Skip to main content
SpringerLink
Log in

Confidence estimation for tolerance intervals

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

The post-data performances of normal tolerance intervals are studied. Under a robust Bayesian predictive scheme, we establish the ordering and bounds of the confidence estimators. It is found that the nominal confidence coefficient tends to be extreme yet coincides with the limiting Bayes estimators in some scenarios. A remark on the choice of beta priors is also given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aitchison, J., Dunsmore, I.R. (1975). Statistical prediction analysis. Cambridge University Press.

  • Berger, J.O. (1984). The robust Bayesian viewpoint. In: J.B. Kadane (Ed.), Robustness of Bayesian analyses. New York: Elsevier, pp. 63–124.

    Google Scholar 

  • Berger, J.O. (1988). An alternative: the estimated confidence approach. In: S.S. Gupta, J.O. Berger, (Eds.), Statistical decision theory and related topics IV (Vol. 1, pp. 85–90) Berlin Heidelberg, New York: Springer.

  • Berger J.O. (1994). An overview of robust Bayesian analysis. Test, 3: 5–124

    Article  MATH  MathSciNet  Google Scholar 

  • Bernardo J.M., Smith A.F.M. (1993) Bayesian theory. Wiley, New York

    Google Scholar 

  • Carrol R.J., Ruppert D. (1999) Prediction and tolerance intervals with transformation and/or weighting. Technometrics 33: 197–210

    Article  Google Scholar 

  • Casella G., Berger R.L. (1987) Reconciling Bayesian and Frequentist evidence in the one-sided testing problem. JASA, 82: 106–111

    MATH  MathSciNet  Google Scholar 

  • Di Bucchianico A.D., Einmahl J., Mushkudiani N. (2001) Smallest nonparametric tolerance regions. Annals of Statistics 29: 1325–1343

    MathSciNet  Google Scholar 

  • de Finetti B. (1991). Theory of probability. WileyInterscience Publishers, New York

    Google Scholar 

  • Geisser S. (1993). Predictive inference: an introduction. Chapman & Hall, New York

    MATH  Google Scholar 

  • Goutis C., Casella G. (1992) Increasing the confidence in Student’s t-interval. Annals of Statistics 20: 1501–1513

    MATH  MathSciNet  Google Scholar 

  • Goutis C., Casella G. (1995) Frequentist post-data inference. International Statistical Review 63: 325–344

    Article  MATH  Google Scholar 

  • Green J.R. (1969). Inference concerning probabilities and quantiles. JRSS B 31: 310–316

    Google Scholar 

  • Guttman I. (1970). Statistical tolerance regions: classical and bayesian. Griffin, London

    MATH  Google Scholar 

  • Hwang J.T.G., Brown L.D. (1991) The estimated confidence approach under the validity constraint criterion. Annals of Statistics 19: 1964–1977

    MATH  MathSciNet  Google Scholar 

  • Jílek M. (1981) A bibliography of statistical tolerance regions. Mathematische Operationsforschung und Statistik, Series Statistics 12: 441–456

    MATH  MathSciNet  Google Scholar 

  • Jílek M., Ackermann H. (1989) A bibliography of statistical tolerance regions II. Statistics, 20: 165–172

    Article  MATH  MathSciNet  Google Scholar 

  • Koziol J.A., Tuckwell H.C. (1999). A Bayesian method for combining statistical tests. Journal of Statistical Planning and Inference 78: 317–323

    Article  MATH  MathSciNet  Google Scholar 

  • Oh H.S., DasGupta A. (1999). Comparison of the P-value and posterior probability. Journal of Statistical Planning and Inference 76: 93–107

    Article  MATH  MathSciNet  Google Scholar 

  • Patel J.K.(1986) Tolerance limits—a review. Communication in Statistics: Theory and Methods 15: 2719–2762

    Article  MATH  Google Scholar 

  • Pearson E.S. (1939) x The probability integral transformation for testing goodness of fit and combining independent tests of significance. Biometrika 30: 134–148

    MathSciNet  Google Scholar 

  • Shaked M., Shanthikumar J.G. (1994). Stochastic orders and their applications. Academic, London

    MATH  Google Scholar 

  • Szekli R. (1995) Stochastic ordering and dependence in applied probability. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  • Tsao C.A. (2002) Assessing post-data weight of evidence (Technical Report).National Dong Hwa University, Taiwan

    Google Scholar 

  • Tsao C.A., Hwang J.T.G. (1999) Generalized Bayes confidence estimators for Fieller’s confidence sets. Statistica Sinica 9: 795–810

    MATH  MathSciNet  Google Scholar 

  • Tsao C.A., Tseng Y.-L. (2004) A statistical treatment of the problem of division. Metrika 59: 289–303

    Article  MathSciNet  MATH  Google Scholar 

  • Tsay J.J., Tsao C.A. (2003) Statistical gambler’s ruin problem. Communications in Statistics: Theory and Methods 32: 1377–1359

    Article  MathSciNet  Google Scholar 

  • Wilks S.S. (1941) Determination of sample size for setting tolerance limits. Annals of Mathematical Statistics 12: 91–96

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, National Dong Hwa University, Hualien, Taiwan

    C. Andy. Tsao & Yu-Ling Tseng

Authors
  1. C. Andy. Tsao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu-Ling Tseng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu-Ling Tseng.

About this article

Cite this article

Tsao, C.A., Tseng, YL. Confidence estimation for tolerance intervals. Ann Inst Stat Math 58, 441–456 (2006). https://doi.org/10.1007/s10463-005-0008-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-005-0008-6

Keywords

Search

Navigation