ORIGINAL ARTICLE Estimation of stress–strength reliability for Maxwell distribution under progressive type-II censoring scheme Sachin Chaudhary 1 • Sanjeev K. Tomer 1 Received: 8 July 2017 / Revised: 4 December 2017 Ó The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and Maintenance, Lulea University of Technology, Sweden 2018 Abstract This paper deals with the estimation of stress– strength reliability P ¼ P½Y \X, when the strength X and stress Y both follow Maxwell distribution with different parameters. We obtain maximum likelihood and Bayes estimates of P using progressive type-II censored samples. We also provide procedures to evaluate asymptotic and bootstrap confidential intervals, as well as, Bayesian credible and highest posterior density intervals for P. We present simulation study and analyze a real data set for numerical illustrations. Keywords Asymptotic confidence intervals  Bayes estimator  Bootstrap  Credible intervals  Highest posterior density  Maximum likelihood estimator  Stress– strength model 1 Introduction A random variable (rv) X is said to follow Maxwell dis- tribution with scale parameter h (MW( h)), if its probability density function (pdf) is given by f ðx; hÞ¼ 4 ffiffiffi p p 1 h 3=2 x 2 e  x 2 h ; 0  x\1; h [ 0: ð1Þ The distribution function of X comes out to be FðxÞ¼ 1  2 ffiffiffi p p C 3 2 x 2 h  ; ð2Þ where C a ðwÞ¼ R 1 w u a1 e u du. The hazard rate of (1) is an increasing function of time (Tomer and Panwar 2015), which justifies its applications to study problems of aging of units that often occur in life testing and reliability esti- mation. The MW distribution is extensively studied in lit- erature. For a brief review of literature on the distribution, including reliability estimation, one may refer to Tyagi and Bhattacharya (1989), Bekker and Roux (2005), Krishna and Malik (2009), Krishna and Malik (2012) and Panwar et al. (2015). Let the strength of a system is denoted by rv X which follows MW( h 1 ) and the stress imposed on it is denoted by rv Y which follows MW( h 2 ). A measure of system relia- bility under the stress–strength set-up is P ¼ PðY \XÞ ¼ Z 1 y¼0 Z 1 x¼y f ðyÞf ðxÞdxdy ¼ 4 p 1 h 3=2 2 Z 1 0 z 1 2 e  z h 2 C 3 2 z=h 1 ð Þdz: ð3Þ Using a result from Gradshteyn and Ryzhik (1965, pp 663,x6:45), we write (3) as follows. P ¼ 16 3p ffiffiffiffiffiffiffiffiffi h 1 h 2 p h 1 þ h 2   3 2 F 1 1; 3; 5 2 ; h 1 h 1 þ h 2   ; ð4Þ where 2 F 1 is Gauss hypergeometric function. Since the expression of P in (3) does not come in a closed form, presentation of the same in the form of (4) makes the calculation of P very convenient. The stress–strength reliability P is often used as a measure of performance when a system of strength X is & Sanjeev K. Tomer sktomer73@gmail.com 1 Department of Statistics, Banaras Hindu University, Varanasi 221005, India 123 Int J Syst Assur Eng Manag https://doi.org/10.1007/s13198-018-0709-x