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Journal of Statistical Theory and Practice

, Volume 1, Issue 2, pp 205–226 | Cite as

Improved Estimation of Inverse Gaussian Shape Parameter and Measure of Dispersion with Prior Information

  • Housila P. Singh
  • Sheetal Pandit
Article

Abstract

This paper is intended to propose an improved class of shrinkage estimators for estimating the general parameter P(α) = λα (α, being an integer) of the Inverse Gaussian distribution using prior information λ0α of the parameter P(α) = λα under investigation. The properties of the suggested class of shrinkage estimators are studied and compared with the usual unbiased estimator, minimum mean squared error (MMSE) estimator and Singh and Pandit’s (2006) estimator. Realistic condition have been obtained under which the proposed class of shrinkage estimators are better than these estimators. In particular, the study has been concentrated over the problem of estimation of shape parameter P(1) = λ and the measure of dispersion P(−1) =1/λ of the Inverse Gaussian distribution. Numerical illustrations are given in the support of the proposed study.

AMS Subject Classification

62G32 

Keywords

Inverse Gaussian distribution Shrinkage estimator Bias Mean squared error 

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References

  1. Antoniadis, A., Besbeas, P., Sapatinas, T., 2001. Wavelet shrinkage for natural exponential families with cubic variance functions, Sankhyā, 63, A, (3), 309–327.MathSciNetzbMATHGoogle Scholar
  2. Banerjee, A.K., Bhattacharya, G.K., 1976. A purchase incidence model with Inverse Gaussian interpurchase times, Journal of American Statistical Association, 71, 823–829.CrossRefGoogle Scholar
  3. Chhikara, R.S., Folks, J.L., 1974. Estimation of the Inverse Gaussian distribution function, Journal of American Statistical Association, 69, 250–254.CrossRefGoogle Scholar
  4. Chhikara, R.S., 1975. Optimum test for the comparison of two Inverse Gaussian means, Journal of American Statistical Association., 17, 77–83.MathSciNetzbMATHGoogle Scholar
  5. Chhikara, R.S., Folks, J.L., 1975. Statistical distribution related to the Inverse Gaussian, Communication in Statistics, 4, 1081–1091.MathSciNetCrossRefGoogle Scholar
  6. Chhikara, R.S., Folks, J.L., 1976. Optimum procedure for the mean of the first passage time in Brownian motion with positive drift, Technometric, 18, 189–193.CrossRefGoogle Scholar
  7. Chhikara, R.S., Folks, J.L., 1977. The Inverse Gaussian distribution as a lifetime model, Technometrics, 19, 461–468.CrossRefGoogle Scholar
  8. Chhikara, R.S., Folks, J.L., 1989. The Inverse Gaussian distribution theory, methodology and application — Statistics, Textbooks and Monographs, 95, New York, Marcel Dekker.Google Scholar
  9. Ebrahimi, N., Hosmane, B., 1987. On shrinkage estimation of the exponential location parameter, Communication in Statistics, Theory and Methods, 16(9), 2623–2637.MathSciNetCrossRefGoogle Scholar
  10. Folks, J.L., Chhikara, R.S., 1978. The Inverse Gaussian distribution and its application, a review, Journal of the Royal Statistical Society, B, 40, 263–289.MathSciNetzbMATHGoogle Scholar
  11. Hasofer, A.M., 1964: A dam with Inverse Gaussian input, Proc.Camb.Phil.Soc., 60,931–933.MathSciNetCrossRefGoogle Scholar
  12. Hawkins, D., Olwell, D., 1997. Inverse Gaussian cumulative sum control charts for location and shape, Journal of the Royal Statistical Society, 46,D, 323.Google Scholar
  13. Hirano, K., Iwase, K., 1989. Minimum risk scale equivariant estimator — Estimating the mean of an Inverse Gaussian distribution with known coefficient of variation, Communication in Statistics, 18 (1), 189–197.MathSciNetCrossRefGoogle Scholar
  14. Howlader, H.A., 1985. Approximate Bayes estimation of reliability of two parameter Inverse Gaussian distribution, Communication in Statistics, Theory and Methods, 14(4), 937–946.MathSciNetCrossRefGoogle Scholar
  15. Iwase, K., Seto, N., 1983. UMVUE’s for the Inverse Gaussian distribution, Journal of American Statistical Association, 78, 660–663.MathSciNetCrossRefGoogle Scholar
  16. Iwase, K., Seto, N., 1985. UMVU estimators of the mode and limits of an interval for the Inverse Gaussian distribution, Communication in Statistics, Theory and Methods, 14, 1151–1161.MathSciNetCrossRefGoogle Scholar
  17. Iwase, K., 1987. UMVU estimation for the Inverse Gaussian distribution I (μ,cμ 2) with known C, Communication in Statistics, 16, 5, 1315–1320.MathSciNetCrossRefGoogle Scholar
  18. Iyenger, S., Patwardhan, G., 1988. Handbook of Statistics, 7, ( Krishnaiah, Ed.) Academic press.Google Scholar
  19. James, W., Stein, C., 1961. (A basic paper on Stein — type estimators), Proceedings of the 4th Berkeley Symposium on Mathematical Statistics,1, University of California Press, Berkeley, CA.361–379.Google Scholar
  20. Jani, P.N., 1991. A class of shrinkage estimators for the scale parameter of exponential distribution, IEEE Transaction on Reliability, 40, 60–70.CrossRefGoogle Scholar
  21. Joshi, S., Shah, M., 1991. Estimating the mean of an Inverse Gaussian distribution with known coefficient of variation, Communication in Statistics, Theory and Methods, 20(9), 2907–2912.MathSciNetCrossRefGoogle Scholar
  22. Korwar, R.M., 1980. On the UMVUE’s of the variance and its reciprocal of an Inverse Gaussian distribution, Journal of American Statistical Association, 75, 734–735.MathSciNetCrossRefGoogle Scholar
  23. Kourouklis, S., 1994. Estimation in the two parameter exponential distribution with prior information, IEEE Transaction on reliability, 43(3), 446–450.CrossRefGoogle Scholar
  24. Lancaster, A., 1972. A stochastic model for the duration of a strike, Journal of the Royal Statistical Society, A, 135, 257–271.CrossRefGoogle Scholar
  25. Mehta, J.S., Srinivasan, S.R., 1971. Estimation of the mean by shrinkage to a point; Journal of American Statistical Association, 66, 86–90.CrossRefGoogle Scholar
  26. Padmanabhan, P., 1978. Application of the Inverse Gaussian distribution in evaluation and estimation of conversion probabilities for convertible securities. MBA research paper, Mc.Gill University Montreal, Quebec.Google Scholar
  27. Padgett, W.J., 1979. Confidence bounds on reliability of the Inverse Gaussian model, IEEE Transaction on Reliability, R-28, 165–168.CrossRefGoogle Scholar
  28. Padgett, W.J., 1981. Bayes estimation of reliability for the Inverse Gaussian models, IEEE Transaction on Reliability,R- 30, 384–385.CrossRefGoogle Scholar
  29. Pandey, B.N., Malik, H.J., 1987. Some improved estimators for a measure of dispersion of an Inverse Gaussian distribution, Communication in Statistics, Theory and Methods, A7, (11), 3935–3949.MathSciNetCrossRefGoogle Scholar
  30. Pandey, B.N., Malik, H.J., 1988. Some improved estimators for a measure of dispersion of an Inverse Gaussian distribution, Communication in Statistics-Theory and Methods, 17(11), 3935–3949.MathSciNetCrossRefGoogle Scholar
  31. Pandey, B.N., Malik, H.J., 1989. Estimation of the mean and the reciprocal of the mean of the Inverse Gaussian distribution, Communication in Statistics and Simulation, 18(3), 1187–1201.MathSciNetCrossRefGoogle Scholar
  32. Patel, M.N., 1998. Progressively censored sample from Inverse Gaussian distribution, Aligarh Journal of Statistics, 17 & 18, 28–34.MathSciNetGoogle Scholar
  33. Pandit, S., 2004. Estimation of parameters of Inverse Gaussian distribution with prior information, M.Phil.Thesis submitted to Vikram University, Ujjain, (M.P.), India.Google Scholar
  34. Saxena, S., Singh, H.P., 2004. Estimating various measures in Normal population through a single class of estimators, Journal of the Korean Statistical Society, 33, 3, 323–337.MathSciNetGoogle Scholar
  35. Schrödinger, E., 1915. Zur Theorie Der Fall-und Steigversu che an Teilchenmit Brownscher Bewegung, Physikali Sche Zeitschrift, 16, 289–295.Google Scholar
  36. Seshadri, R., 1999. The Inverse Gaussian distribution, Statistical Theory and Application, Springer Verlag, New York.zbMATHGoogle Scholar
  37. Sen, A., Khattree, R., 2000. Revisiting the problem of estimating the Inverse Gaussian parameters, IAPQR Transactions, 25, 2, 63–79.MathSciNetzbMATHGoogle Scholar
  38. Shah, M.M., Joshi, S.M., 2003. Bounds for minimum risk scale equivariant and UMVUE estimators, Journal of Indian Statistical Association, 41, 1, 105–116.MathSciNetGoogle Scholar
  39. Sheppard, C.W., 1962. Basic principles of the Tracer methods, New York, Wiley.Google Scholar
  40. Singh, H.P., Shukla, S.K., 2000. Estimation in the two-parameter Weibull distribution with prior information, IAPQR. Transactions, 25, 2,107–117.MathSciNetzbMATHGoogle Scholar
  41. Singh, H.P., Saxena, S., 2001. Improved estimation in one-parameter Exponential distribution with prior information, Gujarat Statistical Review, 28(1–2), 25–35.Google Scholar
  42. Singh, H.P., Saxena, S., 2002. Improved estimation of Weibull shape parameter with prior information in censored sampling, IAPQR Transactions, 27, 1, 51–61.MathSciNetzbMATHGoogle Scholar
  43. Singh, H.P., Pandit, S., 2006. Estimation of shape parameter and measure of dispersion of Inverse Gaussian distribution using prior information, Communicated to Communication in Statistics -Theory and Methods, USA.Google Scholar
  44. Thompson, J.R., 1968. Some shrinkage technique for estimating the mean, Journal of American Statistical Association, 63, 113–123.MathSciNetGoogle Scholar
  45. Travedi, R.J., Ratani, R.T., 1990. On estimation of reliability function for Inverse Gaussian distribution with known coefficient of variation, IAPQR, Transactions, 5, 2, 29–37.Google Scholar
  46. Tweedie, M.C.K., 1945. Inverse Statistical Variate, Nature, 155, 453.MathSciNetCrossRefGoogle Scholar
  47. Tweedie, M.C.K., 1956. Some statistical properties of Inverse Gaussian distribution, Virginia Journal Science, 7, 160–165.MathSciNetGoogle Scholar
  48. Tweedie, M.C.K., 1957a. Statistical properties of Inverse Gaussian distribution I, Annals of Mathematical Statististics, 28, 326–377.MathSciNetGoogle Scholar
  49. Tweedie, M.C.K., 1957b. Statistical properties of Inverse Gaussian distribution II, Annals of Mathematical Statististics, 28, 696–705.MathSciNetCrossRefGoogle Scholar
  50. Voinov, V.G., 1985. “Unbiased estimation of powers of the inverse of the mean and related problems”, Sankhya, B, 47, 354–364.MathSciNetzbMATHGoogle Scholar
  51. Wald, A., 1947. “Sequential Analysis” New York, John Wiley & Sons.zbMATHGoogle Scholar
  52. Whitmore, G.A., 1976. Management applications of the Inverse Gaussian distribution, International Journal of Management Science, 4, 215–223.Google Scholar
  53. Whitmore, G.A., 1979. An Inverse Gaussian model for labour turnover, Journal of the Royal Statistical Society, A, 142, 468–478.CrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2007

Authors and Affiliations

  1. 1.School of Studies in StatisticsVikram UniversityUjjainIndia
  2. 2.School of Studies in StatisticsVikram UniversityUjjainIndia

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