Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution
American Journal of Theoretical and Applied Statistics
Volume 5, Issue 2-1, March 2016, Pages: 1-6
Received: Sep. 9, 2015;
Accepted: Sep. 10, 2015;
Published: Nov. 30, 2015
Views 4062 Downloads 110
Nicholas A. Nechval, Department of Mathematics, Baltic International Academy, Riga, Latvia
Konstantin N. Nechval, Department of Applied Mathematics, Transport and Telecommunication Institute, Riga, Latvia
This paper presents an innovative 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 two-parameter exponential distribution is considered. A practical example is given.
Nicholas A. Nechval,
Konstantin N. Nechval,
Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution, American Journal of Theoretical and Applied Statistics. Special Issue: Novel Ideas for Efficient Optimization of Statistical Decisions and Predictive Inferences under Parametric Uncertainty of Underlying Models with Applications.
Vol. 5, No. 2-1,
2016, pp. 1-6.
V. Mendenhall, “A bibliography on life testing and related topics,” Biometrika, vol. XLV, pp. 521–543, 1958.
I. Guttman, “On the power of optimum tolerance regions when sampling from normal distributions,” Annals of Mathematical Statistics, vol. XXVIII, pp. 773–778, 1957.
A. Wald and J. Wolfowitz, “Tolerance limits for a normal distribution,” Annals of Mathematical Statistics, vol. XVII, pp. 208–215, 1946.
W. A. Wallis, “Tolerance intervals for linear regression,” in Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 1951, pp. 43–51.
J. K. Patel, “Tolerance limits: a review,” Communications in Statistics: Theory and Methodology, vol. 15, pp. 2719–2762, 1986.
I. R. Dunsmore, “Some approximations for tolerance factors for the two parameter exponential distribution,” Technometrics, vol. 20, pp. 317–318, 1978.
W. C. Guenther, S. A. Patil, and V. R. R. Uppuluri, “One-sided -content tolerance factors for the two parameter exponential distribution,” Technometrics, vol. 18, pp. 333–340, 1976.
M. Engelhardt and L. J. Bain, “Tolerance limits and confidence limits on reliability for the two-parameter exponential distribution,” Technometrics, vol. 20, pp. 37–39, 1978.
W. C. Guenther, “Tolerance intervals for univariate distributions,” Naval Research Logistics Quarterly, vol. 19, pp. 309–333, 1972.
G. J. Hahn and W. Q. Meeker, Statistical Intervals: A Guide for Practitioners. New York: John Wiley & Sons, 1991.
N. A. Nechval, and K. N. Nechval, “Characterization theorems for selecting the type of underlying distribution,” in Proceedings of the 7th Vilnius Conference on Probability Theory and 22nd European Meeting of Statisticians. Vilnius: TEV, 1998, pp. 352– 353.
P. H. Muller, P. Neumann, and R. Storm, Tables of Mathematical Statistics. Leipzig: VEB Fachbuchverlag, 1979.