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On Extended Quadratic Hazard Rate Distribution: Development, Properties, Characterizations and Applications

  • Fiaz Ahmad Bhatti EMAIL logo , G. G. Hamedani , Wenhui Sheng and Munir Ahmad
Published/Copyright: March 30, 2018
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Stochastics and Quality Control
From the journal Stochastics and Quality Control Volume 33 Issue 1

Abstract

In this paper, we propose a flexible extended quadratic hazard rate (EQHR) distribution with increasing, decreasing, bathtub and upside-down bathtub hazard rate function. The EQHR density is arc, right-skewed and symmetrical shaped. This distribution is also obtained from compounding mixture distributions. Stochastic orderings, descriptive measures on the basis of quantiles, order statistics and reliability measures are theoretically established. Characterizations of the EQHR distribution are studied via different techniques. Parameters of the EQHR distribution are estimated using the maximum likelihood method. Goodness of fit of this distribution through different methods is studied.

Keywords: Stochastic Ordering; Reliability; Characterizations; Compounding; Maximum Likelihood Estimation
MSC 2010: 60E05; 62E10; 62E15; 62H12; 68U05; 68U20

A Appendix

Theorem A.1.

Let (Ω,F,P) be a given probability space and let H=[a,b] be an interval for some a<b (a=-∞, b=∞ might as well be allowed). Let X:Ω→H be a continuous random variable with distribution function F. Let q be a real function defined on H such that E[q(X)∣X≥x]=η(x), x∈H, is defined with some real function η. Assume that q∈C1⁢(H), η∈C2⁢(H) and F is a twice continuously differentiable and strictly monotone function on the set H. Finally, assume that the equation η=q has no real solution in the interior of H. Then F is uniquely determined by the functions q and η, particularly

F⁢(x)=∫axC⁢|η′⁢(u)η⁢(u)-q⁢(u)|⁢exp⁡(-s⁢(u))⁢𝑑u,

where the function s is a solution of the differential equation s′=η′η-q and C is the normalization constant, such that ∫H𝑑F=1.

References

[1] T. H. M. Abouelmagd, S. Al-Mualim, M. Elgarhy, A. Z. Afify and M. Ahmad, Properties of the four-parameter Weibull distribution and its applications, Pakistan J. Statist. 33 (2017), 449–466. Search in Google Scholar

[2] A. Al-Hossain and J. G. Dar, Some results on moment of order statistics for the quadratic hazard rate distribution, J. Stat. Appl. 5 (2016), 371–376. 10.18576/jsap/050218Search in Google Scholar

[3] B. C. Arnold, N. Balakrishnan and H. Nagarajah, Records, John Wiley, New York, 1998. 10.1002/9781118150412Search in Google Scholar

[4] H. Athar and Z. Noor, Characterization of probability distributions by conditional expectations of functions of record statistics, J. Egyptian Math. Soc. 22 (2014), 275–279. 10.1016/j.joems.2013.07.008Search in Google Scholar

[5] L. J. Bain, Analysis for the linear failure-rate life-testing distribution, Technometrics 16 (1974), 551–559. 10.1080/00401706.1974.10489237Search in Google Scholar

[6] G. K. Bhattacharyya and R. A. Johnson, Estimation of reliability in a multicomponent stress-strength model, J. Amer. Statist. Assoc. 69 (1974), 966–970. 10.1080/01621459.1974.10480238Search in Google Scholar

[7] F. A. Bhatti and M. Ahmad, Some characterizations of quadratic hazard rate-geometric (Qhr-G) distribution, Int. J. Eng. Res. Gen. Sci. 5 (2017), 164–171. Search in Google Scholar

[8] M. A. El-Damcese and D. A. Ramadan, Reliability analysis using the generalized quadratic hazard rate truncated Poisson maximum distribution, Int. J. Stat. Appl. 5 (2015), 302–316. Search in Google Scholar

[9] I. Elbatal and N. S. Butt, A new generalization of quadratic hazard rate distribution, Pak. J. Stat. Oper. Res. 9 (2014), 343–361. 10.18187/pjsor.v9i4.678Search in Google Scholar

[10] W. Glänzel, A characterization theorem based on truncated moments and its application to some distribution families, Mathematical Statistics and Probability Theory, D. Reidel, Dordrecht (1987), 75–84. 10.1007/978-94-009-3965-3_8Search in Google Scholar

[11] W. Glänzel, Some consequences of a characterization theorem based on truncated moments, Statistics 21 (1990), 613–618. 10.1080/02331889008802273Search in Google Scholar

[12] G. G. Hamedani, On certain generalized gamma convolution distributions II, Technical Report 484, Marquette University, 2013. Search in Google Scholar

[13] M. Kayid, I. Elbatal and F. Merovci, A new family of generalized quadratic hazard rate distribution with applications, J. Testing Eval. 44 (2016), 1733–1744. 10.1520/JTE20140324Search in Google Scholar

[14] A. H. Khan and A. A. Alzaid, Characterization of distributions through linear regression of non-adjacent generalized order statistics, J. Appl. Statist. Sci. 13 (2004), 123–136. Search in Google Scholar

[15] F. Merovci and I. Elbatal, The beta quadratic hazard rate distribution, Pakistan J. Statist. 31 (2015), 427–446. Search in Google Scholar

[16] D. Murthy, M. Xie and R. Jiang, Weibull Models, Wiley Ser. Probab. Stat., John Wiley & Sons, New York, 2004. Search in Google Scholar

[17] A. Mustafa, A new bivariate distribution with generalized quadratic hazard rate marginals distributions, Journal of Math. Stat. 12 (2016), 255–270. 10.3844/jmssp.2016.255.270Search in Google Scholar

[18] H. Nagarajah, Some characterizations of continuous distributions based on regressions of adjacent order statistics and record values, Sankhya A 50 (1988), 70–73. Search in Google Scholar

[19] H. M. Okasha, M. Kayid, M. A. Abouammoh and I. Elbatal, A new family of quadratic hazard rate-geometric distributions with reliability applications, J. Testing Eval. 44 (2015), 1937–1948. 10.1520/JTE20150116Search in Google Scholar

[20] R. Roozegar and S. Nadarajah, The quadratic hazard rate power series distribution, J. Testing Eval. 45 (2016), 1058–1072. 10.1520/JTE20150506Search in Google Scholar

[21] A. M. Sarhan, Generalized quadratic hazard rate distribution, Int. J. Appl. Math. Stat. 14 (2009), 94–106. Search in Google Scholar

[22] M. H. Tahir, G. M. Cordeiro, M. Mansoor and M. Zubair, The Weibull–Lomax distribution: Properties and applications, Hacet. J. Math. Stat. 44 (2015), 461–480. 10.15672/HJMS.2014147465Search in Google Scholar

Received: 2018-1-13
Revised: 2018-3-5
Accepted: 2018-3-8
Published Online: 2018-3-30
Published in Print: 2018-6-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/eqc-2018-0002
Keywords for this article
Stochastic Ordering; Reliability; Characterizations; Compounding; Maximum Likelihood Estimation

Articles in the same Issue

  1. Frontmatter
  2. Comparing Short-Memory Charts to Monitor the Traffic Intensity of Single Server Queues
  3. Some Characterizations of the Log-Logistic Distribution
  4. Topp–Leone Linear Exponential Distribution
  5. On Extended Quadratic Hazard Rate Distribution: Development, Properties, Characterizations and Applications
  6. Reliability Test Plans for Percentiles Based on the Harris Generalized Linear Exponential Distribution
  7. Reliability Test Plan for the Gumbel-Uniform Distribution
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