the Estimation of Stress-Strength Model Reliability Parameter of Power Rayleigh Distribution

The aim of this paper is to estimate a single reliability system (R = P, Z > W) with a strength Z subjected to a stress W in a stress-strength model that follows a power Rayleigh distribution. It proposes, generates and examines eight methods and techniques for estimating distribution parameters and reliability functions. These methods are the maximum likelihood estimation(MLE), the exact moment estimation (EMME), the percentile estimation (PE), the least-squares estimation (LSE), the weighted least squares estimation (WLSE) and three shrinkage estimation methods (sh1) (sh2) (sh3). We also use the mean square error (MSE) Bias and the mean absolute percentage error (MAPE) to compare the estimation methods. Both theoretical comparison, simulation and real data are used. The results in light of this distribution show the advantage of the proposed methods.

Where k is the shape parameter and ʓ is the scale parameter. The rest of the paper is organized as follows: In Section 2, we provide the single reliability system PR(ʓ,k) in the Stress-Strength model. We also derived the eight estimation methods. The numerical studies (simulation and real data) are presented in Sections 3 and 4. Finally, the conclusions and discussions appear in Section 5.

Reliability of the Systems for PR Stress-Strength Models
Let Z be the strength and W be the stress random variable independent and each other by PR(ʓ, 1 ) and PR(ʓ, 2 ) respectively, with two different parameters when = 1 ʓ 2 . Then ].

Maximum Likelihood Estimator (MLE)[3]
The MLE carefully estimates the parameters of an entire sample power Rayleigh distribution. Let 1 , 2 , … , 1 be a random sample for PR(ʓ , k).When ʓ is known and the shape parameter k is unknown then the likelihood function f( ; ; ʓ) in equation (2) is Taking the logarithm of both sides, then 1 ln = nln 1 − 2nlnʓ + (2 1 − 1) ∑ − 1 2ʓ 2 ∑( ) 2 1 =1 =1 (6) The partial derivative of equation (6) with regard to 1 and the equivalence of the results to zero then we get In the same way, let 1 , 2 , … , 2 be a random sample from the stress w which is distributed as power Rayleigh distribution when ʓ is known and shape parameter 2 unknown then the likelihood function f( ; 2 ; ʓ) in equation (2) is done, then the MLE method is presented by Where n and m are the size of Z and W samples, respectively. Now we substitute equations (7) and (8) into equation (5) we obtain. ].

The Exact Estimators of Moments Method (EMME)[11]
We provide the expectation and variance of the power Rayleigh distribution as follows: (1 1) And then the coefficient of variation is given by From the equation (12) and (13) in (5) we obtain. ].

The Percentile Estimator (PE)
Let F( ,ʓ, 1 ) and F( ,ʓ, 2 ) be two c. d. f. for the random variables of the strength and the stress, respectively.

The Least Squares Estimator Method (LSE).
By minimizing the sum of square error between the value and its predicted value, least squares technique estimators can be created. The LS approach is frequently used to fit models and solve mathematical and engineering issues, in particular, in linear and non-linear regression [5].

=1
when ∈ = ( − ʓ − ), Now, we solve the two equations (24) and (25) so we get where Z is the PR distribution's strength random variable with sample size n, and W is the PR distribution's stress random variable with sample size m. A distribution function is derived (CDF) we obtain. Now from equation (26), the LS method for the shape parameter k can get the LS estimators of ̂1 is presented by: And We substitute equations (27)and (28)into equation (5) we obtain. ].

Weighted least squares (LSE) estimators
The weighted least squares estimators of and 2 are ̂2 and ̂2 , respectively. They can be obtained by minimizing from the equation as follows: ]. (30) We substitute equations (32)and (33)into equation (10) we obtain ̂1 ].

The Shrinkage Estimator (Sh) [4].
The shrinkage estimation method can be thought of as a Bayesian strategy that relies on prior knowledge. Thompson had introduced the main arguments for utilizing previous estimating [4]. The parameter was utilized as a starting value 0 where [ 0 = ∓∈], ∈= 0.001 from the past in the shrinkage estimation method, and the normal estimator(̂) was employed to them by shrinkage weight factor (̂), 0≤ Ω (k) ≤ 1, which can be written as: [9] The weight shrinking function will be considered. In this subsection the function form is denoted by (k) =[

Monte Carlo Simulation Study and it's Results
Simulation is a numerical technique of performing experiments on a computer while the Monte Carlo simulation is a computer experiment that involves random sampling of probability distributions. In order to verify the performance of the proposed estimation method that was introduced for estimating the single component reliability system, the Monte Carlo simulation was used. The proposed eight estimation methods are implemented using diverse samples (25, 50, 75, 100). Statistical results for each sample are based on bias, mean absolute percentage error and mean squared error criteria with 1000 replicates. Therefore, the following steps explain the Monte Carlo simulations for each model.
Step 3: The ʓ is considered the known parameter as the mean of the sample and is considered as the unknown parameter. The MLE estimators ̂1 and ̂2 have been calculated respectively from equations (7) and (8).
Step 7: The weighted least squares estimators ̂1 and ̂2 have been calculated from equations (31) and (32), respectively.  [2]. In this part, we have tested all the results above from a real data, as shown below  We investigate strength data, which was originally reported by Badar and Priest [1] and represent strength that is measured in the GPA of mono and flooded 1000 -carbon fiber. Single fibers are tested under pressure at gauge lengths of 20 mm (data set 1) and 10 mm (data set 2), with sample sizes n = 74 and m = 63, with 1 = 1, 2 = 1.5 , and ʓ = 1 , respectively. The data is shown in Tables 1 and 2. Several authors have analysed these data sets such as Surles and Padgett [10], Neck and Kondo [8].

5.Conclusion
In Tables (1-6) for the random data, we conclude the following: 1-The value of MSE decreases with increasing sample size (n, m) for all the factors under study. 2-When the value of k decreases, the value of the estimated reliability decreases. 3 -When k1 < k2, the MLE method is the best possible. 4 -When k1 = k2, the (Sh3) method is the best possible 5-When k1 > k2, the estimation methods alternate with each other according to the sample size, but the best methods are (Sh3, MLE, LE, WLS, LS), respectively. In Table (7) the practical example, we conclude the following: 1 -When k1 > k2, MLE is the best possible method. 2 -When k1 < k2 the (LS) method is the best possible. 3 -When k1=k2, the LS method is the best it can be. In general, the (LS) method is the best possible in the applied example.
In general, the Monte Carlo simulations are used to compare reliability estimates for small samples, with the MLE approach under k1 and k2 estimators providing the best results, and using three criteria (Bias, Mse, Mape).