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Moments Inequalities for NBRUrp Distributions with Hypotheses Testing Applications

International Journal of Computer Applications
Foundation of Computer Science (FCS), NY, USA
Year of Publication: 2017
S. M. El-Arishy, L. S. Diab, E. S. El-Atfy

S M El-Arishy, L S Diab and E S El-Atfy. Moments Inequalities for NBRUrp Distributions with Hypotheses Testing Applications. International Journal of Computer Applications 159(2):34-40, February 2017. BibTeX

	author = {S. M. El-Arishy and L. S. Diab and E. S. El-Atfy},
	title = {Moments Inequalities for NBRUrp Distributions with Hypotheses Testing Applications},
	journal = {International Journal of Computer Applications},
	issue_date = {February 2017},
	volume = {159},
	number = {2},
	month = {Feb},
	year = {2017},
	issn = {0975-8887},
	pages = {34-40},
	numpages = {7},
	url = {},
	doi = {10.5120/ijca2017912879},
	publisher = {Foundation of Computer Science (FCS), NY, USA},
	address = {New York, USA}


In this paper, we presented a new test statistic for testing exponentiality against new better than renewal used in the RP order NBRUrp based on moment inequality. Pitman's asymptotic efficiency, The Pitman asymptotic relative efficiency (PARE) are studied for other testes. Critical values are tabulated for sample size n=5(1)30(5)50 , the power of the test are calculate. Also we proposed a test for testing exponentiality versus NBRUrp for right censored data and the power estimates of this test are also simulated for some commonly used distributions in reliability. Finally, real data are given to elucidate the use of the proposed test statistic in the reliability analysis.


  1. Abdul Aziz, A. A., On testing exponentiality against RNBRUE alternatives, AppliedMathematical Science, 1, 1725-1736 (2007).
  2. Abouammoh, A.M, Abdulghani, S.A and Qamber, I.S (1994) On partial orderings and testing of new better than renewal used classes. Reliability Eng Syst Safety 43,pp. 37.41.
  3. Abu-Youssef, S. E. (2002). A moment inequality for decreasing (increasing) mean residual life distributions with hypothesis testing application, Statist. Probab. Lett., 57, 171-177.
  4. Ahmad, I. A. (2001). Moments inequalities of ageing families of distribution with hypothesis testing applications, J. Statist. Plan. Inf., 92,121-132.
  5. Barlow, R. E. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing, To BeginWith, Silver Spring,Md, USA.
  6. Bhattacharjee, M C., Abouammoh, A. M., Ahmed, A. N. and Barry, A. M. .Preservation results for life distributions based on comparisons with asymptotic remaining life under replacements,. Journal of Applied Probability, vol. 37, no. 4, pp. 999. 1009, 2000.
  7. Bon, J.-L. and Illayk, A. .A note on some new renewal ageing notions,.Statistics & Probability Letters, vol. 57, no. 2, pp. 151.155, 2002.
  8. Bon, J.-L. and Illayk, A. .Ageing properties and series systems,.Journal of Applied Probability, vol. 42, no. 1, pp. 279.286, 2005.
  9. Chen, Y., Lü, J., Yu, X. and Lin, Z. .Consensus of discrete-time second-order multiagent systems based on in.nite products of general stochastic matrices,.SIAMJournal on Control and Optimization, vol. 51, no. 4, pp. 3274.3301, 2013.
  10. Diab L.S. et al (2009). .Moments inequalities for NBUL distributions with hypotheses testing applications., Contem. Engin. Sci., 2, 319-332.
  11. Diab,L.S. A new approach to moments inequalities for NRBU and RNBU classes with hypothesis testing applications, International Journal of Basic & Applied Sciences IJBAS-IJENS 13(06) (2013), 7-13.
  12. EL-Arishy, S. M., Diab, L. S. and Mahmoud, M. A. W. (2003). .Moment inequalities for testing new renewal better than used and renewal new better than used classes, Int. J. Rel. Appl., 4, 97-123.
  13. Grubbs, F. E. (1971). Fiducial bounds on reliability for the two parameter negative exponential distribution. Technomet., 13, pp. 873-876.
  14. Kaplan, E. L. and Meier, P.(1958). .Nonparametric estimation from incomplete observation., J. Amer. Statist.Assoc., 53, pp.457-481.
  15. Kayid, M., Ahmad, I. A, Izadkhah, S. and Abouammoh, A. M. .Further results involving the mean time to failure order, and the decreasingmean time to failure class.,IEEE Transactions on Reliability, vol. 62, no. 3, pp. 670.678, 2013.
  16. Kayid, M., Izadkhah, S. and Alshami, 1, S. (2014)..Residual Probability Function, Associated Orderings, and Related Aging Classes. Mathematical Problems in Engineering
  17. Lai, C-D . and Xie, M . (2006) Stochastic Ageing and Dependence for Reliability, Springer, New York, NY, USA.
  18. Lawless, J. F. (1982). Statistical Models & Methods for lifetime Data, John Wiley & sons, New York.
  19. Li, X. and Xu, M. .Reversed hazard rate order of equilibrium distributions and a related aging notion,. Statistical Papers, vol. 49, no. 4, pp. 749.767, 2008.
  20. Lü, J. and Chen, G..Atime-varying complex dynamical network model and its controlled synchronization criteria,.IEEE Transactions on Automatic Control, vol. 50, no. 6, pp. 841.846, 2005.
  21. Mahmoud, M. A.W. and Abdul Alim, A. N., A Goodness of .t approach to for testing NBUFR (NWUFR) and NBAFR (NWAFR) Properties, International Journal of Reliability Application, 9, 125-140 (2008).
  22. Mi, J..Some comparison results of system availability,.Naval Research Logistics, vol. 45, no. 2, pp. 205.218, 1998.
  23. Mugdadi, A. R. and Ahmed, I. A. Moment inequalities derived from comparing life with its equilibrium form, Journal of Statistical Planning and Inference, 134, 303-317 (2005).
  24. Pena, A. E. (2002).Goodness of fit tests with censored data. actronel
  25. Susarla, V. and Vanryzin, J (1978). Empirical bayes estimations of a survival function right censored observation. Ann. Statist., 6, pp. 710-755.
  26. Tan, S. and Lü, J..Characterizing the effect of population heterogeneity on evolutionary dynamics on complex networks,.Scienti. Reports, vol. 4, p. 5034, 2014.
  27. Zardasht, V. and Asadi, M. (2010)..Evaluation of P (X_t > Y_t) when both X_t and Y_t are residual lifetimes of two systems. Statistics Neer landica, vol. 64, no. 4, pp. 460.481.
  28. Zhou, J. and Lü, J.(2006) .Adaptive synchronization of an uncertain complex dynamical network,.IEEE Transactions on Automatic Control, vol. 51, no. 4, pp. 652.656.


Life distributions, NBRUrp aging class, moment inequalities, exponentiality U-statistic, asymptotic normality, efficiency, Monte Carlo method, power and censored data.