ON ESTIMATION OF STRESS-STRENGTH RELIABILITY USING LOWER RECORD VALUES FROM ODD GENERALIZED EXPONENTIAL-EXPONENTIAL DISTRIBUTION

This research paper aims to find the estimated values closest to the true values of the reliability function under lower record values, and to know how to obtain these estimated values using point estimation methods or interval estimation methods. This helps researchers later in obtaining values of the reliability function in theory and then applying them to reality which makes it easier for the researcher to access the missing data for long periods such as weather. We evaluated the stress–strength model of reliability based on point and interval estimation for reliability under lower records by using Odd Generalize Exponential–Exponential distribution (OGEE) which has an important role in the lifetime of data. After that, we compared the estimated values of reliability with the real values of it. We analyzed the data obtained by the simulation method and the real data in order to reach certain results. The Numerical results for estimated values of reliability supported with graphical illustrations. The results of both simulated data and real data gave us the same coverage.


INTRODUCTION
The lower record values have an important role in solving a lot of problems that concern the studying of missing data for long periods, for example, weather, phenomenon, and health care studies. The statistical study of lower record was introduced. In article [1], to obtain estimators of R(t) and P they don't require Rao-Blackwel Simulation studies and an example based on real data considered as an illustration. for Bayesian comparison of record values based on generalized exponential distribution were considered in [2]. Also, authors found Bayesian analysis for record data Based on Generalized Inverted Exponential Model which considered in [3]. For finding interval estimation for Inverse Rayleigh Distribution based on lower record see [4]. For estimating the reliability for a family of life time distribution based on records see [5]. In [6] they estimated reliability for burr distribution in case of record data. For general class of distribution [7] studied the reliability with lower record. In [8] they found UMVUE of reliability in case of record values and the data has proportional reversed hazard family. In this article the authors studied the statistical inferences for linedly distribution for record data, see [9]. for other form of linedly distribution was studied by [10] for record data. For bathtub-shaped distribution and record data was studied by [11]. There are more articles about finding reliability as [12] and [13]. This raises some questions: What does the lower record mean? How to study it in different statistical models? And where lower record value can be repressed as xi if its values are less than all previous observations xi<xj for i>j?. To answer these questions, let us see the following definition; Let X1, X2, …. be an infinite sequence of identically and independently distributed (iid) random variables. An observation Xi is called an lower record if Xi<Xj for every i>j. We shall assume that occurs at time i, then the record time sequence is defined as L n-max{i:X i <X Ln-1 }. The lower record sequence R 1 ,R 2 ,….,R nis defined as R n =X Ln ,n∈N . The joint probability density function (pdf) of first n lower records is given by: In a model of Reliability of stress-strength, the service still provides until strength is more than stress. The probability of stress-strength model includes two random variables: X and Y, which indicate strength and stress respectively where R=P(Y<X), and both X and Y are independent. a lot of researches were published in this part of the study of R which explains the major role of probability in statics. for wide application of R=P [Y<X]. A lot of papers studied this model in different situations. The estimation of reliability for parallel system is considered in [13]. In addition, the reliability of the model of stress-strength based on Poisson-exponential distribution which discussed the reliability model based on simulated data [14]. This paper tends to estimate stress-strength model R=(Y<X) where strength and strength are two independent lower record values with OGEE distribution. Assume those scale parameters are known. The importance of OGEE distribution is the flexibility in modeling lifetime data for better representation of the phenomenon contained in the data set. For more information on OGEE distribution see [15]. According to paper [15], the new distribution OGEE can be represented for its Pdf and Cdf The maximum likelihood estimate and exact confidence interval of R are derived. Besides, Bayes estimator of R is derived, and all of these estimators are obtained based on mean square errors. The paper is organized as follows. In section (2), maximum and exact C.I, the Bayes estimator and Bootstrap C.I are obtained. For simulation, the studies' proposal is shown in section (3). A real data example is obtained in section (4). Results and discussions are obtained in section (5). tables and figures are represented in section (6). Finally, conclusions appear in section (7).

Non-Bayesian method
In this section maximum likelihood estimate (MLE) and the exact confidence interval of R is obtained.

MLE of the Reliability Function R
Let X be the strength of a system or component which is subjected to the stress that X~OGEED (θ 1 ,β) and Y~OGEED (θ 2 ,β); therefore, the following reliability function is obtained Let r=(r 0 ,r 1 ,r 2 ,…,r n ) be a set of first observed lower record values of size (n+1) from OGEED with parameter (θ 1 ,β) and s=(s 0 ,s 1 ,s 2 ,…,s m ) be an independent set of the observed first lower record values of size (m+1) form OGEED With parameters (θ 2 ,β), where β assumed to be Learned. The likelihood functions for both observed r and s are given, respectively The likelihood function of the observed r obtained ( ) ( ) ( ) ( ) ( ) ( ) l λ ,θ|r =Ln L = n+ ln λ + n+ ln θ + Therefore, the probability density functions of so the probability density function of Z 1 is given as: is recognized as the inverted gamma distribution with [(2,λ 1 (n+1))]. Similar, for Z 2 has the inverted gamma distribution with [(2,λ 2 (m+1))]. Therefore Pdf of the reliability (R) can be obtained as follow: Considering, Z 1 /Z 2 , it is easy through the properties of gamma distribution to show that: Since Z 1 and Z 2 are independent, then it can be shown that Exact distribution of R written as: Are the lower and upper α/2 th percentile point of F 2((m+1),2(n+1) .

Bayes estimate of R Based on MSE
In this section, the Bayes estimator of R is obtained under mean squared errors. Firstly, the Bayes estimator for λ 1 and λ 2 are obtained, and the non-informative priors for λ 1 and λ 2 can be found from the equation of fisher Information as follows: The posterior density λ 1 and λ 2 , denoted by π 1 * λ 1 and π 2 * (λ 2 ), are obtained by combining the equations of likelihood (3), (4) and the priors of λ 1 and λ 2 respectively The bayes estimators of λ 1 and λ 2 under squared errors loss function, denoted by λ̂1 (SE) and λ̂2 (SE) , are the posterior means which can be obtained as follow: Therefore the bayes estimator of the R under square error loss function, denoted by R (SE) can obtained from Compensation in Eq. (3) as follow:

Bootstrap of Bayes C.I
To find the Bayes confidence interval, use the method of the bootstrap Confidence interval, see [16](Kotz et al., 2003) [15]. In proposes the bootstrap method as an alternative way to construct a confidence interval. Algorithm of the (1-α)% confidence interval for α by using bootstrap method is illustrated below: 1. 1 Use the estimators value λ̂1 (SE) and λ̂2 (SE ) to generate N=5000 the bootstrap sample X 1 * , X 2 * ,….,X N * and Y 1 * ,Y 2 * ,..,Y N * , then compute the estimated value of R (SE) by Bayes which shown in Eq. (17).

Calculate the bootstrap MSE by
Where N=5000 3. The asymptotic (1-α)% confidence interval is given by

Monte-Carlo Simulation
In this section a simulation study is designed to find and compare the values of estimated Methods.

Real Data examples
In 1, the authors use an example of strengths of glass fibers data set which obtained from Smith and Naylor (1987) see [17], in table (I), which was proved that, this data follows to OGEE distribution, in this section, we will study this real data as follow: 1. The set of real data gas size 63 2. Select from each vector the first (n+1), n=2(1)9, lower record values, lower record values r 0 , r 1 , …, r n for the values of strength random variables X under the assumption that θ is known 3. Repeat step No.2 but for stress r.v. Y 4. The MLE of λ̂1 and λ̂2 are obtained from Eq. (7) and Eq. (8), then the MLE of R is obtained is obtained from Eq. (9) The exact confidence intervals of R is evaluate with confidence level at α=0.05. 5. The Bayes estimates of R under mean square loss function is obtained by obtain the Bayes estimates for λ̂1 (SE) , λ̂2 (SE) from Eq. (16), then find R (SE) estimates value from Eq. (17). Also, find Bootstrap C.I of R (SE) from Eq. (18).

Results and discussion
Simulation results were tabulated in tables (1-6) for estimated values and tables (7,8) for confidence intervals, and represented through figures (1, 2), for real data the results are tabulated in tables (9)(10)(11)(12)(13)(14) for estimated values and tables (16,17) for confidence intervals. First: For simulated data Tables (1-8) 1. The percentage for MLE is better than the percentage for Bayes. 2. As the size of the sample increase, the MSE becomes lesser. 3. The average length of the bootstrap Bayes confidence interval is smaller than the exact confidence interval, according to tables (7,8). 4. As the sample size increases, the mean confidence interval length becomes lower, except for some points, the average confidence length becomes longer, according to tables (7,8). 5. For some points when the value of m more than n, the percentage increase. Second: For real data Tables (9-16) 1. The percentage for MLE is better than the percentage for Bayes. 2. As the size of the sample increase, the MSE becomes lesser. 3. The average length of the bootstrap Bayes confidence interval is smaller than the exact confidence interval, according to tables (15,16). 4. As the sample size increases, the mean confidence interval length becomes lower, except for some points, the average confidence length becomes longer, according to tables (15,16). 5. For some points when the value of m more than n, the percentage increase.

Figure 2: MSEs OF MLE AND BAYES ESTIMATORS WHEN R=0.75 Shows that the MSEs of MLE and BAYES estimators decrease as n and m are increase at R=0.75, also, MSEs of MLE is smallest than MSEs of Bayes
For Real data figures [3,4]