New closed-form efficient estimator for multivariate gamma distribution. (English) Zbl 07778734
Summary: Maximum likelihood estimation is used widely in classical statistics. However, except in a few cases, it does not have a closed form. Furthermore, it takes time to derive the maximum likelihood estimator (MLE) owing to the use of iterative methods such as Newton-Raphson. Nonetheless, this estimation method has several advantages, chief among them being the invariance property and asymptotic normality. Based on the first approximation to the solution of the likelihood equation, we obtain an estimator that has the same asymptotic behavior as the MLE for multivariate gamma distribution. The newly proposed estimator, denoted as \(\mathrm{MLE}_{\mathrm{CE}}\), is also in closed form as long as the \(\sqrt{n}\)-consistent initial estimator is in the closed form. Hence, we develop some closed-form \(\sqrt{n}\)-consistent estimators for multivariate gamma distribution to improve the small-sample property. \(\mathrm{MLE}_{\mathrm{CE}}\) is an alternative to MLE and performs better compared to MLE in terms of computation time, especially for large datasets, and stability. For the bivariate gamma distribution, the \(\mathrm{MLE}_{\mathrm{CE}}\) is over 130 times faster than the MLE, and as the sample size increasing, the \(\mathrm{MLE}_{\mathrm{CE}}\) is over 200 times faster than the MLE. Owing to the instant calculation of the proposed estimator, it can be used in state-space modeling or real-time processing models.
© 2023 Netherlands Society for Statistics and Operations Research.
© 2023 Netherlands Society for Statistics and Operations Research.
Keywords:
closed-form estimator; maximum likelihood estimator; \(\sqrt{n}\)-consistent estimator; real-time processing model; state-space modelReferences:
| [1] | Brouste, A., Dutang, C., & Mieniedou, D. N. (2021). OneStep: Le Cam’s one‐step estimation procedure. The R Journal, 13, 366-377. |
| [2] | Chiu, S. N., & Liu, K. I. (2009). Generalized cram [( {}^{\prime } \]\) er‐von mises goodness‐of‐fit tests for multivariate distributions. Computational Statistics and Data Analysis, 53, 3817-3834. · Zbl 1453.62070 |
| [3] | Everitt, B. S., & Hand, D. J. (1981). Finite mixture distributions. New York, NY: Methuen. · Zbl 0466.62018 |
| [4] | Ferguson, T. S. (2002). A course in large sample theory. Boca Raton, FL: CRC Press. |
| [5] | Kim, H.‐M., & Jang, Y.‐H. (2021). New closed‐form estimators for weighted Lindley distribution. Journal of the Korean Statistical Society, 50, 580-606. · Zbl 1485.62023 |
| [6] | Kim, H.‐M., Kim, S. B., Jang, Y.‐H., & Zhao, J. (2022). New closed‐form estimator and its properties. Journal of the Korean Statistical Society, 51, 47-64. · Zbl 1485.62024 |
| [7] | Lehmann, E. L. (1999). Elements of large‐sample theory. New York, NY: Springer‐Verlag. · Zbl 0914.62001 |
| [8] | Lehmann, E. L., & Casella, G. (1998). Theory of point estimation (2nd ed.). New York, NY: Springer‐Verlag. · Zbl 0916.62017 |
| [9] | Lindsay, B. G. (1995). Mixture models: Theory, geometry and applications. Hayward, CA: Institute for Mathematical Statistics. · Zbl 1163.62326 |
| [10] | Mathai, A. M., & Moschopoulos, P. G. (1992). A form of multivariate gamma distribution. Annals of the Institute of Statistical Mathematics, 44(1), 97-106. · Zbl 0760.62049 |
| [11] | McLachlan, G. J., & Basford, K. E. (1988). Mixture models. New York, NY: Marcel Dekker. · Zbl 0697.62050 |
| [12] | Nawa, V. M., & Nadarajah, S. (2023a). New closed form estimators for a bivariate gamma distribution. Statistics, 57(1), 150-160. · Zbl 07659657 |
| [13] | Nawa, V. M., & Nadarajah, S. (2023b). Closed form estimators for a multivariate gamma distribution. Statistics, 57(2), 482-495. https://doi.org/10.1080/02331888.2023.2183204 · Zbl 07687636 · doi:10.1080/02331888.2023.2183204 |
| [14] | Titterington, D. M. (1997). Mixture distributions (update). In S.Kotz (ed.), C. B.Read (ed.), & D. L.Banks (ed.) (Eds.), Encyclopedia of statistical science, update (Vol. 1, pp. 399-407). New York, NY: John Wiley & Sons. |
| [15] | Titterington, D. M., Smith, A. F. M., & Makov, U. E. (1985). Statistical analysis of finite mixture distributions. New York, NY: John Wiley & Sons. · Zbl 0646.62013 |
| [16] | Toomet, O., & Henningsen, A. (2008). Sample selection models in R: Package sample Selection. Journal of Statistical Software, 27, 1-23. |
| [17] | Ye, Z., & Chen, N. (2017). Closed‐Form estimators for the gamma distribution derived from likelihood equations. The American Statistician, 71, 177-181. · Zbl 07671797 |
| [18] | Yee, T., & Moler, C. (2022). C. VGAM: Vector generalized linear and additive models. Retrieved from. https://CRAN.R‐project.org/package=VGAM |
| [19] | Yue, S. (2001). A bivariate gamma distribution for use in multivariate flood frequency analysis. Hydrological Processes, 15, 1033-1045. |
| [20] | Zhao, J., Jang, Y.‐H., & Kim, H.‐M. (2022). Closed‐form and bias‐corrected estimators for the bivariate gamma distribution. Journal of Multivariate Analysis, 191, 1-12. · Zbl 1493.62329 |
| [21] | Zhao, J., Kim, S. B., & Kim, H.‐M. (2021). Closed‐form estimators and bias‐corrected estimators for the Nakagami distribution. Mathematics and Computers in Simulation, 185, 308-324. · Zbl 1540.62036 |
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