Abstract
Based on a doubly type-II censored sample, a joint confidence region with minimum area is provided for the location and scale parameter of the exponential distribution. In the type-II right censored case, explicit formulas for the associated (expected) minimum area and the coverage probabilities of false parameters are derived. For both quality measures, area and coverage probabilities, comparisons are made to confidence regions known from the literature. Finally, a real data application is presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bain LJ, Engelhardt M (1991) Statistical analysis of reliability and life-testing models, 2nd edn. Marcel Dekker, New York
Balakrishnan N, Basu AP (eds) (1995) The exponential distribution: theory, methods and applications. Gordon and Breach Publishers, Amsterdam
Balakrishnan N, Hayter AJ, Liu W, Kiatsupaibul S (2015) Confidence intervals for quantiles of a two-parameter exponential distribution under progressive type-II censoring. Commun Stat Theory Methods 44(14):3001–3010. https://doi.org/10.1080/03610926.2013.813051
Bedbur S, Lennartz JM, Kamps U (2019) Confidence regions for Pareto parameters from a single and independent samples. Commun Stat Theory Methods. https://doi.org/10.1080/03610926.2018.1476706
Chan PS, Ng HKT, Su F (2015) Exact likelihood inference for the two-parameter exponential distribution under type-II progressively hybrid censoring. Metrika 78(6):747–770. https://doi.org/10.1007/s00184-014-0525-5
Czarnowska A, Nagaev AV (2001) Confidence regions of minimal area for the scale-location parameter and their applications. Appl Math 28(2):125–142
David HA, Nagaraja HN (2003) Order statistics, 3rd edn. Wiley, Hoboken
Dharmadhikari S, Joag-Dev K (1988) Unimodality, convexity, and applications. Academic Press, Boston
Engelhardt M, Bain LJ (1978) Tolerance limits and confidence limits on reliability for the two-parameter exponential distribution. Technometrics 20(1):37–39. https://doi.org/10.1080/00401706.1978.10489615
Epstein B (1956) Simple estimators of the parameters of exponential distributions when samples are censored. Ann Inst Stat Math 8(1):15–26. https://doi.org/10.1007/BF02863562
Epstein B (1960) Estimation of the parameters of two parameter exponential distributions from censored samples. Technometrics 2(3):403–406. https://doi.org/10.1080/00401706.1960.10489907
Epstein B, Sobel M (1954) Some theorems relevant to life testing from an exponential distribution. Ann Math Stat 25(2):373–381
Fernández AJ (2004) On estimating exponential parameters with general type II progressive censoring. J Stat Plan Inference 121(1):135–147. https://doi.org/10.1016/S0378-3758(03)00083-1
Fernández AJ (2012) Minimizing the area of a Pareto confidence region. Eur J Oper Res 221(1):205–212. https://doi.org/10.1016/j.ejor.2012.03.007
Fernández AJ (2013) Smallest Pareto confidence regions and applications. Comput Stat Data Anal 62:11–25. https://doi.org/10.1016/j.csda.2012.12.016
Grubbs FE (1971) Approximate fiducial bounds on reliability for the two parameter negative exponential distribution. Technometrics 13(4):873–876. https://doi.org/10.1080/00401706.1971.10488858
Jeyaratnam S (1985) Minimum volume confidence regions. Stat Probab Lett 3(6):307–308. https://doi.org/10.1016/0167-7152(85)90061-6
Jiang F, Zhou J, Zhang J (2019) Restricted minimum volume confidence region for Pareto distribution. Stat Pap. https://doi.org/10.1007/s00362-018-1018-9
Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, 2nd edn. Wiley, New York. http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471584959.html
Kambo NS (1978) Maximum likelihood estimators of the location and scale parameters of the exponential distribution from a censored sample. Commun Stat Theory Methods 7(12):1129–1132. https://doi.org/10.1080/03610927808827696
Lawless JF (2003) Statistical models and methods for lifetime data, 2nd edn. Wiley, Hoboken
Sarhan AE, Greenberg BG (eds) (1962) Contributions to order statistics. Wiley, New York
Sun X, Zhou X, Wang J (2008) Confidence intervals for the scale parameter of exponential distribution based on type II doubly censored samples. J Stat Plan Inference 138(7):2045–2058. https://doi.org/10.1016/j.jspi.2007.08.006
Wu SF (2007) Interval estimation for the two-parameter exponential distribution based on the doubly type II censored sample. Qual Quant 41(3):489–496. https://doi.org/10.1007/s11135-006-9008-8
Wu SF (2010) Interval estimation for the two-parameter exponential distribution under progressive censoring. Qual Quant 44:181–189. https://doi.org/10.1007/s11135-008-9187-6
Zhang J (2018) Minimum volume confidence sets for two-parameter exponential distributions. Am Stat 72(3):213–218. https://doi.org/10.1080/00031305.2016.1264315
Acknowledgements
The authors would like to thank the associate editor and the referees for their careful reading and constructive comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lennartz, J.M., Bedbur, S. & Kamps, U. Minimum area confidence regions and their coverage probabilities for type-II censored exponential data. Stat Papers 62, 171–191 (2021). https://doi.org/10.1007/s00362-019-01087-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-019-01087-x
Keywords
- Two-parameter exponential distribution
- Type-II censoring
- Confidence region
- Expected volume
- Coverage probability