Abstract
There is no closed form maximum likelihood estimator (MLE) for some distributions. This might cause some problems in real-time processing. Using an extension of Box–Cox transformation, we develop a closed-form estimator for the family of distributions. If such closed-form estimators exist, they have the invariance property like MLE and are equal in distribution with respect to the transformation. Specifically, the joint exact and asymptotic distributions of the closed-form estimators are the same irrespective of the transformation parameter, which is useful for statistical inference. For the gamma related and weighted Lindley related distributions, the closed-form estimators achieve strong consistency and asymptotic normality similar to MLE. That is, the closed-form estimators from the family of distributions obtained from an extension of the Box–Cox transformation for the gamma and weighted Lindley distributions as the initial distributions achieve strong consistency and asymptotic normality. A bias-corrected closed-form estimator that is also independent of the transformation is derived. In this sense, the closed-form estimator and the bias-corrected closed-form estimator are invariant with respect to the transformation. Some examples are provided to demonstrate the underlying theory. Some simulation studies and a real data example for the inverse gamma distribution are presented to illustrate the performance of the proposed estimators in this study.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cheng, J., & Beaulieu, N. C. (2001). Maximum-likelihood based estimation of the Nakagami \(m\) parameter. IEEE Communications Letters, 5, 101–103.
Cordeiro, G. M., & Klein, R. (1994). Bias correction in ARMA models. Statistics and Probability Letters, 19, 169–176.
Cox, D. R., & Reid, N. (1987). Parameter orthogonality and approximate conditional inference. Journal of the Royal Statistical Society: Series B (Methodological), 49, 1–39.
Cox, D. R., & Snell, E. J. (1968). A general definition of residuals. Journal of the Royal Statistical Society: Series B (Methodological), 30, 248–275.
Cramér, H. (1946). Mathematical Methods of Statistics. Princeton: Princeton University Press.
Ghitany, M., Alqallaf, F., Al-Mutairi, D., & Husain, H. (2011). A two-parameter weighted Lindley distribution and its applications to survival data. Mathematics and Computers in Simulation, 81, 1190–1201.
Glen, A. G. (2011). On the inverse gamma as a survival distribution. Journal of Quality Technology, 43, 158–166.
Kim, H.-M., & Jang, Y.-H. (2020). New closed-form estimators for weighted Lindley distribution. Journal of the Korean Statistical Society (accepted).
Li, Y., Olama, M., Djouadi, S., Fathy, A., & Kuruganti, T. (2009). Stochastic UWB Wireless Channel Modeling and Estimation from Received Signal Measurements (pp. 195–198). San Diego: IEEE Radio and Wireless Symposium.
Lieblein, J., & Zelen, M. (1956). Statistical Investigation of the fatigue life of deep-groove ball bearings. Journal of Research of the National Bureau of Standards, 57, 273–316.
Louzada, F., Ramos, P. L., & Nascimento, D. (2018). The inverse Nakagami-m distribution: A novel approach in reliability. IEEE Transactions on Reliability, 67, 1030–1042.
Louzada, F., Ramos, P. L., & Ramos, E. (2019). A note on bias of closed-form estimators for the gamma distribution derived from likelihood equations. The American Statistician, 73, 195–199.
Ramos, P. L., Louzada, F., & Ramos, E. (2016). An efficient, closed-Form MAP estimator for Nakagami-\(m\) fading parameter. IEEE Communications Letters, 20, 2328–2331.
Ramos, P. L., Louzada, F., Shmizu, T. K. O., & Luiz, A. O. (2019). The inverse weighted Lindley distribution: Properties, estimation and an application to a failure time data. Communications in Statistics: Theory and Methods, 48, 2372–2389.
Song, K. (2008). Globally convergent algorithms for estimating generalized gamma distributions in fast signal and image processing. IEEE Transactions on Image Processing, 17, 1233–1250.
Toomet, O., & Henningsen, A. (2008). Sample selection models in R: Package sample selection. Journal of Statistical Software, 27, 1–23.
Wilson, E. B., & Hilferty, M. M. (1931). The distribution of Chi-square. Proceedings of the National Academy of Sciences of the United States of America, 17, 684–688.
Ye, Zhi-Sheng, & Chen, Nan. (2017). Closed-form estimators for the gamma distribution derived from likelihood equations. The American Statistician, 71, 177–181.
Zhao, Jun, Kim, Sung Bum, & Kim, Hyoung-Moon. (2021). Closed-form estimators and bias-corrected estimators for the Nakagami distribution. Mathematics and Computers in Simulation, 185, 308–324.
Acknowledgements
This paper was written as part of Konkuk University’s research support program for its faculty on sabbatical leave in 2020. The authors thank Barry C. Arnold for valuable comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kim, HM., Kim, S., Jang, YH. et al. New closed-form estimator and its properties. J. Korean Stat. Soc. 51, 47–64 (2022). https://doi.org/10.1007/s42952-021-00118-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42952-021-00118-4