## Estimation of Stress Strength Reliability P [Y < X < Z] of Lomax Distribution under Different Sampling Scheme

Neethu Jacob *

Department of Statistics, Nirmala College, Muvattupuzha, Kerala, India.

Anjana E. J.

Department of Mathematics, Rheinische Friedrich-Wilhelms-Universit ¨ at Bonn, Bonn-53115, Germany.

*Author to whom correspondence should be addressed.

### Abstract

Lomax distribution can be considered as the mixture of exponential and gamma distribution. This distribution is an advantageous lifetime distribution in reliability analysis. The applicability of Lomax distribution is not restricted only to the reliability field, but it has broad applications in Economics, actuarial modelling, queuing problems,biological sciences, etc. Initially, Lomax distribution was proposed by Lomax in 1954, and it is also known as Pareto Type II distribution. Many statistical methods have been developed for this distribution; for a review of Lomax Distribution, see [1] and the references. The stress strength model plays an important role in reliability analysis. The term stress strength was first introduced by [2]. In the context of reliability, R is defined as the probability that the unit strength is greater than stress, that is, R = P (X > Y ), where X is the random strength of the unit, and Y is the instant stress applied to it. Thus, estimation of R is very important in Reliability Analysis.The estimates of R discussed in the context of Lomax distribution are limited to the study of a single stress strength model with upper stress. But in real life, there are situations where we have to consider not only the upper stress Neethu and Anjana; Curr. J. Appl. Sci. Technol., vol. 42, no. 42, pp. 36-66, 2023; Article no.CJAST.107238 but also the lower stress. Accordingly, in the present paper, the estimation of stress strength model R = P (Y < X < Z) represents the situation where the strength X should be greater than stress Y and smaller than stress Z for Lomax distribution, Shrinkage maximum likelihood estimate and Quasi likelihood estimate are obtained both under complete and right censored data. We have considered the asymptotic confidence interval (CI) based on MLE and bootstrap CI for R. Monte Carlo simulation experiments were performed to compare the performance of estimates obtained.

Keywords: Lomax distribution, stress strength reliability, maximum likelihood estimator, quasi likelihood estimator, confidence interval

**How to Cite**

*Current Journal of Applied Science and Technology*42 (42):36-66. https://doi.org/10.9734/cjast/2023/v42i424271.

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### References

Abd Ellah AH. Bayesian one sample prediction bounds for the Lomax distribution. Indian Journal of Pure and Applied Mathematics. 2003;34(1):101–109.

Church JD, Harris B. The Estimation of Reliability from Stress-Strength Relationships. Technometrics. 1970;12:49-54.

Raqab MZ. Inferences for Generalized Exponential Distribution Based on Record Statistics.

Journal of Statistical Planning and Inference. 2002;104(2):339-350.

Panahi H, Asadi S. Inference of Stress Strength Model for Lomax Distribution. World Academy of Science, Engineering and Technology, International Journal of Mathematical and

Computational Sciences. 2011;5(7):937-940.

Abdul–Moniem IB, Abdel–Hameed HF. On Exponentiated Lomax Distribution. International Journal of Mathematical Archive. 2012;5(3):1-7.

Ali HM. The System Reliability of Stress Strength Model for Lomax Distribution . M.Sc Thesis, ALMustansiriya University; 2013.

Attia AF, Shaban SA, Amer YM. Parameter estimation for the bivariate Lomax distribution based on censored samples. Applied Mathematical Sciences. 2014;8(35):1711-21.

Mahmoud MA, El-Sagheer RM, Soliman AA, Abd Ellah AH. Bayesian Estimation of P(Y < X) Based on Record Values from the Lomax distribution and MCMC technique. Journal of Modern Applied Statistical Methods. 2016;15(1):488-510.

Singh SK, Singh U, Yadav AS. Reliability Estimation for Inverse Lomax Distribution Under

type II Censored Data using Markov chain Monte Carlo method. International Journal of Mathematics and Statistics. 2016;17(1):128-146.

Parviz N. Estimation Parameter of R = P(Y < X) for Lomax Distribution with presence of outliers. International Mathematical Forum. 2016;11(5):239-248.

Rady EHA, Hassanein WA, Elhaddad TA. The Power Lomax Distribution with an Application to Bladder Cancer Data. SpringerPlus. 2016;5(1)5:1838.

Yadav AS, Singh SK, Singh U. Bayesian Estimation of Stress–Strength Reliability for Lomax Distribution Under Type-II Hybrid Censored Data Using Asymmetric Loss Function. Life Cycle Reliability and Safety Engineering. 2019;8:257- 267.

Almetwally EM, Almongy HM. Parameter estimation and stress-strength model of power Lomax distribution: classical methods and Bayesian estimation. Journal of Data Science. 2020 1;18(4):718-38.

Abd El-Monsef MME, Marei GA, Kilany NM. Estimating Stress-Strength Model for Weighted Lomax Distribution. Asian Journal of Probability and Statistics. 2021;12(2):19-31.

Karam NS, Jani HH. Reliability of Multi- Component Stress-Strength System Estimation for Generalized Lomax Distribution. In AIP Conference Proceedings. 2019;2190(1):020038.

Salman AN, Hamad AM. On Estimation of the Stress–Strength Reliability Based on Lomax Distribution. In IOP Conference Series: Materials Science and Engineering. 2019;571(1):012038. IOP Publishing.

Neethu J, Jeevanand ES. Semi Parametric Estimation of Stress Strength Reliability P(X > Y ) of Lomax distribution. Far East Journal of Mathematical Sciences. 2021;61(2):95-107.

Neethu J, Jeevanand ES. Bayesian Estimation of Stress Strength Reliability P(X > Y ) of Lomax and Exponential Distribution. Journal of Information Storage and Processing Systems. 2021;20:182-189.

Baklizi A, Saadati Nik A, Asgharzadeh A. Likelihood and Bayesian Inference in the Lomax Distribution under Progressive Censoring. Mathematics and Statistics. 2022;10(3):615-623.

Neethu J, Jeevanand ES. Bayesian Estimation of Stress Strength Reliability P(X > Y ) of Lomax and Exponential Distribution based on right censored sample. International Journal of Statistics and Applied Mathematics. 2021;6(2):27- 30.

Singh N. On the estimation of P (X1 < Y < X2). Communications in Statistics-Theory and Methods. 1980;9(15):1551-1561.

Dutta K, Sriwastav GL. An n –Standby System with P(X < Y < Z). IAPQR Transaction. 1986;12:95-

65 Neethu and Anjana; Curr. J. Appl. Sci. Technol., vol. 42, no. 42, pp. 36-66, 2023; Article no.CJAST.107238

Ivshin VV. On the Estimation of Probabilities of a Double Linear Inequality in the Case of Uniform and Two Parameter Exponential Distributions. Journal of Mathematical Science. 1998;88(6):819- 827.

Hassan AS, Elsayed AE, Rania MS. On the Estimation of for Weibull Distribution in the Presence of k Outliers. International Journal of Engineering Research and Applications. 2013;3(6):1727-1733.

Hameed BA, Salman AN, Kalaf BA. On Estimation of P (Y1 < X < Y2) in cased Inverse Kumaraswamy Swamy Distribution. Iraqi Journal of Science. 2020;61(4):845-853.

Karam NS, Attia AM. Stress strength Reliability for P(T < X < Z) using Dagum Distribution. Ibn Al-Haitham International Conference for Pure and Applied Sciences(IHICPS). Journal of Physics Conference Series. 2021;1879032004. DOI:10.1088/1742-6596/1879/3/032004, 1-11.

Khaleel AH. Reliability of One Strength- Four Stresses for Lomax Distribution. In Journal of Physics: Conference Series. 2021;1879(3):032015.

Neethu J, Anjana EJ. Shrinkage Estimation of Stress Strength Reliability P(Y < X < Z) for Lomax distribution based on records. International Journal of Statistics and Applied Mathematics. 2023;8(1):107-123.

Thompson JR. Some Shrinkage Techniques for Estimating the Mean JASA. 1968;63:113-122.

Mehta JS, Srinivasan R. Estimation of the Mean by Shrinkage to a Point. Journal of the American Statistical Association. 1971;66(333):86-90.

Abu-Salih MS, Ali MA, Yousef MA. On Some Shrinkage Techniques for Estimating the Parameters of Exponential Distribution. Journal of Information and Optimization Sciences. 1988 May 1;9(2):207-14.

Siu-Keung T, Geoffrey T. Shrinkage Estimation of Reliability for Exponentially Distributed Lifetimes. Communications in Statistics-Simulation and Computation. 1996;25(2):415-430.

Zakerzadeh H, Jafari AA, Karimi M. Preliminary Test and Shrinkage Estimations of Scale Parameters for Two Exponential Distributions based on Record Values. J. Statist. Res. 2016;1343-1358.

Nooghabi MJ. Shrinkage estimation of P(Y < X) in the Exponential Distribution Mixing with

Exponential Distribution. Communications in Statistics-Theory and Methods. 2016;45(5):1477-

Glifin F, Anjana EJ, Jeevanand ES. Shrinkage Estimation of Strength Reliability for Geometric Distribution Using Record Values. Acta Scientific Computer Sciences. 2022;4(4):26-30.

McNolty F, Doyle J, Hansen E. Properties of the Mixed Exponential Failure Process. Technometrics. 1980;22(4):555-565.

Wasserman L. All of Statistics: A Concise Course in Statistical Inference, first ed. Springer, New

York; 2003.

Soliman AA, Abd-Ellah AH, Abou-Elheggag NA, Ahmed EA. Reliability Estimation in

Stress–Strength Models: An MCMC Approach. Statistics. 2013;47(4):715-728.

Dhanya M, Jeevavand ES. Stress-Strength Reliability of Power Function Distribution based on Records. J. Stat. Appl. Prob. 2018;7(1):39-48.

Khan MJS, Khatoon B. Statistical Inferences of R = P(X < Y ) for Exponential Distribution based

on Generalized Order Statistics. Annals of Data Science. 2019;7(3):525-545.

Wedderburn RW. Quasi-Likelihood Functions, Generalized Linear Models and the Gauss—Newton Method. Biometrika. 1974;61(3):439-447.

Efron B. The Jackknife, the Bootstrap and Other Resembling Plans. SIAM, Philadelphia; 1982.