.svg){kind=logo}
BOOTSTRAP TOLERANCE INTERVALS FOR LINEAR REGRESSION MODELS
Keywords:
bootstrap, coverage probability, expected interval width, tolerance interval, linear regressionDOI:
https://doi.org/10.17654/0972361723032Abstract
Weissberg and Beatty [28] and Young [30] introduced two approaches for constructing tolerance intervals for simple linear regression models. This paper introduces two approaches for constructing bootstrap tolerance intervals for linear regression models. One approach is based on resampling errors from a known distribution function and the other approach focuses on resampling residuals resulting from the original regression model. The performance of these two approaches is evaluated and compared with the two methods that were introduced by Weissberg and Beatty [28] and Young [30] in terms of coverage probability and expected interval width using extensive simulations. The four approaches are illustrated by an application to a real data example. SAS 9.4 package and SAS/IML matrix language were used to carry out simulation studies and real data analysis.
Received: March 6, 2023
Accepted: April 20, 2023
References
M. Chvostekova, Two-sided tolerance intervals in a simple linear regression, Mathematica 52(2) (2013), 31-41.
B. Efron, Bootstrap methods: Another look at the jackknife, The Annals of Statistics 7 (1979), 1-26.
B. Efron and R. J. Tibshirani, Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy, Statistica Sinica 1(1) (1986), 54-77.
B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap, Chapman and Hall Inc., 1993.
B. E. Ellison, On two-sided tolerance intervals for a normal distribution, The Annals of Mathematical Statistics 35 (1964), 762-772.
D. A. S. Fraser and I. Guttman, Tolerance regions, The Annals of Mathematical Statistics 27(1) (1956), 162-179.
D. A. Freedman and S. C. Peters, Bootstrapping a regression equation: Some empirical results, Journal of the American Statistical Association 79(385) (1984), 97-106.
S. Goncalves and H. White, Bootstrap standard error estimates for linear regression, Journal of the American Statistical Association 100(471) (2005), 970-979.
I. Guttman, Construction of beta-content tolerance regions at confidence level for large samples from the k-variate normal distribution, The Annals of Mathematical Statistics 41 (1970), 376-400.
M. S. Haq and S. Rinco, beta-expectation tolerance regions for a generalized multivariate model with normal error variables, Journal Multivariate Analysis 6 (1976), 414-421.
W. G. Howe, Two-sided tolerance limits for normal populations-some improvements, Journal of the American Statistical Association 64(326) (1969), 610-620.
S. John, A tolerance region for multivariate normal distributions, Sankhya, Series A 25 (1963), 363-368.
S. John, A central tolerance region for the multivariate normal distribution, Journal of the Royal Statistical Society, Series B 30 (1968), 599-601.
K. Krishnamoorthy, P. M. Kulkarni and T. Mathew, Multiple use one-sided hypotheses testing in univariate linear calibration, Journal of Statistical Planning and Inference 93 (2001), 211-223.
K. Krishnamoorthy and T. Mathew, Comparison of approximation methods for computing tolerance factors for a multivariate normal population, Technometrics 41 (1999), 234-249.
K. Krishnamoorthy and T. Mathew, Statistical Tolerance Regions: Theory, Applications, and Computation, Hoboken, Wiley, NJ, 2009.
K. Krishnamoorthy and S. Mondal, Improved tolerance factors for multivariate normal distributions, Communications in Statistics-Simulation and Computation 35(2) (2006), 461-478.
K. Krishnamoorthy and S. Mondal, Tolerance factors in multiple and multivariate linear regressions, Communications in Statistics-Simulation and Computation 37(3) (2008), 546-559.
Y.-T. Lee and T. Mathew, Tolerance regions in multivariate linear regression, Journal of Statistical Planning and Inference 126 (2004), 253-271.
H. Li and G. S. Maddala, Bootstrap variance estimation of nonlinear functions of parameters: An application to long-run elasticities of energy demand, Review of Economics and Statistics 81 (1999), 728-733.
G. J. Lieberman and R. G. Miller, Simultaneous tolerance intervals in regression, Biometrika 50(1-2) (1963), 155-168.
M. Limam and D. R. Thomas, Simultaneous tolerance intervals for the linear regression model, Journal of the American Statistical Association 83(403) (1988), 801-804.
T. Rebafka, S. Clemencon and M. Feinberg, Bootstrap-based tolerance intervals for application to method validation, Chemometrics and Intelligent Laboratory Systems 89(2) (2007), 69-81.
B. Thilakarathne, Finding a Better Confidence Interval for a Single Regression Change Point using Different Bootstrap Confidence Interval Procedures, Electronic Theses and Dissertations, Georgia Southern University, 2012.
A. Wald and J. Wolfowitz, Tolerance limits for a normal distribution, The Annals of Mathematical Statistics 17 (1946), 208-215.
W. A. Wallis, Tolerance intervals for linear regression, Symposium on Mathematical Statistics and Probability 2 (1951), 43-51.
R. Wehrens, H. Putter and L. M. C. Buydens, The bootstrap: A tutorial, Chemometrics and Intelligent Laboratory Systems 54 (2000), 35-52.
A. Weissberg and G. H. Beatty, Tables of tolerance-limit factors for normal distributions, Technometrics 2(4) (1960), 483-500.
A. L. Wilson, An approach to simultaneous tolerance intervals in regression, Annals of Mathematical Statistics 30 (1967), 1536-1540.
D. S. Young, Regression tolerance intervals, Communications in Statistics-Simulation and Computation 42(9) (2013), 2040-2055.
D. S. Young, Chapter 8 - Computing Tolerance Intervals and Regions using R, Handbook of Statistics 32 (2014), 309-338.
Downloads
Published
Issue
Section
License
Copyright (c) 2023 Pushpa Publishing House, Prayagraj, India
This work is licensed under a Creative Commons Attribution 4.0 International License.
____________________________
Attribution: Credit Pushpa Publishing House as the original publisher, including title and author(s) if applicable.
No Derivatives: Modifying or creating derivative works not allowed without written permission.
Contact Pushpa Publishing House for more info or permissions.
Journal Impact Factor: 
