Automatic Control and Computer Sciences

, Volume 50, Issue 6, pp 423–431 | Cite as

Tolerance limits on order statistics in future samples coming from the Pareto distribution

  • N. A. NechvalEmail author
  • K. N. Nechval
  • S. P. Prisyazhnyuk
  • V. F. Strelchonok


It is often desirable to have statistical tolerance limits available for the distributions used to describe time-to-failure data in reliability problems. For example, one might wish to know if at least a certain proportion, say β, of a manufactured product will operate at least T hours. This question cannot usually be answered exactly, but it may be possible to determine a lower tolerance limit L(X), based on a random sample X, such that one can say with a certain confidence γ that at least 100β% of the product will operate longer than L(X). Then reliability statements can be made based on L(X), or, decisions can be reached by comparing L(X) to T. Tolerance limits of the type mentioned above are considered in this paper, which presents a new approach to constructing lower and upper tolerance limits on order statistics in future samples. Attention is restricted to invariant families of distributions under parametric uncertainty. The approach used here emphasizes pivotal quantities relevant for obtaining tolerance factors and is applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. It does not require the construction of any tables and is applicable whether the past data are complete or Type II censored. The proposed approach requires a quantile of the F distribution and is conceptually simple and easy to use. For illustration, the Pareto distribution is considered. The discussion is restricted to one-sided tolerance limits. A practical example is given.


order statistics F distribution lower tolerance limit upper tolerance limit Pareto distribution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Mendenhall, V., A bibliography on life testing and related topics, Biometrika, 1958, vol. 45, pp. 521–543.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Guttman, I., On the power of optimum tolerance regions when sampling from normal distributions, Ann. Math. Stat., 1957, vol. 28, pp. 773–778.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Wald, A. and Wolfowitz, J., Tolerance limits for a normal distribution, Ann. Math. Stat., 1946, vol. 17, pp. 208–215.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Wallis, W.A., Tolerance intervals for linear regression, Second Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, 1951, pp. 43–51.Google Scholar
  5. 5.
    Patel, J.K., Tolerance limits: A review, Commun. Stat.: Theory Methodol., 1986, vol. 15, pp. 2719–2762.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dunsmore, I.R., Some approximations for tolerance factors for the two parameter exponential distribution, Technometrics, 1978, vol. 20, pp. 317–318.CrossRefzbMATHGoogle Scholar
  7. 7.
    Guenther, W.C., Patil, S.A., and Uppuluri, V.R.R., One-sided ß-content tolerance factors for the two parameter exponential distribution, Technometrics, 1976, vol. 18, pp. 333–340.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Engelhardt, M. and Bain, L.J., Tolerance limits and confidence limits on reliability for the two-parameter exponential distribution, Technometrics, 1978, vol. 20, pp. 37–39.CrossRefzbMATHGoogle Scholar
  9. 9.
    Guenther, W.C., Tolerance intervals for univariate distributions, Nav. Res. Logist. Q., 1972, vol. 19, pp. 309–333.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hahn, G.J. and Meeker, W.Q., Statistical Intervals: A Guide for Practitioners, New York: John Wiley & Sons, 1991.CrossRefzbMATHGoogle Scholar
  11. 11.
    Nechval, N.A. and Nechval, K.N., Tolerance limits on order statistics in future samples coming from the twoparameter exponential distribution, Am. J. Theor. Appl. Stat., 2016, vol. 5, pp. 1–6.CrossRefGoogle Scholar
  12. 12.
    Arnold, B.C., Pareto Distributions, Burtonsville: International Cooperative Publishing House, 1983.zbMATHGoogle Scholar
  13. 13.
    Beaumont, G.P., Intermediate Mathematical Statistics, London: Chapman and Hall, 1980.CrossRefzbMATHGoogle Scholar
  14. 14.
    Evans, M., Hastings, N., and Peacock, B., Statistical Distributions, New York: Wiley, 2000, 3rd ed.zbMATHGoogle Scholar
  15. 15.
    Aban, I.B., Meerschaert, M.M., and Panorska, A.K., Parameter estimation for the truncated Pareto distribution, J. Am. Stat. Assoc., 2006, vol. 101, pp. 270–277.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Asimit, A.V., Furman, E., and Vernic, R., On a multivariate Pareto distribution, Insur.: Math. Econ., 2010, vol. 46, pp. 308–316.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Rytgaard, M., Estimation in the Pareto distribution, ASTIN Bull., 1990, vol. 20, pp. 201–216.CrossRefGoogle Scholar
  18. 18.
    Gelb, J.M., Heath, R.E., and Tipple, G.L., Statistics of distinct clutter classes in midfrequency active sonar, IEEE Oceanic Eng., 2010, vol. 35, pp. 220–229.CrossRefGoogle Scholar
  19. 19.
    Chakravarthi, P.R. and Ozturk, A., On determining the radar threshold from experimental data, IEEE Proceed., 1991, pp. 594–598.Google Scholar
  20. 20.
    Piotrkowski, M., Some preliminary experiments with distribution-independent EVTCFAR based on recorded radar data, IEEE, 2008.Google Scholar
  21. 21.
    Farshchian, M. and Posner, F.I., The Pareto distribution for low grazing angle and high resolution X-band sea clutter, IEEE Radar Conf., 2010, pp. 789–793.Google Scholar
  22. 22.
    Nechval, N.A. and Nechval, K.N., Characterization theorems for selecting the type of underlying distribution, Proceedings of the 7th, Vilnius: Conference on Probability Theory and 22nd European Meeting of Statisticians, Vilnius, 1998, pp. 352–353.Google Scholar
  23. 23.
    Muller, P.H., Neumann, P., and Storm, R., Tables of Mathematical Statistics, Leipzig: VEB Fachbuchverlag, 1979.Google Scholar

Copyright information

© Allerton Press, Inc. 2016

Authors and Affiliations

  • N. A. Nechval
    • 1
    Email author
  • K. N. Nechval
    • 2
  • S. P. Prisyazhnyuk
    • 3
  • V. F. Strelchonok
    • 1
  1. 1.Department of MathematicsBaltic International AcademyRigaLatvia
  2. 2.Department of Applied MathematicsTransport and Telecommunication InstituteRigaLatvia
  3. 3.Department of Geoinformation SystemsSaint-Petersburg National Research University of Information Technologies, Mechanics and OpticsSt.-PetersburgRussia

Personalised recommendations