2, pp. 219–238, 2002.

V. D. Barnett, "Evaluation of the maximum likelihood estimator when thelikelihood equation has multiple roots," Biometrika, vol. 53, pp. 151–165,1996a.
http://dx.doi.org/10.1093/biomet/53.1-2.151

D. C. Vaughan, "On the Tiku-Suresh method of estimation," Communicationsin statistics, vol. 21, pp. 451–469, 1992.

M. L. Tiku, D. Aysen, and Akkaya, Robust Estimation and Hypothesis Testing,2nd ed. New York: New Age, 2004.

S. Puthenpura and N. K. Sinha, "Modified maximum likelihood method forthe robust estimation of system parametrs from very noisy data," Automatica,vol. 22, pp. 231–235, 1986.
http://dx.doi.org/10.1016/0005-1098(86)90085-3

M. L. Tiku and R. P. Suresh, "A new method of estimation for location andscale parameters," J. Stat. Plann, vol. 30, pp. 281–292, 1992.
http://dx.doi.org/10.1016/0378-3758(92)90088-A

M. L. Tiku, "Estimating the mean and Standard Deviation from a censoredNormal Sample," Biometrika, vol. 54, no. 1, pp. 155–165, 1967a.
http://dx.doi.org/10.1093/biomet/54.1-2.155

M. L. Tiku, "Monte Carlo Study of Some Simple Estimators in Censored NormalSamples," Biometrika, vol. 57, pp. 207–211, 1970.
http://dx.doi.org/10.1093/biomet/57.1.207

G. K. Bhattacharyya, "The Asymptotics of Maximum Likelihood and RelatedEstimators Based on Type II Censored data," Journal of the AmericanStatistical Association, vol. 80, no. 390, pp. 398–404, 1970.
http://dx.doi.org/10.1080/01621459.1985.10478130

R. L. Smith, "Maximum likelihood estimation in a class of nonregular cases,"Biometrika, no. 72, pp. 67–90, 1985.
http://dx.doi.org/10.1093/biomet/72.1.67

M. L. Tiku, W. K. Wong, D. C. Vaughan, and G. Bian, "Time series modelsin non-normal situations: symmetric innovations," J. Time Series Analysis,vol. 21, pp. 571–596, 2000.
http://dx.doi.org/10.1111/1467-9892.00199

M. Alejandro and B. Alexander, "Secante hiperbolica generalizada y un metodode estimacion de sus parametros: maxima verosimilitud modificada,"Ingenieria y Ciencia, vol. 9, no. 18, pp. 93–106, 2013.

L. Hamilton, Regression With Graphics, 1st ed. Brooks/Cole PublishingCompany, 1992.

D. Hand, F. Daly, A. Lunn, K. McConway, and E. Ostrowski, Small DataSets, 1st ed. Springer-Science Business, 1994.