Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmed, A., Reshi, J.A., Mir, K.A.: Structural properties of size biased Gamma distribution. IOSR J. Math. 5(2), 55–61 (2013)
Akaike, H.: A new look at the statistical model identification. IEEE Trans. Autom. Control 19, 716–723 (1974)
Ata, N., Demirhan, H.: A simulation study on cox regression with weighted estimations. Electron. J. Appl. Stat. Anal. 7(1), 26–36 (2014)
Bozdogan, H.: Model selection and Akaike’s information criterion (AIC): the general theory and its analytical extensions. Psychometrika 52, 345–370 (1987)
Caroni, C.: The correct ‘‘ball bearings’’ data. Lifetime Data Anal. 8, 395–399 (2002)
Cramér, H.: On the composition of elementary errors. Scand. Actuar. J. 1928(1), 13–74 (1928)
David, H.A., Nagaraja, H.N.: Order statistics, 3rd edn. Wiley, Hoboken (2003)
George, D., George, S.: Marshall–Olkin Esscher transformed Laplace distribution and processes. Braz. J. Prob. Stat. 27(2), 162–184 (2013)
Heggland, K., Lindqvist, B.H.: A nonparametric monotone maximum likelihood estimator of time trend for repairable systems data. Reliab. Eng. Syst. Saf. 92, 575–584 (2007)
Lawless, J.F.: Statistical models and methods for lifetime data. Wiley, New York (1982)
Mdlongwa, P., Oluyede, B., Amey, A., Huang, S.: The Burr XII modified Weibull distribution: model, properties and applications. Electron. J. Appl. Stat. Anal. 10(1), 118–145 (2017)
Mudasir, S., Ahmad, S.P.: Structural properties of length biased Nakagami distribution. Int. J. Mod. Math. Sci. 13(3), 217–227 (2015)
Patil, G.P., Rao, C.R.: Weighted distributions and size-biased sampling with applications to wildlife populations and human families. Biometrics 34, 179–189 (1978)
Perveen, Z., Ahmed, Z., Ahmad, M.: On size-biased double weighted exponential distribution (SDWED). Open J. Stat. 2016(6), 917–930 (2016)
Rama, S., Shukla, K., Fesshaye, H.: On weighted Lindley distribution and its applications to model lifetime data. Jacobs J. Biostat. 1(1), 002 (2016)
Rao, C.R.: On discrete distributions arising out of methods of ascertainment. In: Patil, G.P. (ed.) Classical and Contagious Discrete Distributions, pp. 320–322. Pergamum Press and Statistical Publishing Society, Calcutta (1965)
Rao, C.R., Patil, G.P.: The weighted distributions: a survey of their applications. Appl. Stat. 383, 383–405 (1977)
Rényi, A.: On measures of entropy and information, pp. 547–561. University of California Press, Berkeley (1961)
Safi, S., AlSheikh Ahmed, R.: Generalized order statistics from generalized exponential distributions in explicit forms. Electron. J. Appl. Stat. Anal. 6(2), 222–237 (2013)
Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6, 461–464 (1978)
Shanker, R.: Rama distribution and its application. Int. J. Stat. Appl. 7(1), 26–35 (2017)
Shanker, R., Shukla, K.K.: Ishita distribution and its applications. Biom. Biostat. Int. J. 5(2), 1–9 (2017)
Stephens, M.A.: EDF statistics for goodness of fit and some comparisons. J. Am. Stat. Assoc. 69, 730–737 (1974)
Warren, W.G. (1975) Statistical distribution of forestry and forest products research. In: A Modern Course on Statistical Distributions in Scientific Work. NATO Advanced Study Institutes Series, vol. 17, pp. 369–384
Zelen, M.: Problems in Cell Kinetics and Early Detection of Disease. Reliability and Biometry, pp. 701–726. SIAM, Philadelphia (1974)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Al-Omari, A.I., Al-Nasser, A.D. & Ciavolino, E. A size-biased Ishita distribution and application to real data. Qual Quant 53, 493–512 (2019). https://doi.org/10.1007/s11135-018-0765-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11135-018-0765-y