Doubly Type II Censoring of Two Stress-Strength System Reliability Estimation for Generalized Exponential-Poisson Distribution

In this paper, a Bayesian analysis is made to estimate the Reliability of two stress-strength model systems. First: the reliability 𝑅 1 of a one component strengths X under stress Y. Second, reliability 𝑅 2 of one component strength under three stresses. Where X and Y are independent generalized exponential-Poison random variables with parameters (α,λ,θ) and (β,λ,θ) . The analysis is concerned with and based on doubly type II censored samples using gamma prior under four different loss functions, namely quadratic loss function, weighted loss functions, linear and non-linear exponential loss function. The estimators are compared by mean squared error criteria due to a simulation study. We also find that the mean square error is the best performance of the estimator from that found in quadratic, weighted, linear and non-linear exponential loss functions.


AbdAwon and Karam
Iraqi Journal of Science, 2023, Vol. 64, No. 4, pp: 1869-1880 1870 [2]. introduced in the same shape a two-parameter distribution known as exponential distribution with the Poisson distribution. In )2009) [3] Barreto-Souza and Cribari-Neto generalized exponential-Poisson with decreasing or increasing failure rate. The two parameter exponential-Poisson (EP) with cumulative distribution function (c.d.f) is given as follows: [4] The Generalized Exponential-Poisson distribution (GEP) for the random variable X with parameters (α,λ,θ ) is given as follows: [3] ( ) = [ The paper is organized as in section 2, the general expression of 1 and 2 are given for one stress-strength and one strength composed under three stresses. The Bayesian estimators are found for 1 and 2 under four different loss functions are given in section 3. Finally, in section 4 the performance of the estimators is illustrating by experiment simulation study.

Reliability of the systems for GEP Stress-Strength Models
The purpose of this section is to obtain the reliabilities expression of two different systems for stress-strength models.

One component system reliability.
The reliability 1 of a component operating under stress -strength system given by [5] 1 = ( < ) (3) Let the strength random variable ~( , , ) and the stress random variable ~( , , ) when X and Y are independent but not identical.then the (c.d.f) for Y can be written as follows: Now, the reliability 1 from equation (3) can be given as follows: (2), then we can use the following: So we get

Three stress-one strength
The reliability of a component that has X strength is exposed to three independent stresses namely 1 , 2 and 3 . The stress-strength reliability can be obtained as follows: be strength random variable and 1~( 1 , , ), 2~( 2 , , ) and 3~( 3 , , )are the stresses random variables, then the reliability can be obtained as follows [6]: Since X, 1 , 2 3 are non-identical independently distributed, we can get: By using equation (4( we get 2 = + 1 + 2 + 3 (6)

Bayes analysis
In this section, the Bayes estimators of reliabilities 1 and 2 are given based on a doubly type II censored sample using gamma prior under quadratic, weighted, linear and non-linear exponential loss functions.

Doubly type II censored sample
When we have units that are subject to testing and we want to censor the work of units, where m=s-r+1 and r < < ,any censoring data ( , … , ) ,so in this case ,it cannot be determined the random variable in the time and thus stop the test up to get units of censored and upon arrival to the s , The likelihood function for this type of data [7]: Continuing from equation (7), we find that:  (8), (9)and(10)in(7( we get

Bayes procedure
By using the Bayes method to find the posterior function under gamma prior function which is given as follows: The posterior function can be found form the following relation: By using (11)and (12) the posterior function becomes: Then, we get

Quadratic loss function.
The Bayes estimator for α using the quadratic loss function is given as follows: [8] ̂= By compensating (14) in equation (15) we can get By equation (13) we get Then, the estimates will be as follows: .
And the reliabilities estimation in equation (5)and (6), we get:

Weighted loss function
The Bayes estimator for α, β using the weighted loss function which is given as follows: By compensating (16) in (17), we get And the reliabilities estimation in equation (5) and (6) we get:

Linear exponential loss function
The Bayes estimator for α, β using the linear exponential function is given as follows: And the reliabilities estimation in equation (5)and (6), we get:

Simulation Study
In this section, Monte Carlo simulation is performed to compare the performance of different estimations of 1 and 2 . The simulation study has been carried out for four samples size = 15,30,50,100. The values r=5,10,16,34 and s=9,20,30,50, respectively, and the values of c=1 ,the parameter values of the shape parameter α, the prior distribution a=2 and b=3 the results have been sufficiently replicated for L=1000 time for each experiment. The result presented some numerical experiment to compare the performance of the Bayes estimators under gamma prior distribution and four loss functions proposed in the previous sections. we have presented the simulation results using MATLAB(2013) program .A simulation results are conducted to examine and compare the performance of the estimates for parameter to the MSE. The estimator has the smallest value of MSE as well as doubly type II censored sample as it is shown in the last column in Tables 3-8.