Characterizations of gamma, inverse Gaussian, and negative binomial distributions via their length-biased distributions
Received: 07 January 1992 Revised: 12 October 1992 DOI:
Cite this article as: Ahmed, A.N. & Abouammoh, A.M. Statistical Papers (1993) 34: 167. doi:10.1007/BF02925538 Abstract
A general form for characterizing inverse Gaussian and Wald distributions, based on their respective length-biased distributions, is introduced. Further results for characterizations of the gamma distribution, the negative binomial distribution and some mixtures of them by using their lengthbiased distributions are establised.
Key words Length biased distribution Negative binomial distribution Gamma distribution Inverse Gaussain distribution Chi-square distribution Mixed distributions References
Ahasnullah, M. and Kirmani, S. (1984). A characterization of Wald distribution. Naval Research Logistics Quarterly, Vol. 31, pp. 155–158.
Chikara, R. and Folks, J. (1977). The inverse Gaussian distribution as a lifetime model. Technometrics, Vol. 19, pp. 461–468.
Cox, D. (1962). Renewal Theory, Methuen and Co. Ltd.
Gupta, R. and Keating, J. (1986). Relations for reliability measures under length-biased sampling. Scandinavian Journal of statistics 13, ppl 49–56.
Khattree, R. (1989). Characterization of inverse-Gaussian and Gamma distributions through their length-biased distributions. IEEE. Trans. on Reliability, Vol. 38, No. 5, pp. 610–611.
Patil, P. and Rao, R. (1977). The weighted distributions: A survey of their applications. In Applications in Statistics (ed. Krishnaiah, R.). North Holland, pp. 383–405.
Ross, S. (1983). Stochastic Processes. John Wiley & Sons, Inc., New York.