Abstract
Independent samples from different exponential distributions are available in many statistical applications, for example, in queuing or reliability models, where the random variables describe waiting times and lifetimes, respectively. When two samples are observed, inference is then carried out for two location and two scale parameters. In multivariate setups including common location and common scale parameter assumptions, we provide confidence regions for the parameters of interest, which have minimum Lebesgue measure among all those based on the usual pivotal quantities and with the same or higher confidence level; in particular, they improve in terms of area (volume) upon the standard ‘trapezoidal’ confidence regions being constructed by combining independent univariate pivot statistics. The proposed confidence regions do not require any factorization of the overall confidence level, and their calculations need simple Monte Carlo simulations, only. Although focusing on two complete samples, generalizations of the results to more than two and doubly type-II censored samples are possible.
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References
Antle CE, Bain LJ (1969) A property of maximum likelihood estimators of location and scale parameters. SIAM Rev 11(2):251–253
Balakrishnan N, Basu AP (1995) The exponential distribution: theory, methods and applications. Taylor&Francis, Amsterdam
Balakrishnan N, Cramer E (2014) The art of progressive censoring. Birkhäuser, New York
Bedbur S, Kamps U, Lennartz JM (2019) On a smallest confidence region for a location-scale parameter in progressively type-II censored lifetime experiments. Stat Probab Lett 154:108545
Bedbur S, Lennartz JM, Kamps U (2013) Confidence regions in models of ordered data. J Stat Theory Pract 7(1):59–72
Bedbur S, Lennartz JM, Kamps U (2019) Confidence regions for Pareto parameters from a single and independent samples. Commun Stat Theory Methods 48(13):3341–3359
Czarnowska A, Nagaev AV (2001) Confidence regions of minimal area for the scale-location parameter and their applications. Appl Math (Warsaw) 28(2):125–142
Dharmadhikari S, Joag-Dev K (1988) Unimodality, convexity, and applications. Academic Press, Boston
Fernández AJ (2013) Smallest Pareto confidence regions and applications. Comput Stat Data Anal 62:11–25
Fernández AJ (2014) Computing optimal confidence sets for Pareto models under progressive censoring. J Comput Appl Math 258:168–180
Jeyaratnam S (1985) Minimum volume confidence regions. Stat Probab Lett 3(6):307–308
Jiang F, Zhou J, Zhang J (2018) 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
Lawless JF (2003) Statistical models and methods for lifetime data, 2nd edn. Wiley, Hoboken
Lennartz JM, Bedbur S, Kamps U (2019) Minimum area confidence regions and their coverage probabilities for type-II censored exponential data. Stat Pap. https://doi.org/10.1007/s00362-019-01087-x
Littell RC, Louv WC (1981) Confidence regions based on methods of combining test statistics. J Am Stat Assoc 76(373):125–130
Nelson W (1982) Applied life data analysis. Wiley, Hoboken
Sarhan AE, Greenberg BG (eds) (1962) Contributions to order statistics. Wiley, New York
Shetty BN, Joshi PC (1987) Estimation of parameters of \(k\) exponential distributions in doubly censored samples. Commun Stat Theory Methods 16(7):2115–2123
Viveros R, Balakrishnan N (1994) Interval estimation of parameters of life from progressively censored data. Technometrics 36(1):84–91
Wu S-F (2007) Interval estimation for the two-parameter exponential distribution based on the doubly type II censored sample. Qual Quant 41(3):489–496
Wu S-F (2010) Interval estimation for the two-parameter exponential distribution under progressive censoring. Qual Quant 44(1):181–189
Zhang J (2018) Minimum volume confidence sets for two-parameter exponential distributions. Am Stat 72(3):213–218
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Lennartz, J.M., Bedbur, S. & Kamps, U. Minimum Volume Confidence Regions for Parameters of Exponential Distributions from Different Samples. J Stat Theory Pract 14, 27 (2020). https://doi.org/10.1007/s42519-020-00096-6
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DOI: https://doi.org/10.1007/s42519-020-00096-6