A New Approach to Constructing Tolerance Limits on Order Statistics in Future Samples Coming from a Normal Distribution

  • Nicholas A Nechval Department of Mathematics, Baltic International Academy, Riga, Latvia
  • Konstantin N. Nechval Department of Applied Mathematics, Transport and Telecommunication Institute, Riga, Latvia
  • Vladimir F. Strelchonok Department of Mathematics, Baltic International Academy, Riga, Latvia
Keywords: Order Statistics, F Distribution, Lower Tolerance Limit, Upper Tolerance Limit, Normal Distribution

Abstract

Although the concept of statistical tolerance limits has been well recognized for long time, surprisingly, it seems that their applications remain still limited. Analytic formulas for the tolerance limits are available in only simple cases, for example, for the upper or lower tolerance limit for a univariate normal population. Thus it becomes necessary to use new or innovative approaches which will allow one to construct tolerance limits on future order statistics for many populations. In this paper, a new approach to constructing lower and upper tolerance limits on order statistics in future samples is proposed. 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 normal distribution is considered. The discussion is restricted to one-sided tolerance limits.  A practical example of finding a warranty assessment of image quality is given.

References

(1) Mendenhall, V., A bibliography on life testing and related topics. Biometrika, 1958. XLV: p. 521-543.

(2) Guttman, I., On the power of optimum tolerance regions when sampling from normal distributions. Annals of Mathematical Statistics, 1957. XXVIII: p. 773-778.

(3) Wald, A. and J. Wolfowitz, Tolerance limits for a normal distribution. Annals of Mathematical Statistics, 1946. XVII: p. 208-215.

(4) Wallis, W. A., Tolerance intervals for linear regression, in Second Berkeley Symposium on Mathematical Statistics and Probability, 1951. Berkeley: University of California Press, p. 43-51.

(5) Patel, J. K., Tolerance limits: a review. Communications in Statistics: Theory and Methodology, 1986. 15: p. 2719-2762.

(6) Dunsmore, I. R., Some approximations for tolerance factors for the two parameter exponential distribution. Technometrics, 1978. 20: p. 317-318.

(7) Guenther, W. C., S. A. Patil, and V. R. R. Uppuluri, One-sided -content tolerance factors for the two parameter exponential distribution. Technometrics, 1976. 18: p. 333-340.

(8) Engelhardt, M. and L. J. Bain, Tolerance limits and confidence limits on reliability for the two-parameter exponential distribution. Technometrics, 1978. 20: p. 37-39.

(9) Guenther, W. C., Tolerance intervals for univariate distributions. Naval Research Logistics Quarterly, 1972. 19: p. 309-333.

(10) Hahn, G. J. and W. Q. Meeker, Statistical Intervals: A Guide for Practitioners. 1991. New York: John Wiley & Sons.

(11) Nechval, N. A. and K. N. Nechval, Tolerance limits on order statistics in future samples coming from the two-parameter exponential distribution. American Journal of Theoretical and Applied Statistics, 2016. 5: p. 1-6.

(12) Johnson, N. L., S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, vol. 1. Second edition, 1994. New York: John Wiley & Sons.

(13) Patel, J. K. and C. B. Read, Handbook of the Normal Distribution. Second edition, 1996. New York: Marcel Dekker.

(14) Balakrishnan, N. and V. B. Nevzorov, A Primer on Statistical Distributions. 2003. New Jersey: Wiley.

(15) Kotz, S. and D. Vicari, Survey of developments in the theory of continuous skewed distributions. Metron, 2005. 63: p. 225-261.

(16) Nechval, N. A. and E. K. Vasermanis, Improved Decisions in Statistics. 2004. Riga: Izglitibas soli.

(17) Nechval, N. A., K. N. Nechval, and E. K. Vasermanis, Effective state estimation of stochastic systems. Kybernetes (An International Journal of Systems & Cybernetics), 2003. 32: p. 666-678.

(18) Nechval, N. A., G. Berzins, M. Purgailis, and K. N. Nechval, Improved estimation of state of stochastic systems via invariant embedding technique. WSEAS Transactions on Mathematics, 2008. 7: p. 141-159.

(19) Nechval, N. A., M. Purgailis, K. N. Nechval, and V. F. Strelchonok, Optimal predictive inferences for future order statistics via a specific loss function. IAENG International Journal of Applied Mathematics, 2012. 42: p. 40-51.

(20) Rao, C. R., Linear Statistical Inference and its Applications. 1965. New York: Wiley.

(21) Searle, S. R., Linear Models. 1971. New York: Wiley.

(22) Daly S., The visual difference predictor: an algorithm for the assessment of image fidelity, in Human Vision, Visual Processing, and Digital Display, Proc. SPIE, vol. 1666, 1992, San Jose, CA, p. 2-15.

(23) Heeger, D. J. and P. C. Teo, A model of perceptual image fidelity, in Proc. IEEE International Conference on Image Processing, vol. 2, 1995, p. 343-345.

(24) Malo, J., A. M. Pons, and 5. M. Artigas, Subjective image fidelity metric based on bit allocation of the human visual system in the DCT domain. Image und Vision Computing, 1997. 15: p. 535-548.

(25) Watson, A. B., J. Hu, and J. F. McGowm, Digital video quality metric based on human vision. J. of Electronic Imaging, 2001. 10: p. 20-29.

(26) Lai, Y. K. and C.-C. J. Kuo, A Haar wavelet approach to compressed image quality measurement. J. VisuaI Communication und Image Repres., 2000. 11: p. 17-40.

(27) Beghdadi, A. and B. Pesquet-Popescu, A new image distortion measure based on wavelet decomposition, in Proc. Seventh Intern. Symp. Signal Proces. its Applications (ISSPA-2003), vol. 1, 2003, Paris, p. 485-488.

(28) Beghdadi, A. and R. Iordache, Image quality assessment using the joint space/spatial-frequency representation. EURASIP Journal on Applied Signal Processing, 2006. 6: p. 1-8.

(29) Wang, Z., A. C. Bovik, and L. Lu, Why is image quality assessment so difficult? In Proc. IEEE Inter. Conference Acoustics, Speech, and Signal Processing (ICASSP-2002), vol. 4, 2002, Orlando FL, p. 3313-3336.

(30) Dodson, B., Weibull Analysis. 1994. Milwaukee: ASQ Quality Press.

Published
2016-06-24
How to Cite
Nechval, N. A., Nechval, K. N., & Strelchonok, V. F. (2016). A New Approach to Constructing Tolerance Limits on Order Statistics in Future Samples Coming from a Normal Distribution. European Journal of Applied Sciences, 4(3), 01. https://doi.org/10.14738/aivp.42.2014