New efficient estimators for the Weibull distribution

Abstract

For some distributions, there is no available closed-form maximum likelihood estimator (MLE). This could be a problem when estimating the parameters of state-space models or of real-time processing models, because of time consuming iterations required to obtain an MLE. A simple method of obtaining a closed-form and efficient estimators for the univariate Weibull distribution is presented. Furthermore, an alternative efficient estimator for bivariate Weibull distribution parameters other than the MLE is also suggested. For the bivariate Weibull distribution, one estimator is not in closed-form, but the others are all in closed-form. In general, if we have a closed-form $\sqrt{n}$-consistent estimator for each parameter, then we have a closed-form solution set which is efficient and has asymptotic normality. This is quite striking since we can obtain such a closed-form and efficient estimator using a very simple strategy. Roughly speaking, our suggested estimator is approximately 11 times as fast as the MLE for univariate case and 85 times for bivariate case. A simulation study is done for univariate and bivariate Weibull distributions to find some small sample properties. A real data illustration is also provided.

Keywords:

• Closed-form estimator
• efficient estimator
• MLE
• real-time processing model
• state-space model

2020 Mathematics Subject Classification:

Additional information

Funding

The corresponding author’s research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2018R1D1A1B07045603) and the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (2021R1A4A5032622). Zhao’s research was supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ23A010012.