Kybernetika 55 no. 6, 943-960, 2019

Exact simultaneous location-scale tests for two shifted exponential samples

Amitava Mukherjee, Zhi Lin Chong and Marco MarozziDOI: 10.14736/kyb-2019-6-0943

Abstract:

The failure time distribution for various items often follows a shifted (two-parameter) exponential model and not the traditional (one-parameter) exponential model. The shifted exponential is very useful in practice, in particular in the engineering, biomedical sciences and industrial quality control when modeling time to event or survival data. The open problem of simultaneous testing for differences in origin and scale parameters of two shifted exponential distributions is addressed. Two exact tests are proposed using maximum likelihood estimators. They are based on the combination of two statistics following a maximum-type and a distance-type approach. The exact null distributions of the respective test statistics are derived analytically. Small sample type-one error rate and power of the tests are studied numerically. We showed that the test based on the maximum type combination (the Max test) should be preferred being generally more powerful than the test based on the distance type combination (the Distance test). An application to a biomedical experiment is discussed.

Keywords:

hypothesis testing, failure time model, simultaneous testing, shifted exponential, type-one error rate, power

Classification:

62F03, 62N05

paper.pdf

References:

  1. J. Ahmadi and S. M. T. K. MirMostafaee: Prediction intervals for future records and order statistics coming from two parameter exponential distribution. Stat. Probab. Lett. 79 (2009), 977-983.   DOI:10.1016/j.spl.2008.12.002
  2. A. Baklizi: Interval estimation of the stress-strength reliability in the two-parameter exponential distribution based on records. J. Stat. Comput. Simul. 84 (2014), 2670-2679.   DOI:10.1080/00949655.2013.816307
  3. A. Chaturvedi and V. Sharma: A note on the estimation of P(Y$>$X) in two-parameter exponential distributions. Statistics 44 (2010), 73-75.   CrossRef
  4. A. C. Cohen and F. R. Helm: Estimation in the exponential distribution. Technometrics 15 (1973), 415-418.   DOI:10.1080/00401706.1973.10489054
  5. N. Ebrahimi: Estimating the parameters of an exponential. J. Stat. Plan. Inference 14 (1986), 255-261.   DOI:10.1016/0378-3758(86)90163-1
  6. M. Engelhardt and L. J. Bain: Tolerance limits and confidence limits on reliability for the two-parameter exponential distribution. Technometrics 20 (1978), 37-39.   CrossRef
  7. A. Ganguly, S. Mitra, D. Samanta and D. Kundu: Exact inference for the two-parameter exponential distribution under Type-II hybrid censoring. J. Stat. Plan. Inference 142 (2012), 613-625.   DOI:10.1016/j.jspi.2011.08.001
  8. S. Huang, A. Mukherjee and J. Yang: Two CUSUM schemes for simultaneous monitoring of parameters of a shifted exponential time to events. Qual. Reliab. Eng. Int. 34 (2018), 6, 1158-1173.   CrossRef
  9. N. L. Johnson and S. Kotz: Distributions in Statistics, Vol. 1: Continuous Univariate Distributions. Houghton Mifflin, Boston 1970.   CrossRef
  10. S. C. Kao: Normalization of the origin-shifted exponential distribution for control chart construction. J. Appl. Stat. 37 (2010), 1067-1087.   DOI:10.1080/02664760802571333
  11. K. Krishnamoorthy and Y. Xia: Confidence intervals for a two-parameter exponential distribution: one- and two-sample problems. Commun. Stat. Theory Methods 47 (2018), 935-952.   CrossRef
  12. K. Krishnamoorthy, S. Mukherjee and H. Guo: Inference on reliability in two-parameter exponential stress-strength model. Metrika 65 (2007), 261-273.   DOI:10.1007/s00184-006-0074-7
  13. J. Li, W. Song and J. Shi: Parametric bootstrap simultaneous confidence intervals for differences of means from several two-parameter exponential distributions. Stat. Probab. Lett. 106 (2015), 39-45.   DOI:10.1016/j.spl.2015.07.002
  14. A. Mukherjee, A. K. McCracken and S. Chakraborti: Control Charts for simultaneous monitoring of parameters of a shifted exponential distribution. J. Qual. Technol. 47 (2015), 176-192.   DOI:10.1016/j.spl.2015.07.002
  15. M. Pal, M. A. Masoom and J. Woo: Estimation and testing of $P(Y > X)$ in two parameter exponential distributions. Statistics 39 (2005), 415-428.   CrossRef
  16. M. Z. Raqab: Approximate maximum likelihood predictors of future failure times of shifted exponential distributions under multiple type II censoring. Stat. Methods Appl. 13 (2004), 43-54.   DOI:10.1007/s10260-004-0084-4
  17. A. Roy and T. Mathew: Reliability function of a two-parameter exponential distribution. J. Stat. Plan .Inference 128 (2005), 509-517.   DOI:10.1016/j.jspi.2003.11.012
  18. S. Sangnawakij and S. Niwitpong: Confidence intervals for coefficients of variation in two-parameter exponential distributions. Commun. Stat. Simul. Comp. 46 (2017), 6618-6630.   DOI:10.1080/03610918.2016.1208236
  19. N. Schenk, M. Burkschat, E. Cramer and U. Kamps: Bayesian estimation and prediction with multiply Type-II censored samples of sequential order statistics from one-and two-parameter exponential distributions. J. Stat. Plan. Inference 141 (2011), 1575-1587.   DOI:10.1016/j.jspi.2010.11.009
  20. P. Singh and A. Abebe: Comparing several exponential populations with more than one control. Stat. Methods Appl. 18 (2009), 359-374.   DOI:10.1007/s10260-008-0092-x
  21. E. A. Tanis: Linear forms in the order statistics from an exponential distribution. Ann. Math. Stat. 35 (1964), 270-276.   DOI:10.1214/aoms/1177703749
  22. R. van Zyl and A. J. van der Merwe: Schemes for the two-parameter exponential distribution. Commun. Stat. Theory Methods DOI:10.1080/03610926.2018.1440307.   CrossRef
  23. S. D. Varde: Life testing and reliability estimation for the two parameter exponential distribution. J. Amer. Stat. Assoc. 64 (1969), 621-631.   DOI:10.1080/01621459.1969.10501000
  24. Z. Wang and H. K. T. Ng: A comparative study of tests for paired lifetime data. Lifetime Data Anal. 12 (2006), 505-522.   DOI:10.1007/s10985-006-9026-9
  25. I. Weissman: Sum of squares of uniform random variables. Stat. Probab. Lett. 129 (2017), 147-154.   DOI:10.1016/j.spl.2017.05.018
  26. J.-W. Wu. H.-M. Lee and C.-L. Lei: Computational testing algorithmic procedure of assessment for lifetime performance index of products with two-parameter exponential distribution. Appl. Math. Comput. 190 (2007), 116-125.   DOI:10.1016/j.amc.2007.01.010
  27. S.-F. Wu: Interval estimation for the two-parameter exponential distribution under progressive censoring. Qual. Quant. 44 (2010), 181-189.   DOI:10.1007/s11135-008-9187-6