Skip to main content
SpringerLink
Log in

The use of the disguised Wishart distribution in a Bayesian approach to tolerance region construction

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

Summary

In attacking the problem of this paper (see Section 1), the authors were confronted with finding the distribution of a (k×k) matrix of random variablesR=P′VP, wherePP′=Σ -1, and where the (k×k) symmetric matrix Σ-1 has the Wishart distribution, matrix [(n−1)V]−1, and degrees of freedom (n−1), withV a (k×k) symmetric positive definite matrix of constants. This distribution (whenP is lower triangular with positive diagonal elements), and a related result, has recently been found by the authors and given in Tan and Guttman [7]. In this paper we use these results (stated here without proof in Theorems 1.1 and 1.2) to help us construct a β-expectation tolerance region, when sampling is from thek-variate normal,N(μ,Σ), where Σ is positive definite.

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

  1. Aitchison, J. and Sculthorpe, Diane (1965). Some problems of statistical prediction,Biometrika,52, 469–483.

    Article  MATH  MathSciNet  Google Scholar 

  2. Deemer, W. L. and Olkin, I. (1951). The Jacobians of certain matrix transformations useful in multivariate analysis. Based on lectures of P. L. Hsu at the University of North Carolina, 1947,Biometrika,38, 345–367.

    Article  MATH  MathSciNet  Google Scholar 

  3. Fraser, D. A. S. and Guttman, Irwin (1956). Tolerance regions,Ann. Math. Statist.,27, 162–179.

    MATH  MathSciNet  Google Scholar 

  4. Guttman, Irwin (1971). A Bayesian analogue of Paulson's Lemma and its use in tolerance region construction when sampling from the multivariate normal,Ann. Inst. Statist. Math.,23, 67–76.

    MATH  MathSciNet  Google Scholar 

  5. Paulson, E. (1943). A note on tolerance limits,Ann. Math. Statist.,14, 90–93.

    MATH  MathSciNet  Google Scholar 

  6. Tiao, G. C. and Guttman, Irwin (1965). The inverted Dirichlet distribution with applications,J. Amer. Statist. Ass.,60, 793–805.

    Article  MATH  Google Scholar 

  7. Tan, W. Y. and Guttman, Irwin (1971). A disguised Wishart variable and related theorems,J. R. Statist. Soc., Ser. B,30, 147–152.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Montreal, Montreal, Canada

    Irwin Guttman & W. Y. Tan

  2. University of Wisconsin, Wisconsin, USA

    Irwin Guttman & W. Y. Tan

Authors
  1. Irwin Guttman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. Y. Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was partially supported by the National Institute of Health under Grant No. GM 15422, and the Wisconsin Research Foundation.

About this article

Cite this article

Guttman, I., Tan, W.Y. The use of the disguised Wishart distribution in a Bayesian approach to tolerance region construction. Ann Inst Stat Math 25, 549–556 (1973). https://doi.org/10.1007/BF02479398

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02479398

Keywords

Search

Navigation