MISCLASSIFICATION IN SIZE-BIASED MODIFIED POWER SERIES DISTRIBUTION AND ITS APPLICATIONS

  • Received : 2008.08.02
  • Accepted : 2009.03.15
  • Published : 2009.03.25

Abstract

A misclassified size-biased modified power series distribution (MSBMPSD) where some of the observations corresponding to x = c + 1 are misclassified as x = c with probability $\alpha$, is defined. We obtain its recurrence relations among the raw moments, the central moments and the factorial moments. Discussion of the effect of the misclassification on the variance is considered. To illustrate the situation under consideration some of its particular cases like the size-biased generalized negative binomial (SBGNB), the size-biased generalized Poisson (SBGP) and sizebiased Borel distributions are included. Finally, an example is presented for the size-biased generalized Poisson distribution to illustrate the results.

References

  1. Cohen, A.C.: Estimating the parameters of a modified Poisson distribution, Amer. Statist. Assoc. 55 (1960), 139-143. https://doi.org/10.1080/01621459.1960.10482054
  2. Cohen, A.C.: Estimation in the Poisson distribution when sample value of c+1 is sometimes erroneously reported as c, Ann.Inst.Stat.Math.9 (1960),189-193.
  3. Cohen, A.C.: Misclassified data from binomial distribution, Technometrics, 2(1960), 109-113. https://doi.org/10.1080/00401706.1960.10489884
  4. Jani, P.N. and Shah, S.M.: Misclassification in modified power series distribution in which the value one is sometimes reported as zero and some of its applications, Metron, 37(1979), 121-136.
  5. Patel, A.I. and Patel, I.D.: Recurrence relations for moments of the so called misclassified distribution, Assam Statistical Review, 10(1996), 107-119.
  6. Patel, A.I. and Patel, I.D.: Misclassification in modified Power Series Distribution and its applications, Assam Statistical Review, 15 (2001), 55-69.
  7. Bortkiewicz, L. von: Das Gesetz der Kleinen Zahlen, Leipzig: Teubner, 1898.
  8. Williford, W. O., and Bingham, S. F.: Bayesian estimation of the parameters in two modified Poisson distributions. Commun. Statist.-Theory Methods, A8 13(1979), 1315-1326.
  9. Blumenthal, S.: Proportional sampling in life length studies, Technometrics, 9(1967), 205-218 https://doi.org/10.1080/00401706.1967.10490456
  10. Scheaffer, R.L: Size biased sampling, Technometrics, 14(1972), 635-644. https://doi.org/10.1080/00401706.1972.10488952
  11. Gupta, R.C.: Some characterizations of discrete distributions. Communications in Statistics, A5 1 (1975), 45-48.
  12. Gupta, R.C.: Some characterizations of distributions by properties of their forward and backward recurrence times in a renewal process, Scand. J.Stat., Theory Appl., 3(1976), 215-216.
  13. Gupta, R.C: Waiting time paradox and size biased sampling, Communications in Statistics, A8 13 (1979), 601-607
  14. Gupta, R.C: Some characterizations of renewal densities with emphasis in reliability, Mathematische operations- forschung und Statistics, 15 (1984), 571-579.
  15. Gupta, R.C. and Tripathi, R.C: A Comparison between the ordinary and the length -biased modified power series distributions with applications, Commun.Statist.-Theory Methods,16(1987),1195-1206. https://doi.org/10.1080/03610928708829431
  16. Gupta, R.C. and Tripathi, R.C.: Statistical inference based on the length-biased data for the modified power series distributions, Commun. Statist.-Theory Methods, 21(1992), 519-537. https://doi.org/10.1080/03610929208830793
  17. Gupta, R.C.: Modified Power Series Distribution and some of its applications, Sankhya, Series B, 36(1974), 288-298.
  18. Patil, G. P.: Certain properties of the generalized power series distributions, Ann. Inst. Stat. Math. Tokyo.14 (1962), 179-182. https://doi.org/10.1007/BF02868639
  19. Noack, A: A class of random variables with discrete distributions. Ann.Math.Stat., 21(1950), 127-132. https://doi.org/10.1214/aoms/1177729894
  20. Jain, G.C. and Consul, P.C.: A generalized negative binomial distribution, SIAM J.Appl.Math.21 (1971), 501-513. https://doi.org/10.1137/0121056
  21. Consul, P.C and Jain, G.C.: A generalization of the Poisson distribution, Technometrics, 15(1973), 791-799. https://doi.org/10.1080/00401706.1973.10489112
  22. Shoukri, M.M. and Consul, P.C.: Bayesian analysis of generalized Poisson distribution, Commun.Statist.-Theory Methods, 18 (1989), 1465-1480.
  23. Borel, E.: Sur l'emploi du theoreme de Bernoulli pour faciliter le calcul d'un infinite de coefficients. Application au probleme de l' attente a un guichet, comptes rendus, Acedemic des sciences, Paris, series A, 214(1942), 452-456.
  24. Mishra, A. and Singh, S.K.: A size biased Lagrangian Poisson Distribution and its use in zero-truncated cases, Ranchi university Mathematical Journal,24(1993), 21-28.
  25. Williams, C.B.: The number of publications written by biologist, Annals of Eugenics, 12(1944), 193-196.