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Research Article

Open Access Peer-reviewed

Taruna Kumari, Anupam Pathak^{ }

Received June 11, 2019; Revised July 14, 2019; Accepted July 29, 2019

In this article, we consider a multicomponent stress-strength model which has k independent and identical strength components X_{1}, X_{2}, …, X_{k} and each component is exposed to a common random stress Y. Both stress and strength are assumed to have proportional hazard rate model with different unknown power parameters. The system is regarded as operating only if at least s out of k(1≤s≤k) strength variables exceeds the random stress. Reliability of the system is estimated by using maximum likelihood, uniformly minimum variance unbiased and Bayesian methods of estimation. The asymptotic confidence interval is constructed for the reliability function. The performances of these estimators are studied on the basis of their mean squared error through Monte Carlo simulation technique.

In the reliability context, the single-component stress-strength model is the probability which represents the reliability of an item or system of random strength subject to random stress In the single-component stress-strength model, a system fails if and only if at any given time, the applied stress exceeds the strength. Single-component stress-strength models have great utility in the fields of genetics, psychology, engineering and in so many others. A lot of work has been done in the literature for the classical and Bayesian estimation of single-component stress-strength reliability. For a brief review, one may refer to Basu ^{ 1}, Kelley et al. ^{ 2}, Awad and Gharraf ^{ 3}, Tyagi and Bhattacharya ^{ 4}, Chaturvedi and Kumar ^{ 5}, Chaturvedi and Pathak ^{ 6, 7}, Chaturvedi et al. ^{ 8} and others. Inferences have been drawn for single-component stress-strength reliability for some families of lifetime distributions by Chaturvedi and Pathak ^{ 9}, Chaturvedi and Kumari ^{ 10, 11, 12} and Kumari et al. ^{ 13}.

The reliability in a multicomponent stress-strength model was developed by Bhattacharyya and Johnson ^{ 14}. Panday and Uddin ^{ 15} assumed the parameters not involved in reliability as known using Bayesian estimation. Rao and Kantam ^{ 16} studied the estimation of reliability in a multicomponent stress-strenght model for log-logistic distribution and Rao ^{ 17} developed an estimation procedure for reliability in multicomponent stress-strength based on generalized exponential distribution. Recently, Rao et al. ^{ 18} studied the estimation of reliability in a multicomponent stress-strength model for Burr-XII distribution and Kizilaslan and Nadar ^{ 19} developed an estimation procedure for reliability in multicomponent stress-strength based on Weibull distribution.

The multicomponent stress-strength system consists of independent and identical strengths component and a common stress, functions when or more of the components simultaneously survive. This model corresponds to the *s*-out-of-*k*: system. Multicomponent stress-strength models have great applications range from communication and industrial systems to logistic and military systems. For example, in suspension bridges, the deck is supported by a series of vertical cables hung from the towers. Suppose a suspension bridge consisting of number of vertical cable pairs. The bridge will only survive if a minimum s number of vertical cable through the deck is not damaged when subjected to stresses due to wind loading, heavy traffic, corrosion etc. For extensive reviews of -out-of- and related systems one may refer to Kuo and Zuo ^{ 20}.

Let the random samples be independent, be the continuous distribution function of and be the common continuous distribution function of The reliability in a multicomponent stress-strength model is given by ^{ 14}.

(1.1) |

(1.2) |

where are identically independently distributed (*iid*) strength variables and subjected to common random stress, . The probability in (1.1) is termed as reliability in a multicomponent stress-strength model ^{ 14}.

The random variable (*r**v*) X follows the proportional hazard rate (PHR) model if the cumulative distribution function (*cdf *) of X have the following form

(1.3) |

where is the survival function of the baseline *rv* and is power parameter. The probability density function (*pdf*) corresponding to the *cdf* (1.3) is given by

(1.4) |

where is the first derivative of with respect to A *rv* following PHR model with power parameter will be denoted by PHR().

The family represented by and is well known in lifetime experiments as the PHR model ^{ 21}. Ahmadi et al. ^{ 22}, Wang and Shi ^{ 23} and Wang ^{ 24}, mentioned some of it's particular cases such as exponential, Pareto, Lomax, Burr XII and others.

In this study, we consider the multicomponent stress-strength model which has independent and identical strength components and a common stress. We assume that the strength variables and stress variable follow PHR model. The system functions if or more of the components simultaneously survive. The estimation of reliability for this system is obtained under the classical and Bayesian framework. The Lindley's approximation technique is carried out to obtain Bayesian estimates. Explicit expression for Bayes estimator of reliability is also obtained. Moreover, the asymptotic confidence interval (ACI) for reliability function is constructed.

The rest of the paper is organized as follows: In Section 2, the maximum likelihood (ML) estimator and ACI of are obtained. In Section 3, uniformly minimum variance unbiased (UMVU) estimator of is provided. In Section 4, Bayes estimator of is developed in both approximate and explicit forms under squared error loss function (SELF) ^{ 7}. In Section 5, simulation study is carried out to compare the estimates of by using Monte Carlo simulation technique and findings are illustrated by tables and plots. Finally, conclusions on the paper are provided in Section 6.

This section deals with the ML estimation of . Here, we assume that be independent; PHR() and PHR(). Therefore from (1.1) and (1.3), is given by

(2.1) |

where

In order to obtain the estimators of , suppose systems are put on life-testing experiment from the strength population and systems are from the stress population. In this case, we obtain the following observed data: and and Then, the likelihood function of the observed sample is given by

(2.2) |

and the log-likelihood function is given by

(2.3) |

From (2.3), the ML estimators of and are given by

(2.4) |

where and

Hence, the ML estimator of is obtained from (2.1) and (2.4) by using the invariance property of ML estimators

(2.5) |

where

To obtain the ACI interval for we proceed as follows:

The Fisher information matrix of is given as

The ML estimator of is asymptotically normal with mean and variance

where

and

Therefore, an asymptotic confidence interval of is given by

(2.6) |

where is the upper quantile of the standard normal distribution and is the value of at the ML estimate of the parameteres.

This section deals with the UMVU estimation of . To find the UMVU estimator of , say, it is enough to find UMVU estimator of by using the linearity property of UMVU estimators. From (2.2), it is seen that is a complete sufficient statistics for Moreover, and . Let

where and Obviously, have exponential distribution with means and , respectively. Then, is an unbiased estimator of .

The UMVU estimator of , say, can be obtained by using Lehmann-Scheffe Theorem and is given by

where Notice that, and

Thus,

(3.1) |

The integral in (3.1) is considered in two cases, i.e., and .

When the integral (3.1) can be expressed as

(3.2) |

Similarly, when the double integral in (3.1) can be expressed as

Proceeding on the similar lines as earlier, we get

(3.3) |

Therefore, can be obtained by using (2.1), (3.2) and (3.3) as

(3.4) |

This section deals with the Bayesian estimation of . To obtain the Bayes estimator of we have considered two independent non-informative priors, and say, where

(4.1) |

Looking at (2.2) and (4.1), the joint posterior density of comes out to be

(4.2) |

Therefore, the Bayes estimator of under SELF is given by

(4.3) |

In this section, we consider the Lindley's approximation technique for the estimation of . To find the Bayes estimator of say, using the Lindley's approximation technique, consider the posterior expectation is expressible in the form of ratio of integral as given below

(4.4) |

where is the log of joint prior of and , given by

(4.5) |

If and are sufficiently large, according to Lindley ^{ 25}, can be approximately evaluated as

Thus

(4.6) |

where,

and

Therefore,

Hence,

(4.7) |

Following Kotz et al. ^{ 26} and Ventura and Racugno ^{ 27}, we obtain the posterior *pdf* of by mean of a one-to-one transformation of the type So, putting , taking into account that the Jacobian of the transformation is , by (4.2) the joint *pdf *of is

(4.8) |

Consequently, we can obtain the posterior *pdf* of marginalizing (4.8) with respect to , i.e.,

(4.9) |

where .

The Bayes estimator of , say, without using Lindley's approximation technique, under SELF can be easily obtained by using a result of Gradshteyn and Ryzhik (^{ 28}, p.286, section 3.197(3)), as

(4.10) |

where denotes the well-known hypergeometric function [see, for example, Gradshteyn and Ryzhik (^{ 29}, p.1005, Eq. 9.111)].

Therefore, can be obtained by using (2.1) and (4.10) as

(4.11) |

It is seen that the marginal densities of and are gamma distribution with parameters and , respectively. To obtain the Bayes estimate of under SELF the following algorithm is used:

(1) Set *i*=1

(2) Generate from

(3) Generate from

(4) Compute at

(5) Set *i*=*i*+1

(6) Repeat steps 2-5, times and get the posterior sample , .

Then the Bayes estimate of under SELF is given by

(4.12) |

This section deals with some experimental results to examine the behavior of the proposed methods for different parametric values and sample sizes. Simulation study is carried out by using Monte Carlo simulation technique and comparisons are made on the basis of mean squared errors (MSEs) of different estimates. Throughout the simulation, we have considered exponential distribution by taking All the computations are done on *statistical software-R*.

In order to obtain the we ran a MCMC chain. We generate 30000 iteration and to diminish the effect of the starting distribution, we discard first 5000 observations and focus on the remaining.

We have generated 3000 random samples each of size from strength and of size from the stress populations for different vales of and . For =2 and =1(1)4, we have computed , average values of , , , and their corresponding MSEs. We have also computed the ACI and length of the ACI. For different values of and =(1,3), (2,3), these results are reported in Table 1. Under the same set-up for =3, =1(1)4 and different values of , and =(1,4), (2,4), the results are presented in Table 2.

In order to compare the performances of different estimators of, graphically, for different values of , , we have conducted the simulation experiment based on the above mentioned procedure. For =2 and =1(1)4, we have computed the MSEs and the biases corresponding to the different estimators of . For different values of and =(1,3), obtained MSEs and biases have scaled by multiplying , thereafter these results are plotted in Figure 1-3, respectively.

The Table 1 - Table 2 and Figure 1 - Figure 3, illustrate the following:

(1) Minimum MSE is depicted by the Bayes estimators of ,

(2) both the Bayes estimators of depicting the same behavior of MSEs and biases,

(3) the MSEs and biases of the estimates decreases when the sample size increases,

(4) all the estimates come close to each other when the sample size increases,

(5) when is around 0.5 the corresponding MSEs are maximum,

(6) when is small or large the corresponding MSEs are minimum for all estimates, and

(7) length of the ACI decreases as sample size increase.

Here, we have studied the multicomponent system, which has independent and identical strength components and each component is exposed to a common random stress, where the underlying distribution of stress and strength variables is assumed to be PHR model. Using Monte Carlo simulation technique UMVU, ML and Bayes estimates are obtained and compared on the basis of their corresponding MSEs. Further this paper give the clear picture of comparison of classical and Bayesian methods of estimation for the members of PHR family of distributions.

The authors are extremely thankful to the editor and referee for their useful suggestions.

Both authors declare no competing interest.

[1] | Basu AP. Estimates of reliability for some distributions useful in life testing. Technometrics. 1964; 6: 215-219. | ||

In article | View Article | ||

[2] | Kelly GD, Kelly JA, Schucany WR. E_cient estimation of P(Y<X) in the exponential case. Technometrics. 1976; 18: 359-360. | ||

In article | View Article | ||

[3] | Awad AM, Gharraf MK. Estimation of P (Y < X) in the Burr case: A comparative study. Communication in statistics-Simulation and Computation. 1986; 15(2): 389-403. | ||

In article | View Article | ||

[4] | Tyagi RK, Bhattacharya SK. A note on the MVU estimation of reliability for the Maxwell failure distribution. Estadistica. 1989; 41: 73-79. | ||

In article | |||

[5] | Chaturvedi A, Kumar S. Further remarks on estimating the reliability function of exponential distribution under type I and type II censorings. Brazilian Journal of Probability and Statistics. 1999; 13: 29-39. | ||

In article | |||

[6] | Chaturvedi A, Pathak A. Bayesian estimation procedures for three parameter exponentiated Weibull distribution under entropy loss function and type II censoring. Inter Stat. 2013; interstat.statjournals.net/YEAR/2013/ abstracts/1306001.php. | ||

In article | |||

[7] | Chaturvedi A, Pathak A. Bayesian Estimation Procedures for Three-parameter Exponentiated-Weibull Distribution under Squared-Error Loss Function and Type II Censoring. World Engineering & Applied Sciences Journal. 2015; 6 (1): 45-58. | ||

In article | |||

[8] | Chaturvedi A, Kang SB, Pathak A. Estimation and testing procedures for the reliability functions of generalized half logistic distribution. Journal of the Korean Statistical Society. 2016. | ||

In article | View Article | ||

[9] | Chaturvedi A, Pathak A. Estimating the Reliability Function for a Family of Exponentiated Distributions. Journal of Probability and Statistics. 2014. | ||

In article | View Article | ||

[10] | Chaturvedi A, Kumari T. Estimation and testing procedures for the reliability functions of a family of lifetime distributions. Inter Stat. 2015. Available April 19, 2015 from: http://interstat.statjournals.net/YEAR/2015/abstracts/ 1504001.php; http://interstat.statjournals.net/INDEX/Apr15.html. | ||

In article | |||

[11] | Chaturvedi A, Kumari T. Estimation and Testing Procedures for the Reliability Functions of a General Class of Distributions. Communications in Statistics-Theory and Methods. 2017; 46 (22), 11370-11382. | ||

In article | View Article | ||

[12] | Chaturvedi A, and Kumari T. Estimation and Testing Procedures of the Reliability Functions of Generalized Inverted Scale Family of Distributions. Statistics-A Journal of Theoretical and Applied Statistics. 2018. | ||

In article | View Article | ||

[13] | Kumai T, Chaturvedi A, Pathak A. Estimation and Testing Procedures for the Reliability Functions of Kumaraswamy-G Distributions and a Characterization Based on Records. Journal of Statistical Theory and Practice. 2019. | ||

In article | View Article | ||

[14] | Bhattacharyya GK, Johnson RA. Estimation of reliability in a multicomponent stress strength model. Journal of the American Statistical Association. 1974; 69(348):966-970. | ||

In article | View Article | ||

[15] | Pandey M, Uddin Md. B. Estimation of reliability in multi-component stress-strength model following a Burr distribution. Microelectronics Reliability. 1991; 31(1): 21-25. | ||

In article | View Article | ||

[16] | Rao GS, Kantam RRL. Estimation of reliability in multicomponent stress-strength model: Log-logistic distribution. Electron Journal of Applied Statistical Analysis. 2010; 3(2): 75-84. | ||

In article | |||

[17] | Rao GS. Estimation of reliability in multicomponent stress-strength model based on generalized exponential distribution. Colombian Journal of Statistics. 2012; 35(1): 67-76. | ||

In article | |||

[18] | Rao GS, Kantam RRL, Rosaiah K, Reddy JP. Estimation of reliability in multicomponent stress-strength model based on inverse Rayleigh distribution. Journal of Statistics Applications and Probability. 2013; 3: 261-267. | ||

In article | View Article | ||

[19] | Kizilaslan F, Nadar M. Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based onWeibull Distribution. Revista Colombiana de Estadistica. 2015; 2: 467-484. | ||

In article | View Article | ||

[20] | Kuo W, Zuo MJ. Optimal Reliability Modeling, Principles and Applications. New York, John Wiley & Sons; 2003. | ||

In article | |||

[21] | Basirat M, Baratpour S, Jafar A. Statistical inferences for the proportional hazard models based on progressive Type-II censored samples. Journal of Statistical Computation and Simulation. 2014; http://www.tandfonline.com/loi/gscs20. | ||

In article | |||

[22] | Ahmadi J, Mohammad JJ, Marchand E, Parsian, A. Bayes estimation based on k-record data from a general class of distribution under balanced type loss functions. Journal of Statistical Planning and Inference. 2009; 139(3): 1180-1189. | ||

In article | View Article | ||

[23] | Wang L, Shi YM. Reliability analysis based on progressively first-failure-censored samples for the proportional hazard rate model. Mathematics and Computers in Simulation. 2012; 82: 1383-1395. | ||

In article | View Article | ||

[24] | Wang L. Inference for a general lower-truncated family of distributions under records. Communication in Statistics-Theory and Methods. 2016. | ||

In article | View Article | ||

[25] | Lindley DV. Approximate Bayes Method. Trabajos de Estadistica. 1980; 3: 281-288. | ||

In article | |||

[26] | Kotz S, Lumelskii Y, Pensky M. The Stress-Strength Model and its Generalization: Theory and Applications. Singapore, World Scientefic; 2003. | ||

In article | View Article | ||

[27] | Ventura L, Racugno W. Recent advances on Bayesian inference for P(X < Y). Bayesian Analysis. 2011; 6(3): 411-428. | ||

In article | View Article | ||

[28] | Gradshteyn IS, Ryzhik IM. Tables of Integrals, Series and Products. London, Academic Press; 1980. | ||

In article | |||

[29] | Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products. New York, Academic; 2007. | ||

In article | |||

Published with license by Science and Education Publishing, Copyright © 2019 Taruna Kumari and Anupam Pathak

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Taruna Kumari, Anupam Pathak. On the Comparison of Classical and Bayesian Methods of Estimation of Reliability in Multicomponent Stress-Strength Model for a Proportional Hazard Rate Model. *American Journal of Applied Mathematics and Statistics*. Vol. 7, No. 4, 2019, pp 152-160. http://pubs.sciepub.com/ajams/7/4/5

Kumari, Taruna, and Anupam Pathak. "On the Comparison of Classical and Bayesian Methods of Estimation of Reliability in Multicomponent Stress-Strength Model for a Proportional Hazard Rate Model." *American Journal of Applied Mathematics and Statistics* 7.4 (2019): 152-160.

Kumari, T. , & Pathak, A. (2019). On the Comparison of Classical and Bayesian Methods of Estimation of Reliability in Multicomponent Stress-Strength Model for a Proportional Hazard Rate Model. *American Journal of Applied Mathematics and Statistics*, *7*(4), 152-160.

Kumari, Taruna, and Anupam Pathak. "On the Comparison of Classical and Bayesian Methods of Estimation of Reliability in Multicomponent Stress-Strength Model for a Proportional Hazard Rate Model." *American Journal of Applied Mathematics and Statistics* 7, no. 4 (2019): 152-160.

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[1] | Basu AP. Estimates of reliability for some distributions useful in life testing. Technometrics. 1964; 6: 215-219. | ||

In article | View Article | ||

[2] | Kelly GD, Kelly JA, Schucany WR. E_cient estimation of P(Y<X) in the exponential case. Technometrics. 1976; 18: 359-360. | ||

In article | View Article | ||

[3] | Awad AM, Gharraf MK. Estimation of P (Y < X) in the Burr case: A comparative study. Communication in statistics-Simulation and Computation. 1986; 15(2): 389-403. | ||

In article | View Article | ||

[4] | Tyagi RK, Bhattacharya SK. A note on the MVU estimation of reliability for the Maxwell failure distribution. Estadistica. 1989; 41: 73-79. | ||

In article | |||

[5] | Chaturvedi A, Kumar S. Further remarks on estimating the reliability function of exponential distribution under type I and type II censorings. Brazilian Journal of Probability and Statistics. 1999; 13: 29-39. | ||

In article | |||

[6] | Chaturvedi A, Pathak A. Bayesian estimation procedures for three parameter exponentiated Weibull distribution under entropy loss function and type II censoring. Inter Stat. 2013; interstat.statjournals.net/YEAR/2013/ abstracts/1306001.php. | ||

In article | |||

[7] | Chaturvedi A, Pathak A. Bayesian Estimation Procedures for Three-parameter Exponentiated-Weibull Distribution under Squared-Error Loss Function and Type II Censoring. World Engineering & Applied Sciences Journal. 2015; 6 (1): 45-58. | ||

In article | |||

[8] | Chaturvedi A, Kang SB, Pathak A. Estimation and testing procedures for the reliability functions of generalized half logistic distribution. Journal of the Korean Statistical Society. 2016. | ||

In article | View Article | ||

[9] | Chaturvedi A, Pathak A. Estimating the Reliability Function for a Family of Exponentiated Distributions. Journal of Probability and Statistics. 2014. | ||

In article | View Article | ||

[10] | Chaturvedi A, Kumari T. Estimation and testing procedures for the reliability functions of a family of lifetime distributions. Inter Stat. 2015. Available April 19, 2015 from: http://interstat.statjournals.net/YEAR/2015/abstracts/ 1504001.php; http://interstat.statjournals.net/INDEX/Apr15.html. | ||

In article | |||

[11] | Chaturvedi A, Kumari T. Estimation and Testing Procedures for the Reliability Functions of a General Class of Distributions. Communications in Statistics-Theory and Methods. 2017; 46 (22), 11370-11382. | ||

In article | View Article | ||

[12] | Chaturvedi A, and Kumari T. Estimation and Testing Procedures of the Reliability Functions of Generalized Inverted Scale Family of Distributions. Statistics-A Journal of Theoretical and Applied Statistics. 2018. | ||

In article | View Article | ||

[13] | Kumai T, Chaturvedi A, Pathak A. Estimation and Testing Procedures for the Reliability Functions of Kumaraswamy-G Distributions and a Characterization Based on Records. Journal of Statistical Theory and Practice. 2019. | ||

In article | View Article | ||

[14] | Bhattacharyya GK, Johnson RA. Estimation of reliability in a multicomponent stress strength model. Journal of the American Statistical Association. 1974; 69(348):966-970. | ||

In article | View Article | ||

[15] | Pandey M, Uddin Md. B. Estimation of reliability in multi-component stress-strength model following a Burr distribution. Microelectronics Reliability. 1991; 31(1): 21-25. | ||

In article | View Article | ||

[16] | Rao GS, Kantam RRL. Estimation of reliability in multicomponent stress-strength model: Log-logistic distribution. Electron Journal of Applied Statistical Analysis. 2010; 3(2): 75-84. | ||

In article | |||

[17] | Rao GS. Estimation of reliability in multicomponent stress-strength model based on generalized exponential distribution. Colombian Journal of Statistics. 2012; 35(1): 67-76. | ||

In article | |||

[18] | Rao GS, Kantam RRL, Rosaiah K, Reddy JP. Estimation of reliability in multicomponent stress-strength model based on inverse Rayleigh distribution. Journal of Statistics Applications and Probability. 2013; 3: 261-267. | ||

In article | View Article | ||

[19] | Kizilaslan F, Nadar M. Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based onWeibull Distribution. Revista Colombiana de Estadistica. 2015; 2: 467-484. | ||

In article | View Article | ||

[20] | Kuo W, Zuo MJ. Optimal Reliability Modeling, Principles and Applications. New York, John Wiley & Sons; 2003. | ||

In article | |||

[21] | Basirat M, Baratpour S, Jafar A. Statistical inferences for the proportional hazard models based on progressive Type-II censored samples. Journal of Statistical Computation and Simulation. 2014; http://www.tandfonline.com/loi/gscs20. | ||

In article | |||

[22] | Ahmadi J, Mohammad JJ, Marchand E, Parsian, A. Bayes estimation based on k-record data from a general class of distribution under balanced type loss functions. Journal of Statistical Planning and Inference. 2009; 139(3): 1180-1189. | ||

In article | View Article | ||

[23] | Wang L, Shi YM. Reliability analysis based on progressively first-failure-censored samples for the proportional hazard rate model. Mathematics and Computers in Simulation. 2012; 82: 1383-1395. | ||

In article | View Article | ||

[24] | Wang L. Inference for a general lower-truncated family of distributions under records. Communication in Statistics-Theory and Methods. 2016. | ||

In article | View Article | ||

[25] | Lindley DV. Approximate Bayes Method. Trabajos de Estadistica. 1980; 3: 281-288. | ||

In article | |||

[26] | Kotz S, Lumelskii Y, Pensky M. The Stress-Strength Model and its Generalization: Theory and Applications. Singapore, World Scientefic; 2003. | ||

In article | View Article | ||

[27] | Ventura L, Racugno W. Recent advances on Bayesian inference for P(X < Y). Bayesian Analysis. 2011; 6(3): 411-428. | ||

In article | View Article | ||

[28] | Gradshteyn IS, Ryzhik IM. Tables of Integrals, Series and Products. London, Academic Press; 1980. | ||

In article | |||

[29] | Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products. New York, Academic; 2007. | ||

In article | |||