Two-sided tolerance intervals in a simple linear regression. (English) Zbl 1302.62155

Chvosteková, Martina

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 52, No. 2, 31-41 (2013).

Summary: Numerical results for a simple linear regression indicate that the non-simultaneous two-sided tolerance intervals nearly satisfy the condition of multiple-use confidence intervals, but the numerical computation of the limits of the multiple-use confidence intervals is needed. We modified the Lieberman-Miller method [G. J. Lieberman and R. G. Miller, Biometrika 50, 155–168 (1963; Zbl 0124.35501)] for computing the simultaneous two-sided tolerance intervals in a simple linear regression with independent normally distributed errors. The suggested tolerance intervals are the narrowest of all the known simultaneous two-sided tolerance intervals. The computation of the multiple-use confidence intervals based on the new simultaneous two-sided tolerance intervals is simple and fast.

MSC:

62J05	Linear regression; mixed models
62F25	Parametric tolerance and confidence regions

Keywords:

multiple-use confidence interval; simultaneous two-sided tolerance interval

Citations:

Zbl 0124.35501

References:

[1]	Chvosteková, M.: Simultaneous two-sided tolerance intervals for a univariate linear regression model. Communications in Statistics, Theory and Methods 42 (2013), 1145-1152. · Zbl 1347.62050
[2]	Chvosteková, M.: Determination of two-sided tolerance interval in a linear regression model. Forum Statisticum Slovacum 6 (2010), 79-84.
[3]	Chvosteková, M., Witkovský, V.: Exact likelihood ratio test for the parameters of the linear regression model with normal errors. Measurement Science Review 9 (2009), 1-8.
[4]	Krishnamoorthy, K., Mathew, T.: Statistical Tolerance Regions: Theory, Applications, and Computation. Wiley series in probability and statistics, Wiley, Chichester, 2009. · Zbl 1291.60001
[5]	Lee, Y., Mathew, T.: Advances on Theoretical and Methodological Aspects of Probability and Statistics. Taylor & Francis, London, 2002.
[6]	Lieberman, G. J., Miller, R. G., Jr.: Simultaneous Tolerance intervals in regression. Biometrika 50 (1963), 155-168. · Zbl 0124.35501
[7]	Lieberman, G. J., Miller, R. G., Hamilton, M. A.: Unlimited simultaneous discrimination intervals in regression. Biometrika 54 (1967), 133-145.
[8]	Limam, M. M. T., Thomas, R.: Simultaneous tolerance intervals for the linear regression model. Journal of the American Statistical Association 83 (1988), 801-804. · Zbl 0649.62030
[9]	Mee, R. W., Eberhardt, K. R.: A Comparison of Uncertainty Criteria for Calibration. Technometrics 38 (1996), 221-229. · Zbl 0898.62092
[10]	Mee, R. W., Eberhardt, K. R., Reeve, C. P.: Calibration and simultaneous tolerance intervals for regression. Technometrics 33 (1991), 211-219.
[11]	Scheffé, H.: A statistical theory of calibration. The Annals of Statistics 1 (1973), 1-37. · Zbl 0253.62023
[12]	Wilson, A. L.:: An approach to simultaneous tolerance intervals in regression. The Annals of Mathematical Statistics 38 (1967), 1536-1540. · Zbl 0183.20902
[13]	Witkovský, V.:: On exact multiple-use linear calibration confidence intervals. MEASUREMENT 2013: 9th International Conference on Measurement, Smolenice, Slovakia, 2013, 35-38.