Estimation of the density and cumulative distribution functions of the exponentiated Burr XII distribution

Amal S. Hassan; Salwa M. Assar; Kareem A. Ali; Heba F. Nagy

doi:10.21307/stattrans-2021-044

Estimation of the density and cumulative distribution functions of the exponentiated Burr XII distribution

Amal S. Hassan Department of Mathematical Statistics, Cairo University, Faculty of Graduate Studies for Statistical Research, Egypt ORCID:https://orcid.org/0000-0003-4442-8458 , Salwa M. Assar Department of Mathematical Statistics, Cairo University, Faculty of Graduate Studies for Statistical Research, Egypt ORCID:https://orcid.org/0000-0001-7450-7486 , Kareem A. Ali Department of Mathematical Statistics, Cairo University, Faculty of Graduate Studies for Statistical Research, Egypt ORCID:https://orcid.org/0000-0002-2522-7733 , Heba F. Nagy Department of Mathematical Statistics, Cairo University, Faculty of Graduate Studies for Statistical Research. Egypt ORCID:https://orcid.org/0000-0003-0262- 205X Statistics in Transition new series, vol. 22, 2021, 4, pages: 171-189 Published online: 8 December 2021 DOI 10.21307/stattrans-2021-044

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ABSTRACT

The exponentiated Burr Type XII (EBXII) distribution has wide applications in reliability and economic studies. In this article, the estimation of the probability density function and the cumulative distribution function of EBXII distribution is considered. We examine the maximum likelihood estimator, the uniformly minimum variance unbiased estimator, the least squares estimator, the weighted least squares estimator, the maximum product spacing estimator, the Cramér–von-Mises estimator, and the Anderson–Darling estimator. We derive analytical forms for the bias and mean square error. A simulation study is performed to investigate the consistency of the suggested methods of estimation. Data relating to the wind speed and service times of aircraft windshields are used with the studied methods. The simulation studies and real data applications have revealed that the maximum likelihood estimator performs more efficiently than its remaining counterparts.

KEYWORDS

exponentiated Burr Type XII model, least squares estimator, maximum likelihood estimator, uniform minimum variance unbiased estimator, weighted least squares estimator

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