Skip to main content
SpringerLink
Log in

Using the best two-observational percentile and maximum likelihood methods in a multicomponent stress-strength system to reliability estimation of inverse Weibull distribution

  • Original Research
  • Published:
Life Cycle Reliability and Safety Engineering Aims and scope Submit manuscript

Abstract

In this paper, we propose an estimate of reliability in a multicomponent system. The system has \(k\) components strengths are given by independently and identically distributed random variables \({X}_{1}\), \({X}_{2}\),…, \({X}_{k}\) and each component is exposed to random stress \({\rm Y}\). The reliability of such a system is obtained when strength and stress variables are given by inverse Weibull (IW) distribution with scale parameters \({\lambda }_{1}\),  \({\lambda }_{2}\) and common shape parameter \(\alpha\). The system reliability is estimated using maximum likelihood estimation (MLE) and the best two-observational percentile estimation (BTPE) methods in samples drawn from strength and stress distributions. Also, the asymptotic confidence interval for system reliability is obtained. The reliability estimators obtained from both methods are compared using average bias, mean squares error, and confidence interval length via Monte Carlo simulation. In the end, using two real data sets we illustrate the procedure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Abdulateef EA, Salman AN (2019) On Shrinkage estimation for R (s, k) in case of exponentiated Pareto distribution. Ibn AL-Haitham J Pure Appl Sci 32(1):147–156

    Article  Google Scholar 

  • Alizadeh M, Bagheri SF, Khaleghy MM (2013) Efficient estimation of the density and cumulative distribution function of the generalized Rayleigh distribution. J Stat Res Iran 10(1):1–22

    Article  Google Scholar 

  • Al-Mutairi DK, Gitany ME, Kundu D (2013) Inferences on stress-strength reliability from Lindley distribution. Commun Stat Theory Methods 42(8):1443–1463

    Article  MathSciNet  MATH  Google Scholar 

  • Asgharzadeh A, Valiollahi R, Raqab MZ (2011) Stress-strength reliability of Weibull distribution based on progressively censored samples. Stat Oper Res Trans 35(2):103–124

    MathSciNet  MATH  Google Scholar 

  • Awad M, Gharraf K (1986) Estimation of P(Y < X) in Burr case: a comparative study. Commun Stat Simul Comput 15(2):389–403

    Article  MathSciNet  MATH  Google Scholar 

  • Bader MG, Priest AM (1982) Statistical aspects of fiber and bundle strength in hybrid composites. In: Hayashi T, Kawata K, Umekawa S (eds) Progress in science and engineering composites. ICCM-IV, Tokyo, pp 1129–1136

    Google Scholar 

  • Bagheri SF, Alizadeh M, Baloui Jamkhaneh E, Nadarajah S (2014) Evaluation and comparison of estimations in the generalized exponential—Poisson distribution. J Stat Comput Simul 84(11):2345–2360

    Article  MathSciNet  MATH  Google Scholar 

  • Bhattacharyya GK, Johnson RA (1974) Estimation of reliability in multicomponent stress-strength model. J Am Stat Assoc 69(34):966–970

    Article  MathSciNet  MATH  Google Scholar 

  • Bi Q, Gui W (2017) Bayesian and classical estimation of stress-strength reliability for inverse Weibull lifetime models. Algorithms 10(71):1–16

    MathSciNet  MATH  Google Scholar 

  • Downtown F (1973) The estimation of PY < X in the normal case. Techno Metr 15(3):551–558

    Google Scholar 

  • Drapella A (1993) The complementary weibull distribution: unknown or just forgotten? Qual Reliab Eng Int 9(4):383–385

    Article  Google Scholar 

  • Dubey SD (1967) Some percentile estimators for Weibull parameters. Techno Metr 9(1):119–129

    Article  MathSciNet  Google Scholar 

  • Enis P, Geisser S (1971) Estimation of the probability that Y < X. J Am Stat Assoc 66(333):162–168

    MathSciNet  MATH  Google Scholar 

  • Fulment AK, Josephat PK, Rao GS (2017) Estimation of reliability in multicomponent stress-strength based on Dagum distribution. Stoch Qual Control 32(2):77–85

    MathSciNet  MATH  Google Scholar 

  • Ghitany ME, Al-Mutairi DK, Aboukhamseen SM (2015) Estimation of the reliability of a stress-strength system from power Lindley distributions. Commun Stat Simul Comput 44(1):118–136

    Article  MathSciNet  MATH  Google Scholar 

  • Gusmo FRSD, Ortega EMM, Cordeiro GM (2011) The generalized inverse Weibull distribution. Stat Pap 52(3):591–619

    Article  MathSciNet  MATH  Google Scholar 

  • Hassan MKh, Alohali MI (2018) Estimation of reliability in a multicomponent stress-strength model based on generalized linear failure rate distribution. Hacet J Math Stat 47(6):1634–1651

    MathSciNet  Google Scholar 

  • Jazi MA, Lai CD, Alamatsaz MH (2010) A discrete inverse Weibull distribution and estimation of its parameters. Stat Methodol 7(2):121–132

    Article  MathSciNet  MATH  Google Scholar 

  • Johnson NL, Kotz S, Balakrishnan N (1995) Distributions. In: Johnson NL, Kotz S, Balakrishnan N (eds) Continuous univariate distributions, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  • Kantam RRL, Rao GS (2002) Log-logistic distribution: modified maximum likelihood estimation. Guj Stat Rev 29(1–2):25–36

    Google Scholar 

  • Keller AZ, Kamath ARR, Perera UD (1982) Reliability analysis of CNC machine tools. Reliab Eng 3(6):449–473

    Article  Google Scholar 

  • Keller AZ, Giblin MT, Farnworth NR (1985) Reliability analysis of commercial vehicle engines. Reliab Eng 10(1):15–25

    Article  Google Scholar 

  • Khan MS, King R (2012) Modified inverse Weibull distribution. J Stat Appl Probab 1(2):115–132

    Article  Google Scholar 

  • Khan MS, Pasha GR, Pasha AH (2008) Theoretical analysis of inverse Weibull distribution. WSEAS Trans Math 7(2):30–38

    MATH  Google Scholar 

  • Kizilaslan F, Nadar M (2015) Classical and Bayesian estimation of reliability in multicomponent stress-strength model based on Weibull distribution. Revista Colombiana de Estadística 38(2):467–484

    Article  MathSciNet  MATH  Google Scholar 

  • Kizilaslan F, Nadar M (2018) Estimation of reliability in a multicomponent stress-strength model based on a bivariate Kumaraswamy distribution. Stat Pap 59(1):307–340

    Article  MathSciNet  MATH  Google Scholar 

  • Kohansal A (2019) On estimation of reliability in a multicomponent stress-strength model for a Kumaraswamy distribution based on progressively censored sample. Stat Pap 60(6):2185–2224

    Article  MathSciNet  MATH  Google Scholar 

  • Kundu D, Gupta RD (2006) Estimation of PY < X for Weibull distribution. IEEE Trans Reliab 55(2):270–280

    Article  Google Scholar 

  • Kuo W, Zuo MJ (2003) Optimal reliability modeling, principles and applications. Wiley, New York

    Google Scholar 

  • Li C, Hao H (2017) Reliability of a stress-strength model with inverse Weibull distribution. Int J Appl Math 47(3):1–5

    MathSciNet  Google Scholar 

  • Lio YL, Tsai TR (2012) Estimation of δ = PX > Y for Burr XII distribution based on the progressively first failure-censored samples. J Appl Stat 39(2):465–483

    Article  Google Scholar 

  • Maurya KR, Tripathi MY (2020) Reliability estimation in a multicomponent stress-strength model for Burr XII distribution under progressive censoring. Braz J Probab Stat 34(2):345–369

    Article  MathSciNet  MATH  Google Scholar 

  • Menon MV (1963) Estimation of the shape and scale parameters of the Weibull distribution. Techno Metr 5(2):175–182

    Article  MathSciNet  MATH  Google Scholar 

  • Mohie El-Din MM, Riad FH (2011) Estimation and prediction for the inverse Weibull distribution based on records. J Adv Res Stat Probab 2(3):20–27

    MathSciNet  Google Scholar 

  • Mohie El-Din MM, Sadek A, Elmeghawry ShH (2018) Stress-strength reliability estimation for exponentiated generalized inverse Weibull distribution. J Stat Appl Probab 7(1):127–136

    Article  Google Scholar 

  • Nadar M, Kizilaslan F (2016) Estimation of reliability in a multicomponent stress–strength model based on a Marshall–Olkin bivariate Weibull distribution. IEEE Trans Reliab 65(1):370–380

    Article  Google Scholar 

  • Nandi SB, Aich AB (1994) A note on estimation of PX > Y for some distributions useful in life-testing. IAPQR Trans 19(1):35–44

    MathSciNet  MATH  Google Scholar 

  • Pak A, Gupta AK, Bagheri KN (2018) On reliability in a multicomponent stress-strength model with power Lindley distribution. Revista Colombiana de Estadística 41(2):251–267

    Article  MathSciNet  MATH  Google Scholar 

  • Pandey M, Borhan UMD (1985) Estimation of reliability in multicomponent stress-strength model following Burr distribution. In: Proceedings of the first Asian congress on quality and reliability, New Delhi, India, pp 307–312

  • Pandit PV, Joshi S (2018) Reliability estimation in multicomponent stress-strength model based on generalized Pareto distribution. Am J Appl Math Stat 6(5):210–217

    Article  Google Scholar 

  • Pandit PV, Joshi S (2019) Estimation of multicomponent system reliability for a bivariate generalized Rayleigh distribution. Int J Comput Theor Stat 6(1):51–63

    Article  Google Scholar 

  • Rao CR (1973) Linear statistical inference and its applications. Wiley Eastern Limited, Calcutta

    Book  MATH  Google Scholar 

  • Rao GS (2012a) Estimation of reliability in multicomponent stress-strength model based on generalized exponential distribution. Colomb J Stat 35(1):67–76

    MATH  Google Scholar 

  • Rao GS (2012b) Estimation of reliability in multicomponent stress-strength model based on generalized inverted exponential distribution. Int J Tour Res 4(21):48–56

    Google Scholar 

  • Rao GS (2012c) Estimation of reliability in multicomponent stress-strength model based on Rayliegh distribution. Prob Stat Forum 5:150–161

    MATH  Google Scholar 

  • Rao GS (2013) Estimation of reliability in multicomponent stress-strength model based on inverse exponential distribution. Int J Stat Econ 10(1):28–37

    MathSciNet  MATH  Google Scholar 

  • Rao GS, Kantam RRL (2010) Estimation of reliability in multicomponent stress-strength model: log-logistic distribution. Electron J Appl Stat Anal 3(2):75–84

    Google Scholar 

  • Rao GS, Kantam RRL, Rosaiah K, Reddy PJ (2013) Estimation of reliability in multicomponent stress-strength model based on inverse Rayleigh distribution. J Stat Appl Probab 2(3):261–267

    Article  Google Scholar 

  • Rao GS, Aslam M, Aril OH (2017) Estimation of reliability in multicomponent stress-strength model based on exponentiated Weibull distribution. Commun Stat Simul Comput 46(15):7495–7502

    MATH  Google Scholar 

  • Raqab MZ, Kundu D (2005) Comparison of different estimators of PY < X for a scaled Burr type X distribution. Commun Stat Simul Comput 34(2):465–483

    Article  MATH  Google Scholar 

  • Raqab MZ, Madi MT, Kundu D (2008) Estimation of PY < X for the 3-parameter generalized exponential distribution. Commun Stat Theory Methods 37(18):2854–2864

    Article  MATH  Google Scholar 

  • Sultan KS, Ismail MA, Al-Moisheer AS (2007) Mixture of two inverse Weibull distributions: properties and estimation. Comput Stat Data Anal 51(11):5377–5387

    Article  MathSciNet  MATH  Google Scholar 

  • Travadi RJ, Ratani RT (1990) On estimation of reliability function for inverse Gaussian distribution with known coefficient of variation. IAPQR Trans 5(2):29–37

    MATH  Google Scholar 

  • Von Alven WH (1964) Reliability engineering by ARINC. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  • Yadava AS, Singh SK, Singh U (2018) Estimation of stress–strength reliability for inverse Weibull distribution under progressive type-II censoring scheme. J Ind Prod Eng 35(1):48–55

    Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referees and the associate editor of the journal for evaluating and constructive suggestions.

Author information

Authors and Affiliations

  1. Department of Statistics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran

    Kazem Fayyaz Heidari, Einollah Deiri & Ezzatallah Baloui Jamkhaneh

Authors
  1. Kazem Fayyaz Heidari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Einollah Deiri
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ezzatallah Baloui Jamkhaneh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Einollah Deiri.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Heidari, K.F., Deiri, E. & Jamkhaneh, E.B. Using the best two-observational percentile and maximum likelihood methods in a multicomponent stress-strength system to reliability estimation of inverse Weibull distribution. Life Cycle Reliab Saf Eng 10, 255–265 (2021). https://doi.org/10.1007/s41872-021-00166-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41872-021-00166-z

Keywords

Search

Navigation