Journal of the Korean Statistical Society

The Korean Statistical Society (한국통계학회)
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BAYESIAN INFERENCE FOR THE POWER LAW PROCESS WITH THE POWER PRIOR

  • KIM HYUNSOO (Division of Applied Mathematics, Hanyang University) ;
  • CHOI SANGA (Department of Mathematics, Ajou University) ;
  • KIM SEONG W. (Division of Applied Mathematics, Hanyang University)
  • Published : 2005.12.01
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Abstract

Inference on current data could be more reliable if there exist similar data based on previous studies. Ibrahim and Chen (2000) utilize these data to characterize the power prior. The power prior is constructed by raising the likelihood function of the historical data to the power $a_o$, where $0\;{\le}\;a_o\;{\le}\;1$. The power prior is a useful informative prior in Bayesian inference. However, for model selection or model comparison problems, the propriety of the power prior is one of the critical issues. In this paper, we suggest two joint power priors for the power law process and show that they are proper under some conditions. We demonstrate our results with a real dataset and some simulated datasets.

Keywords

References

  1. BAR-LEV, S., LAVI, I. AND REISER, B., (1992). 'Byesian inference for the power law process', Annals of the Institute of Statistical Mathematics, 44(4), 623-639 https://doi.org/10.1007/BF00053394
  2. BARNARD, G. A., (1953). 'Time intervals between accidents-a note on Maguire, Pearson and Wynn's paper', Biometrika, 40, 212-213
  3. CHEN, M.-H., IBRAHIM, J. G. AND SHAO, Q.-M., (2000). 'Power prior distributions for generalized linear models', Journal of the Statistical Planning and Inference, 84, 121-137 https://doi.org/10.1016/S0378-3758(99)00140-8
  4. CHEN, M.-B., IBRAHIM, J. G., SHAD, Q.-M. AND WEISS, R. E., (1999). 'Prior elicitation for model selection and estimation in generalized linear mixed models', Journal of the Statistical Planning and Inference, 111, 57-76 https://doi.org/10.1016/S0378-3758(02)00285-9
  5. Cox, D. R. AND LEWIS, P. A. W., (1966). The Statistical Analysis of Series of Events, (London: Methuen)
  6. CROW, L. H., (1974). 'Reliability analysis of complex repairable systems, reliability and biometry, Proschan, F. and Serfling, R. J., Eds', SIAM, Philadelphia, 379-410
  7. DUANE, J. T., (1964). 'Learning curve approach to reliability monitoring', IEEE Transaction on Aerospace, 2, 563-566 https://doi.org/10.1109/TA.1964.4319640
  8. GUIDA, M., CALABRIA, R. AND PULCINI, G., (1989). 'Bayesian inference for a nonhomogeneous Poisson process with power intensity law', IEEE Transaction on Reliability, 38(5), 603-609 https://doi.org/10.1109/24.46489
  9. IBRAHIM, J. G. AND CHEN, M.-H., (2000), 'Power prior distributions for regression models', Statistical Science, 15, 46-60 https://doi.org/10.1214/ss/1009212673
  10. IBRAHIM, J. G., CHEN, M.-H. AND RYAN, L. M., (2000). 'Bayesian variable selection for time series count data', Statistical Sinica, 10, 971-987
  11. JARRETT, R. G., (1979). 'A note on the intervals between coal-mining disasters', Biometrika, 66, 191-193 https://doi.org/10.1093/biomet/66.1.191
  12. KIM, S. W. AND SUN, D., (2000). 'Intrinsic priors for model selection using an encompassing model with applications to censored failure time data', Lifetime Data Analysis, 6, 251-269 https://doi.org/10.1023/A:1009641709382
  13. KIM, K., KIM, S. W. AND KIM, H., (2003). 'Intrinsic Bayes factors for model selection in nonhomogeneous Poisson processes', Far East Journal of Theoretical Statistics, 11(1), 15-30
  14. KYPARISIS, J. AND SINGPURWALLA, N., (1985). 'Bayesian inference for the Weibull process with applications to assessing software reliability growth and predicting software failures' , Computer Science and Statistics: Proceeding of the 16th. Symposium on the interface of computer science and statistics, 57-64
  15. LINGHAM, R. T. AND SIVAGANESAN, S., (1997). 'Testing hypotheses about the power law process under failure truncation using intrinsic Bayes factors', Annals of the Institute of Statistical Mathematics., 49. 693-710 https://doi.org/10.1023/A:1003218410136
  16. MAGUIR, B. A., PEARSON, E. S. AND WYNN, A. H. A., (1952). 'The time intervals between in dustrial accidents', Biometrika, 39, 168-180 https://doi.org/10.1093/biomet/39.1-2.168
  17. ZELLNER, A., (1988). 'Optimal information processing and Bayes's theorem (with discussion)', The American Statistician, 42, 278-284 https://doi.org/10.2307/2685143