Modified Maximum Likelihood Estimation in Autoregressive Processes with Generalized Exponential Innovations
Affiliation(s)
1 Department of Statistics, Universidad de Concepción, Concepción, Chile. 2 Department of Mathematics, University Federico Santa María, Valparaíso, Chile.
1 Department of Statistics, Universidad de Concepción, Concepción, Chile. 2 Department of Mathematics, University Federico Santa María, Valparaíso, Chile.
ABSTRACT
We consider a time series following a simple linear regression with first-order autoregressive errors belonging to the class of heavy-tailed distributions. The proposed model provides a useful generalization of the symmetrical linear regression models with independent error, since the error distribution covers both correlated innovations following a Generalized Exponential distribution. Furthermore, we derive the modified maximum likelihood (MML) estimators as an efficient alternative for estimating model parameters. Finally, we investigate the asymptotic properties of the proposed estimators. Our findings are also illustrated through a simulation study.
We consider a time series following a simple linear regression with first-order autoregressive errors belonging to the class of heavy-tailed distributions. The proposed model provides a useful generalization of the symmetrical linear regression models with independent error, since the error distribution covers both correlated innovations following a Generalized Exponential distribution. Furthermore, we derive the modified maximum likelihood (MML) estimators as an efficient alternative for estimating model parameters. Finally, we investigate the asymptotic properties of the proposed estimators. Our findings are also illustrated through a simulation study.
KEYWORDS
Autoregressive Time Series Model, Maximum Likelihood, Modified Maximum Likelihood, Least Squares, Generalized Exponential
Autoregressive Time Series Model, Maximum Likelihood, Modified Maximum Likelihood, Least Squares, Generalized Exponential
Cite this paper
Lagos-Álvarez, B. , Ferreira, G. and Porcu, E. (2014) Modified Maximum Likelihood Estimation in Autoregressive Processes with Generalized Exponential Innovations. Open Journal of Statistics, 4, 620-629. doi: 10.4236/ojs.2014.48058.
Lagos-Álvarez, B. , Ferreira, G. and Porcu, E. (2014) Modified Maximum Likelihood Estimation in Autoregressive Processes with Generalized Exponential Innovations. Open Journal of Statistics, 4, 620-629. doi: 10.4236/ojs.2014.48058.
References
[1] Tiku, M.L., Tan, W.Y. and Balakrishnan, N. (1986) Robust Inference. Marcel Dekker, Inc., New York. [2] Tan, W.Y. and Lin, V. (1993) Some Robust Procedures for Estimating Parameters in an Autoregressive Model. Sankhya B , 55, 415-435. [3] Damsleth, E. and El-Shaarawi, A.H. (1989) ARMA Models with Double Exponentially Distributed Noise. Journal of the Royal Statistical Society B , 51, 61-69. [4] Tiku, M.L. (1980) Robustness of MML Estimators Based on Censored Samples and Robust Test Statistics. Journal of Statistical Planning and Inference , 4, 123-143.
http://dx.doi.org/10.1016/0378-3758(80)90002-6 [5] Tan, W.Y. (1985) On Tiku’s Robust Procedure—A Bayesian Insight. Journal of Statistical Planning and Inference, 11, 329-340.
http://dx.doi.org/10.1016/0378-3758(85)90038-2 [6] Swift, A.L. (1995) Modelling and Forecasting Time Series with a General Non-Normal Distribution. Journal of Forecasting , 14, 45-66.
http://dx.doi.org/10.1002/for.3980140105 [7] Martin, R.D. and Yohai, V.J. (1986) Influence Functionals for Time Series. Annals of Statistics, 14, 781-818.
http://dx.doi.org/10.1214/aos/1176350027 [8] Bhansali, R.J. (1997) Robustness of the Autoregressive Spectral Estimate for Linear Process with Infinite Variance. Journal Time Series Analysis , 18, 213-229.
http://dx.doi.org/10.1111/1467-9892.00047 [9] Davis, R.A. and Resnick, S. (1986) Limit Theory for the Sample Covariance and Correlation Functions of Moving Averages. Annals of the Institute of Statistics , 14, 533-558.
http://dx.doi.org/10.1214/aos/1176349937 [10] Durbin, J. and Koopman, S.J. (1997) Monte Carlo Maximum Likelihood Estimation for Non-Gaussian State Space Models. Biometrika, 84, 669-684.
http://dx.doi.org/10.1093/biomet/84.3.669 [11] Kitagawa, G. (1987) Non-Gaussian State-Space Modelling of Nonstationary Time Series (with Discussion). Journal of the American Statistical Association , 82, 1032-1063. [12] Trindade, A.A. and Zhu, Y. (2010) Time Series Models with Asymmetric Laplace Innovations. Journal of Statistical Computation and Simulation , 80, 1317-1333. [13] Tiku, M.L. (1967) Estimating the Mean and Standard Deviation from Censored Normal Samples. Biometrika, 54, 155- 165.
http://dx.doi.org/10.2307/2283834 [14] Tiku, M.L. (1968) Estimating the Parameters of Log-Normal Distribution from Censored Samples. Journal of the Ame- rican Statistical Association , 63, 134-140.
http://dx.doi.org/10.2307/2283834 [15] Tiku, M.L. and Suresh, R.P. (1992) A New Method of Estimation for Location and Scale Parameters. Journal of Statistical Planning and Inference , 30, 281-292.
http://dx.doi.org/10.1111/1467-842X.00072 [16] Gupta, R.D. and Kundu, D. (1999) Generalized Exponential Distributions. aAustralian and New Zealand Journal of Statistics, 41, 173-188.
http://dx.doi.org/10.1111/1467-842X.00072 [17] Vaughan, D.C. and Tiku, M.L. (2000) Estimation and Hypothesis Testing for a Non-Normal Bivariate Distribution with Applications. Journal of Mathematical and Computer Modelling , 32, 53-67.
http://dx.doi.org/10.1016/S0895-7177(00)00119-9 [18] Bhattacharyya, G.K. (1985) The Asymptotics of Maximum Likelihood and Related Estimators Based on Type II Censored Data. Journal of the American Statistical Association , 80, 398-404.
http://dx.doi.org/10.1080/01621459.1985.10478130 [19] Tiku, M.L. (1970) Some Notes on the Relationship between the Distribution of Central and Non-Central F. Biometrika, 57, 175-179.
http://dx.doi.org/10.1093/biomet/57.1.175 [20] Tiku, M.L. (1970) Monte Carlo Study of Some Simple Estimators in Censored Normal Samples. Biometrika, 57, 207- 210. [21] Gupta, R.D. and Kundu, D. (2007) Generalized Exponential Distribution: Existing Methods and Some Recent Developments. Journal of Statistical Planning and Inference, 137, 3537-3547.
http://dx.doi.org/10.1016/j.jspi.2007.03.030 [22] Balakrishnan, N. and Cohen, A.C. (1991) Order Statistics and Inference. Academic Press, Waltham. [23] Raqab, M.Z. and Ahsanullah, M. (2001) Estimation of Location and Scale Parameters of Generalized Exponential Distribution Based on Order Statistics. Journal of Statistical Computation and Simulation , 69, 109-124. [24] Kendall, M.G. and Stuart, A. (1979) The Advanced Theory of Statistics. Charles Griffin, London. [25] Tiku, M.L. and Akkaya, A.D. (2004) Robust Estimation and Hypothesis Testing. New Age International (P) Publishers, New Delhi, 337. [26] Wong, W.K. and Bian, G. (2005) Estimating Parameters in Autoregressive Models with Asymmetric Innovations. Statistics and Probability Letters , 71, 61-70.
http://dx.doi.org/10.1016/j.spl.2004.10.022 [27] Tiku, M.L., Wong, W.K. and Bian, G. (1999) Time Series Models with Asymmetric Innovations. Communications in Statistics—Theory and Methods , 28, 1331-1360.
http://dx.doi.org/10.1080/03610929908832360
[1] Tiku, M.L., Tan, W.Y. and Balakrishnan, N. (1986) Robust Inference. Marcel Dekker, Inc., New York. [2] Tan, W.Y. and Lin, V. (1993) Some Robust Procedures for Estimating Parameters in an Autoregressive Model. Sankhya B , 55, 415-435. [3] Damsleth, E. and El-Shaarawi, A.H. (1989) ARMA Models with Double Exponentially Distributed Noise. Journal of the Royal Statistical Society B , 51, 61-69. [4] Tiku, M.L. (1980) Robustness of MML Estimators Based on Censored Samples and Robust Test Statistics. Journal of Statistical Planning and Inference , 4, 123-143.
http://dx.doi.org/10.1016/0378-3758(80)90002-6 [5] Tan, W.Y. (1985) On Tiku’s Robust Procedure—A Bayesian Insight. Journal of Statistical Planning and Inference, 11, 329-340.
http://dx.doi.org/10.1016/0378-3758(85)90038-2 [6] Swift, A.L. (1995) Modelling and Forecasting Time Series with a General Non-Normal Distribution. Journal of Forecasting , 14, 45-66.
http://dx.doi.org/10.1002/for.3980140105 [7] Martin, R.D. and Yohai, V.J. (1986) Influence Functionals for Time Series. Annals of Statistics, 14, 781-818.
http://dx.doi.org/10.1214/aos/1176350027 [8] Bhansali, R.J. (1997) Robustness of the Autoregressive Spectral Estimate for Linear Process with Infinite Variance. Journal Time Series Analysis , 18, 213-229.
http://dx.doi.org/10.1111/1467-9892.00047 [9] Davis, R.A. and Resnick, S. (1986) Limit Theory for the Sample Covariance and Correlation Functions of Moving Averages. Annals of the Institute of Statistics , 14, 533-558.
http://dx.doi.org/10.1214/aos/1176349937 [10] Durbin, J. and Koopman, S.J. (1997) Monte Carlo Maximum Likelihood Estimation for Non-Gaussian State Space Models. Biometrika, 84, 669-684.
http://dx.doi.org/10.1093/biomet/84.3.669 [11] Kitagawa, G. (1987) Non-Gaussian State-Space Modelling of Nonstationary Time Series (with Discussion). Journal of the American Statistical Association , 82, 1032-1063. [12] Trindade, A.A. and Zhu, Y. (2010) Time Series Models with Asymmetric Laplace Innovations. Journal of Statistical Computation and Simulation , 80, 1317-1333. [13] Tiku, M.L. (1967) Estimating the Mean and Standard Deviation from Censored Normal Samples. Biometrika, 54, 155- 165.
http://dx.doi.org/10.2307/2283834 [14] Tiku, M.L. (1968) Estimating the Parameters of Log-Normal Distribution from Censored Samples. Journal of the Ame- rican Statistical Association , 63, 134-140.
http://dx.doi.org/10.2307/2283834 [15] Tiku, M.L. and Suresh, R.P. (1992) A New Method of Estimation for Location and Scale Parameters. Journal of Statistical Planning and Inference , 30, 281-292.
http://dx.doi.org/10.1111/1467-842X.00072 [16] Gupta, R.D. and Kundu, D. (1999) Generalized Exponential Distributions. aAustralian and New Zealand Journal of Statistics, 41, 173-188.
http://dx.doi.org/10.1111/1467-842X.00072 [17] Vaughan, D.C. and Tiku, M.L. (2000) Estimation and Hypothesis Testing for a Non-Normal Bivariate Distribution with Applications. Journal of Mathematical and Computer Modelling , 32, 53-67.
http://dx.doi.org/10.1016/S0895-7177(00)00119-9 [18] Bhattacharyya, G.K. (1985) The Asymptotics of Maximum Likelihood and Related Estimators Based on Type II Censored Data. Journal of the American Statistical Association , 80, 398-404.
http://dx.doi.org/10.1080/01621459.1985.10478130 [19] Tiku, M.L. (1970) Some Notes on the Relationship between the Distribution of Central and Non-Central F. Biometrika, 57, 175-179.
http://dx.doi.org/10.1093/biomet/57.1.175 [20] Tiku, M.L. (1970) Monte Carlo Study of Some Simple Estimators in Censored Normal Samples. Biometrika, 57, 207- 210. [21] Gupta, R.D. and Kundu, D. (2007) Generalized Exponential Distribution: Existing Methods and Some Recent Developments. Journal of Statistical Planning and Inference, 137, 3537-3547.
http://dx.doi.org/10.1016/j.jspi.2007.03.030 [22] Balakrishnan, N. and Cohen, A.C. (1991) Order Statistics and Inference. Academic Press, Waltham. [23] Raqab, M.Z. and Ahsanullah, M. (2001) Estimation of Location and Scale Parameters of Generalized Exponential Distribution Based on Order Statistics. Journal of Statistical Computation and Simulation , 69, 109-124. [24] Kendall, M.G. and Stuart, A. (1979) The Advanced Theory of Statistics. Charles Griffin, London. [25] Tiku, M.L. and Akkaya, A.D. (2004) Robust Estimation and Hypothesis Testing. New Age International (P) Publishers, New Delhi, 337. [26] Wong, W.K. and Bian, G. (2005) Estimating Parameters in Autoregressive Models with Asymmetric Innovations. Statistics and Probability Letters , 71, 61-70.
http://dx.doi.org/10.1016/j.spl.2004.10.022 [27] Tiku, M.L., Wong, W.K. and Bian, G. (1999) Time Series Models with Asymmetric Innovations. Communications in Statistics—Theory and Methods , 28, 1331-1360.
http://dx.doi.org/10.1080/03610929908832360