1. bookVolume 19 (2019): Issue 6 (December 2019)
Journal Details
License
Format
Journal
eISSN
1335-8871
First Published
07 Mar 2008
Publication timeframe
6 times per year
Languages
English
access type Open Access

Multiple Use Confidence Intervals for a Univariate Statistical Calibration

Published Online: 21 Nov 2019
Volume & Issue: Volume 19 (2019) - Issue 6 (December 2019)
Page range: 264 - 270
Received: 08 Jul 2019
Accepted: 13 Nov 2019
Journal Details
License
Format
Journal
eISSN
1335-8871
First Published
07 Mar 2008
Publication timeframe
6 times per year
Languages
English
Abstract

The statistical calibration problem treated here consists of constructing the interval estimates for future unobserved values of a univariate explanatory variable corresponding to an unlimited number of future observations of a univariate response variable. An interval estimate is to be computed for a value x of an explanatory variable after observing a response Yx by using the same calibration data from a single calibration experiment, and it is called the multiple use confidence interval. It is assumed that the normally distributed response variable Yx is related to the explanatory variable x through a linear regression model, a polynomial regression is probably the most frequently used model in industrial applications. Construction of multiple use confidence intervals (MUCI’s) by inverting the tolerance band for a linear regression has been considered by many authors, but the resultant MUCI’s are conservative. A new method for determining MUCI’s is suggested straightforward from their marginal property assuming a distribution of the explanatory variable. Using simulations, we show that the suggested MUCI’s satisfy the coverage probability requirements of MUCI’s quite well and they are narrower than previously published. The practical implementation of the proposed MUCI’s is illustrated in detail on an example.

Keywords

[1] Acton, F.S. (1959). Analysis of Straight-Line Data. New York: John Wiley.Search in Google Scholar

[2] Carlstein, E. (1986). Simultaneous Confidence Regions for Predictions. The American Statistician, 40, 277–279.Search in Google Scholar

[3] Eisenhart, C. (1939). The Interpretation of certain regression methods and their use in biological and industrial research. Annals of Mathematical Statistics, 10, 162–186.10.1214/aoms/1177732214Search in Google Scholar

[4] Halperin, M. (1961). Fitting of straight lines and prediction when both variables are subject to error. Journal of the American Statistical Association, 56, 657–669.10.1080/01621459.1961.10480651Search in Google Scholar

[5] Han, Y., Liu, W., Bretz, F., Wan, F., Yang, P. (2016). Statistical calibration and exact one-sided simultaneous tolerance intervals for polynomial regression. Journal of Statistical Planning and Inference, 168, 90–96.10.1016/j.jspi.2015.07.005Search in Google Scholar

[6] Johnson, D., Krishnamoorthy, K. (1996). Combining independent studies in a calibration problem. Journal of the American Statistical Association, 91, 1707–1715.10.1080/01621459.1996.10476742Search in Google Scholar

[7] Krishnamoorthy, K., Mathew, T. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. New Jersey: John Wiley&Sons.Search in Google Scholar

[8] Krishnamoorthy, K., Kulkarni, P.M., Mathew, T. (2001). Multiple use one-sided hypotheses testing in univariate linear calibration. Journal of Statistical Planning and Inference, 93, 211–223.10.1016/S0378-3758(00)00164-6Search in Google Scholar

[9] Lieberman, G. J. (1961). Prediction regions for several predictions from a single regression line. Technometrics, 3, 21–27.10.1080/00401706.1961.10489924Search in Google Scholar

[10] Lieberman, G.J., Miller, R.G., Hamilton, M.A. (1967). Unlimited simultaneous discrimination intervals in regression. Biometrika, 54, 133–145.10.1093/biomet/54.1-2.133Search in Google Scholar

[11] Mandel, J. (1958). A note on confidence intervals in regression problems. Annals of Mathematical Statistics, 29, 903–907.10.1214/aoms/1177706548Search in Google Scholar

[12] Mee, R.W., Eberhardt, K.R. (1996). A comparison of uncertainty criteria for calibration. Technometrics, 38, 221–229.10.1080/00401706.1996.10484501Search in Google Scholar

[13] Mee, R.W., Eberhardt, K.R., Reeve, C.P. (1991). Calibration and simultaneous tolerance intervals for regression. Technometrics, 33, 211–219.10.1080/00401706.1991.10484808Search in Google Scholar

[14] Odeh, R.E., Mee, R. W. (1990). One-sided simultaneous tolerance limits for regression. Communication in statistics-simulation and computation, 19, 663–68.10.1080/03610919008812881Search in Google Scholar

[15] Osborne, C. (1991). Statistical calibration: a review. International Statistical Review, 59, 309–336.10.2307/1403690Search in Google Scholar

[16] Scheffé, H. (1973). A statistical theory of calibration. Annals of Statistics, 1, 1–37.10.1214/aos/1193342379Search in Google Scholar

[17] Witkovský, V. (2014). On the exact two-sided tolerance intervals for univariate normal distribution and linear regression. Austrian Journal of Statistic, 43, 279–292.10.17713/ajs.v43i4.46Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo