Advertisement

Metrika

, Volume 81, Issue 8, pp 1005–1024 | Cite as

On purely sequential estimation of an inverse Gaussian mean

  • Sudeep R. Bapat
Article
  • 46 Downloads

Abstract

The first part of this paper deals with developing a purely sequential methodology for the point estimation of the mean \(\mu \) of an inverse Gaussian distribution having an unknown scale parameter \(\lambda \). We assume a weighted squared error loss function and aim at controlling the associated risk function per unit cost by bounding it from above by a known constant \(\omega \). We also establish first-order and second-order asymptotic properties of our stopping rule. The second part of this paper deals with obtaining a purely sequential fixed accuracy confidence interval for the unknown mean \(\mu \), assuming that the scale parameter \(\lambda \) is known. First-order asymptotic efficiency and asymptotic consistency properties are also built of our proposed procedures. We then provide extensive sets of simulation studies and real data analysis using data from fatigue life analysis to show encouraging performances of our proposed stopping strategies.

Keywords

Fatigue life Inverse Gaussian Purely sequential Fixed-accuracy intervals Point estimation First-order asymptotic efficiency First-order asymptotic consistency 

Mathematics Subject Classification

62L12 62L05 62F12 62P30 

Notes

Acknowledgements

The author would like to sincerely thank the editor in chief, Dr. Hajo Holzmann and the anonymous referee for their valuable and constructive comments which greatly improved an earlier manuscript.

Compliance with ethical standards

Conflicts of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

References

  1. Anscombe FJ (1952) Large sample theory of sequential estimation. Proc Camb Philos Soc 48:600–607MathSciNetCrossRefGoogle Scholar
  2. Banerjee S, Mukhopadhyay N (2016) A general sequential fixed-accuracy confidence interval estimation methodology for a positive parameter: illustrations using health and safety data. Ann Inst Stat Math 68:541–570MathSciNetCrossRefGoogle Scholar
  3. Bapat SR (2018a) Purely sequential fixed accuracy confidence intervals for \(P(X < Y)\) under bivariate exponential models. Am J Math Manag Sci.  https://doi.org/10.1080/01966324.2018.1465867 CrossRefGoogle Scholar
  4. Birnbaum ZW, Saunders SC (1958) A statistical model for life length of material. J Am Stat Assoc 53:151–160MathSciNetCrossRefGoogle Scholar
  5. Birnbaum ZW, Saunders SC (1969) Estimation for a family of life distributions. J Appl Prob 6:319–327MathSciNetCrossRefGoogle Scholar
  6. Chaturvedi A (1996) Correction to sequential estimation of an inverse Gaussian parameter with prescribed proportional closeness. Calcutta Stat Assoc Bull 35:211–212CrossRefGoogle Scholar
  7. Chaturvedi A, Pandey SK, Gupta M (1991) On a class of asymptotically risk-efficient sequential procedures. Scand Actuar J 1:87–96MathSciNetCrossRefGoogle Scholar
  8. Chhikara RS, Folks JL (1989) The inverse gaussian distribution, theory, methodology and applications. Marcel Dekker Inc., New YorkzbMATHGoogle Scholar
  9. Chow YS, Robbins H (1965) On the asymptotic theory of fixed width sequential confidence intervals for the mean. Ann Math Stat 36:457–462MathSciNetCrossRefGoogle Scholar
  10. Edgeman RL, Salzburg PM (1991) A sequential sampling plan for the inverse Gaussian mean. Stat Pap 32:45–53MathSciNetCrossRefGoogle Scholar
  11. Folks JL, Chhikara RS (1978) The inverse Gaussian distribution and its statistical application—a review. J R Stat Soc B40:263–289MathSciNetzbMATHGoogle Scholar
  12. Ghosh M, Mukhopadhyay N (1975) Asymptotic normality of stopping times in sequential analysis. Unpublished ReportGoogle Scholar
  13. Ghosh M, Mukhopadhyay N (1979) Sequential point estimation of the mean when the distribution is unspecified. Commun Stat Ser A 8:637–652MathSciNetCrossRefGoogle Scholar
  14. Ghosh M, Mukhopadhyay N (1981) Consistency and asymptotic efficiency of two-stage and sequential procedures. Sankhya Ser A 43:220–227MathSciNetzbMATHGoogle Scholar
  15. Ghosh M, Mukhopadhyay N, Sen PK (1997) Sequential estimation. Wiley, New YorkCrossRefGoogle Scholar
  16. Johnson N, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1. Wiley, New YorkzbMATHGoogle Scholar
  17. Joshi S, Shah M (1990) Sequential analysis applied to testing the mean of an inverse Gaussian distribution with known coefficient of variation. Commun Stat 19(4):1457–1466MathSciNetCrossRefGoogle Scholar
  18. Lai TL, Siegmund D (1977) A nonlinear renewal theory with applications to sequential analysis I. Ann Stat 5:946–954MathSciNetCrossRefGoogle Scholar
  19. Lai TL, Siegmund D (1979) A nonlinear renewal theory with applications to sequential analysis II. Ann Stat 7:60–76MathSciNetCrossRefGoogle Scholar
  20. Leiva V, Hernandez H, Sanhueza A (2008b) An R package for a general class of inverse Gaussian distributions. J Stat Softw 26(4):1–21CrossRefGoogle Scholar
  21. Mukhopadhyay N (1988) Sequential estimation problems for negative exponential populations. Commun Stat Theory Methods Ser A 17:2471–2506MathSciNetCrossRefGoogle Scholar
  22. Mukhopadhyay N, Banerjee S (2014) Purely sequential and two stage fixed-accuracy confidence interval estimation methods for count data for negative binomial distributions in statistical ecology: one-sample and two-sample problems. Seq Anal 33:251–285MathSciNetCrossRefGoogle Scholar
  23. Mukhopadhyay N, Bapat SR (2016a) Multistage point estimation methodologies for a negative exponential location under a modified linex loss function: illustrations with infant mortality and bone marrow data. Seq Anal 35:175–206MathSciNetCrossRefGoogle Scholar
  24. Mukhopadhyay N, Bapat SR (2016b) Multistage estimation of the difference of locations of two negative exponential populations under a modified linex loss function: real data illustrations from cancer studies and reliability analysis. Seq Anal 35:387–412MathSciNetCrossRefGoogle Scholar
  25. Mukhopadhyay N, Bapat SR (2017a) Purely sequential bounded-risk point estimation of the negative binomial mean under various loss functions: one sample problem. Ann Inst Stat Math.  https://doi.org/10.1007/s10463-017-0620-2 CrossRefzbMATHGoogle Scholar
  26. Mukhopadhyay N, Bapat SR (2017b) Purely sequential bounded-risk point estimation of the negative binomial means under various loss functions: multisample problems. Seq Anal 36(4):490–512CrossRefGoogle Scholar
  27. Mukhopadhyay N, de Silva BM (2009) Sequential methods and their applications. CRC, Boca RatonzbMATHGoogle Scholar
  28. Mukhopadhyay N, Solanky TKS (1994) Multistage selection and ranking procedures. Marcel Dekker Inc., New YorkzbMATHGoogle Scholar
  29. R Core Team (2014) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, AustriaGoogle Scholar
  30. Schrodinger E (1915) Zur theorie der fall und steigversuche an teilchen mit Brown-scher bewegung. Physikalische Zeitschrift 16:289–295Google Scholar
  31. Sen PK (1981) Sequential nonparametrics. Wiley, New YorkzbMATHGoogle Scholar
  32. Seshadri V (1993) The inverse Gaussian distribution—a case study in exponential families. Clarendon Press, OxfordGoogle Scholar
  33. Seshadri V (1999) The invere Gaussian distribution, statistical theory and applications. Springer, New YorkzbMATHGoogle Scholar
  34. Wiener N (1939) The Ergodic theorem. Duke Math J 5:1–18MathSciNetCrossRefGoogle Scholar
  35. Woodroofe M (1977) Second order approximation for sequential point and interval estimation. Ann Stat 5:984–995MathSciNetCrossRefGoogle Scholar
  36. Woodroofe M (1982) Nonlinear renewal theory in sequential analysis, CBMS 39. SIAM, PhiladelphiaCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Statistics and Applied Probability, South HallUniversity of California, Santa BarbaraSanta BarbaraUSA

Personalised recommendations