1. Introduction

It has become a fact that improving the quality of products depends mainly on the on going demand for these products. In addition, staying in the market has become also associated with the reliability of such products. For these reasons, companies need to meet specific measure to compete in world wide markets such as, its ability to high quality, competitive prices, and reliable goods on time. In thisregards, manufactures use formative assessment for there products to achieve reliability which is closely related to durability, accessibility and survival. Failure can beconsidered a random variable since it is so difficult to tell precisely when a specific product will fail under use condition.

Under normal condition, measure of reliability for device becomes too difficult and it requires very long time. So, accelerated life testing induces failures and the failure data at the accelerated conditions are used to estimate the reliability at normal operating conditions when the reliability of a component is "high" and failure data of the component may not be attainable during its expected life. The problem of estimating the reliability arises in the situation of mechanical reliability of a system with strength and stress , and R is a determine of system reliability. The system fails if and only if, at any time the stress exceeds the strength.

Many authors have measured different choices for stress and strength distributions. Johnson (1988)[10] summarized

some of these choices. Awad & Gharraf (1986)[5] considered the case when and are independent and have Burr Type XII random variables. They obtained a maximum likelihood estimator, minimum variance unbiased estimator, and a Bayesian estimator for R. They used a numerical procedure for evaluating their Bayesian estimator. Alam & Roohi (2002)[3] studied stress and strength having exponential distribution but in (2003)[4] have studied the problem of in a different way. They have set up the required parametric values of the assumed distributions as a replacement for finding P (YThe Weibull distribution is one of the most widely used distributions in the reliability and survival studies. Baklizi (2012)[7] considered the stress-strength reliability based on record values from the Weibull distribution. He obtained the Bayes estimator based on squared error loss and the maximum likelihood estimator of the reliability R. Akbar et al (2012) )[2] focus on the inference for the stress- strength reliability R when X and Y are two independent Weibull distributions with the same shape parameter, but having the different scale parameters. They obtained The maximum likelihood and the approximate maximum likelihood estimators of R.

Azzalini (1985)[6] suggested a method of obtaining weighted distributions from independently identically distributed random variables. He used the density function of one random variable and the distribution function of the other random variable as follows:

(1.1)

Recently, Gupta & Kundu, (2009)[9] presented a new class of weighted exponential distributions. Makhdoom (2012)[13] studied the estimation of Stress-Strength reliability when X and Y are two weighted exponential distributions with different parameters. He obtained the MLE of R based on one simple iteration procedure, and he carried out Bayesian estimators of parameters with real data. Saman et al. (2010)[15] proposed The weighted Weibull model based on an idea of Azzalini (1985)[6]. They studied basic properties of the distribution including moments, generating function, hazard rate function and estimation of parameters. The proposing model which in Saman et all (2010)[15] to evaluate the parameters of Weighted Weibull distribution is considered here. Then, the cumulative distribution function is:

(2.1)

and the probability density function is:

(2.1)

hence are shape parameters and is scale parameter. Now, let as the same used manner at Saman et all (2010)[15].

This paper consists of three sections corresponding to sections 2, 3, and 4, respectively. section 2 provides an approach for scheming Stress-Stress reliability system. section 3, discusses the maximum likelihood Estimators, Least Square Estimators and Bayesian estimators of the reliability R. Finally a numerical investigation will be carried out to study the properties of the new estimators.

2. System Reliability

Let be the strength of a system and be the stress acting on it. and are the random variables from Weighted Weibull distributions with parameters and respectively. Therefore, the probability density functions of and are, respectively:

(1.2)

(2.2)

where and are shape parameters and is scale parameter. Then, the reliability function is:

(3.2)

where and .

3. Point Estimator of the Reliability R

3.1. Maximum Likelihood Estimator

Let and be the two independent random samples taken from the Weighted Weibull distributions with parameters and respectively, then, the likelihood and log-likelihood function based on the above samples are respectively given as:

(1.3)

(2.3)

The derivatives of with respect to and are, respectively:

(3.3)

(4.3)

(5.3)

Unfortunately, there is no demand value for in (3.3), (4.3) and (5.3), so with that, Newton–Raphson method in an iterative approach to solve this equations in numerical analysis is considered. It is an iterative method for solving equations where is assumed to have a continuous derivative . Given a function and its derivative , a first guess is initialed. Then, an approximation of is and an approximation of is and so on for number of iterations r or if where is the estimate.

Now, replace which gives from iteration with the parameter after iteration process for estimating the parameter in equations (3.3), (4.3) and (5.3).

Since, Maximum Likelihood Estimators are do not change, So it becomes:

(6.3)

where and .

3.2. Least Square Estimators

Suppose that (2.1) is a linear relation between the two variables and take the logarithm of the two sides as follows:

(7.3)

So, if this function is minimized with respect to , we end up with the Least Square Estimators. This method is known as Least Square method (more details, see Flaih et al (2012)[8].

Now, using the mean rank method, we can estimate from this relation , where are the rank failure times.

Let , so, (7.3) becomes straight line equation. Therefore, the least square estimators of and come from minimizing the following equation:

(8.3)

The first partial derivatives of with respect to and , respectively, are:

(9.3)

(10.3)

Similarly, the first partial derivatives of with respect to is:

(11.3)

If both , and equal zero, then, the Least Square Estimators of , and will result immediately. However, this process is too difficult to be done without numerical solution. So, we adapt the previous technique of , Newton–Raphson method.

3.3. Bayes Estimator of Reliability R

Let and be two independent random samples, drawn from the Weighted Weibull distributions with parameters and respectively, an associate gamma prior distributions for and with parameters and are employed respectively, so that the prior distribution for and are:

(12.3)

(13.3)

where are independent. And the joint density function for data and the parameters and is:

(14.3)

Then, the posterior density function for and based on data is:

(15.3)

Now, by using Posterior Mode Method to obtain the Bayesian estimators of parameters as follows:

The log posterior distribution for the sample taken from Weighted Weibull distribution with Gamma priors and respectively is:

(16.3)

where C does not dependent on the parameters and the parameters and are fixed.

The first derivatives of with respect to and are, respectively:

(17.3)

(18.3)

Let , and .

Then,

(19.3)

(20.3)

(21.3)

where . So, the Hessian matrix of the derivatives I is :

(22.3)

The object of this work is estimate numerically to find where . Solution can be converged using Newton–Raphson algorithm to estimate which are replaced by gleaned from iteration.

4. Simulation Study

Table (1). When       n m RMLE RBN-IF RB-IF MSEMLE MSEBN-IF MSEB-IF 5 5 0.4063 0.3975 0.4094 0.0222 0.0182 0.017 10 0.4179 0.3996 0.4256 0.0161 0.0142 0.0138 20 0.4222 0.3993 0.4324 0.0155 0.0146 0.012 30 0.4233 0.3988 0.4344 0.0119 0.014 0.0115 10 5 0.3939 0.3988 0.3913 0.0156 0.0131 0.149 10 0.4023 0.3977 0.4042 0.0095 0.0085 0.0081 20 0.4059 0.3965 0.4102 0.00617 0.00571 0.0056 30 0.4085 0.3976 0.4136 0.00511 0.00484 0.0048 15 5 0.3852 0.3963 0.3798 0.0122 0.0104 0.0099 10 0.3974 0.3999 0.3961 0.00601 0.00535 0.0051 20 0.401 0.3986 0.402 0.00277 0.00253 0.0024 30 0.4018 0.3979 0.4036 0.00194 0.00179 0.0017 20 5 0.386 0.3996 0.3792 0.0112 0.0092 0.009 10 0.3929 0.3976 0.3905 0.00471 0.00421 0.004 20 0.3955 0.3955 0.3954 0.00182 0.00164 0.0016 30 0.3992 0.3976 0.3999 0.00075 0.00065 0.0006

Table (2). When       n m RMLE RBN-IF RB-IF MSEMLE MSEBN-IF MSEB-IF 25 25 0.5807 0.5895 0.5777 0.0194 0.0234 0.0182 50 0.5916 0.5869 0.5943 0.0139 0.0164 0.123 75 0.5995 0.5881 0.6051 0.0111 0.0129 0.0105 100 0.6027 0.5892 0.6094 0.0105 0.0122 0.0132 50 25 0.5744 0.5926 0.5666 0.0152 0.0173 0.0148 50 0.588 0.5927 0.5861 0.0087 0.0096 0.0083 75 0.5926 0.5903 0.5938 0.0061 0.0068 0.0059 100 0.5964 0.5918 0.5987 0.0051 0.0056 0.0048 75 25 0.5677 0.5907 0.5575 0.0131 0.0141 0.0083 50 0.5814 0.5907 0.5781 0.0067 0.0072 0.0132 75 0.5898 0.5922 0.5887 0.0035 0.0037 0.0034 100 0.5902 0.5902 0.5903 0.0028 0.0029 0.0027 100 25 0.5636 0.5882 0.5526 0.0124 0.0491 0.0126 50 0.5794 0.5903 0.5742 0.0057 0.0061 0.0054 75 0.5842 0.5881 0.5825 0.0027 0.0028 0.0026 100 0.5874 0.5889 0.5846 0.0018 0.0019 0.0017

The computer programs MathCAD (2001) is used to obtain numerical illustration for the last theoretical results. A comparison between the three estimators, MLE, Bayes based on non-informative estimator and Bayes based on informative estimator is performed . 200 samples generated from Weighted Weibull distributions with parameters and are used, respectively, and different values of , and with various sizes for both n and m in table 1. Also, various sizes for them in table 2. Then the means of these replicates are calculated to obtain maximum likelihood estimator of R (R_MLE), Bayes estimator based on non-informative (R_BN-IF) and Bayes estimator based on informative R_B-IF . The mean square error of R_MLE, R_BN-IF and R_B-IF are computed as well.

From Table (1) and (2), note that the mean square error of R_B-IF is smaller than each of the mean square error of R_MLE and R_BN-IF if the sample sizes n, m are small or large. But, in table (2), when the sample sizes are large the mean square error of all estimators are less than the other. For example, when the sample sizes and , we find that the mean square errors of R_MLE and R_BN-IF and R_B-IF are (0.0155, 0.0148 and 0.0120) respectively, while, they are (0.0111, 0.0129 and 0.0105) when the sample sizes are and . This behavior applies on the rest of parameters and as well.

Also, the three estimators R_B-IF, R_MLE and R_BN-IF are increase with decreasing the sample size n and take the sample size m is fixed, and they are increase with decreasing the sample size m and take the sample size n is fixed. On the other hand, they are increase with decreasing both the sample sizes n and m.

5. Conclusions

In this paper, the estimation problem of the reliability of a system R in stress-strength model when and are independent variables distributed as Weighted Weibull Distribution is considered. Maximum Likelihood Estimators, Least Square Estimators and Bayesian Estimators based on non-informative and informative prior distributions were discussed via numerical results.