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On the simple step-stress model for two-parameter exponential distribution |
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| Institution: | 1. Department of Statistics, Universidade Federal do Amazonas, 69077-000, Manaus, AM, Brazil;2. Statistics Department, Universidade Federal de Minas Gerais, 31270-901, Belo Horizonte, Brazil;1. Department of Statistics and Biostatistics, Rutgers University, Piscataway, NJ 08854, USA;2. Department of Biostatistics, Yale University, School of Public Health, New Haven, CT 06511, USA;3. Independent Consultant, Sudbury, MA 01776, USA;1. Department of Applied Statistics, College of Business and Economics, Yonsei University, Seoul 120-749, South Korea;2. Department of Statistics, University of Seoul, Seoul 130-743, South Korea;1. Pediatric Unit Sant''Andrea Hospital, NESMOS Department, Faculty of Medicine and Psychology, “Sapienza” University, Rome, Italy;2. Department of Pharmacology, Faculty of Medicine, Catholic University of Rome, Rome, Italy;3. Department of Drug Sciences, University of Chieti G. D''Annunzio, Chieti, Italy;1. Laboratoire de Mathématiques Appliquées de Compiègne-L.M.A.C., Université de Technologie de Compiègne, B.P. 529, 60205 Compiègne Cedex, France;2. L.S.T.A., Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France;1. Purdue University, United States;2. North Carolina State University, United States;3. Newcastle University, UK |
| Abstract: | In this paper, we consider the simple step-stress model for a two-parameter exponential distribution, when both the parameters are unknown and the data are Type-II censored. It is assumed that under two different stress levels, the scale parameter only changes but the location parameter remains unchanged. It is observed that the maximum likelihood estimators do not always exist. We obtain the maximum likelihood estimates of the unknown parameters whenever they exist. We provide the exact conditional distributions of the maximum likelihood estimators of the scale parameters. Since the construction of the exact confidence intervals is very difficult from the conditional distributions, we propose to use the observed Fisher Information matrix for this purpose. We have suggested to use the bootstrap method for constructing confidence intervals. Bayes estimates and associated credible intervals are obtained using the importance sampling technique. Extensive simulations are performed to compare the performances of the different confidence and credible intervals in terms of their coverage percentages and average lengths. The performances of the bootstrap confidence intervals are quite satisfactory even for small sample sizes. |
| Keywords: | Step-stress model Type-II censoring Two-parameter exponential distribution Maximum likelihood estimates Conditional moment generating function Confidence interval Fisher information matrix Bootstrap confidence interval Bayes estimate |
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