Quality & Quantity

, Volume 53, Issue 1, pp 493–512 | Cite as

A size-biased Ishita distribution and application to real data

  • Amer Ibrahim Al-Omari
  • Amjad D. Al-Nasser
  • Enrico Ciavolino


The present paper offers a new extension to the Ishita distribution called Size-Biased Ishia distribution (SBID). Various structural statistical properties of this distribution are derived such as the jth moment, moment generating function, the coefficients of variation, skewness and kurtosis. Also, the distribution of order statistics, harmonic mean, mode, reliability analysis, maximum likelihood estimation are provided, as well as the Fisher’s information, generalized and Renyi entropies are derived. The main advantage of using sized-based distributions appears when the sample are recorded with unequal probabilities. Accordingly, the superiority of the SBI distribution is illustrated to ball bearings data. It is shown that the SBID is the most appropriate model for this data set as compared to Rama distribution, Ishita distribution and Marshall–Olkin Esscher Transformed Laplace distribution. We believe that the SBID is an alternative distribution to lifetime data analysis.


Ishita distribution Size-biased Garima distribution Order statistics Hazard rate function Reliability function Generalized and Renyi entropies 



The authors would like to acknowledge the time and effort devoted by the editorial team and anonymous reviewers to improving the quality of this article.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceAl al-Bayt UniversityMafraqJordan
  2. 2.Department of Statistics, Science FacultyYarmouk UniversityIrbidJordan
  3. 3.Department of History, Society and Human StudyUniversity of SalentoLecceItaly

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