Comparison Among Three Estimation Methods to Estimate Cascade Reliability Model (2+1) Based On Inverted Exponential Distribution

In this paper, we are mainly concerned with estimating cascade reliability model (2+1) based on inverted exponential distribution and comparing among the estimation methods that are used. The maximum likelihood estimator and uniformly minimum variance unbiased estimators are used to get the parameters of the strengths X and the stress Y ;k=1,2,3 respectively then, by using the unbiased estimators, we propose Preliminary test single stage shrinkage (PTSSS) estimator when a prior knowledge is available for the scale parameter as initial value due past experiences . The Mean Squared Error [MSE] for the proposed estimator is derived to compare among the methods. Numerical results about conduct of the considered estimator are discussed including the study of mentioned expressions. The numerical results are exhibited and put it in tables. Keyword: Inverted exponential distribution, Maximum Likelihood method, Uniformly Minimum Variance Unbiased method, Single Stage Shrinkage Estimator, Mean Squared Error and Cascade Reliability Model (2+1).


Introduction
An -cascade system is defined as a special type of standby system with components and it could be considered as a kind of stress-strength model. The reliability system of a cascade model can be described by a function of parameters of the identical and independent distributions with strength (X) and stress (Y) and the attenuation factor (K) in other words, the stresses on subsequent components are attenuated by a factor 'k', called attenuation factor that is generally assumed to be a constant for all the components or to be a parameter having different fixed values for different components, or it may be simply a random variable. This system was first proposed and studied by Pandit and Sriwastav (1975). Rekha and Chechu Raju (1999) presented a closed form solution of stress attenuated reliability function for ncascade system with exponential stress and standby strengths following Rayleigh and exponential distributions. Sundar (2012)has done a case study of cascade reliability with Rayleigh distribution. Devi (2016) is a prior knowledge(estimation) about the parameter and 0 ∅ 1 is a shrinkage weight factor to assign the degree of belief in ; also is the classical estimator of (MLE or UMVUE). Several authors have studied the estimator defined in (1) for a special distribution for different parameters and suitable regions (R) as well as for estimate the parameters of linear regression model. See [13][14][15][16].

Statistical Model
In life distribution, if a random variable X following an exponential distribution then the variable has an inverted exponential distribution (IE 1,2,3, when activated faces the stresses random variables , and are imposed on the strengths components and followed IED with scale parameter ; 1,2,3. In cascade system, after every failure the stress gets modified by attenuation factor (k) such that: , , … , ; 1 And we suppose a factor (m) to modified the strength such that: , , … , ; m 0 The real reliability function for the (2+1) cascade model is given by: 1 1 So we get (7) , , 1 Then , By the same way finding R R , , Substitution (6), (11) and (13) in (4)

Maximum Likelihood Estimation (MLE)
The Maximum likelihood method is an important and commonly, since it contained properties for good estimate. Suppose ; 1,2, 3 strength random sample follows IED  with  the  sample  size  .The  likelihood  function  is  given  by; Taking logarithm of (16) and then differentiating the result partially with respect to : Equalizing (17) By the same way for the stress random variables ~ ; 1,2, … , , ~ ; 1,2, … , and ~ ; 1,2, … , with samples size , and respectively, the MLE estimator for unknown parameters , and will be as follows: Substitution (19) and (20) in (15) the MLE for Cascade reliability, invariability will be as: (21)

Uniformly Minimum Variance Unbiased Estimators
The UMVUE method depends on minimizing the mean square error among unbiased estimators.
then So the unbiased estimator of ( ) is ( ), therefore according to Lehmann-Scheffe theorem the (UMVUE) of ( ) is By the same way, we can obtain (UMVUE) of ( ) as below: Substituting (25), (26) in (15) to obtain UMVU estimator for cascade reliability model of IED as the following :

Simulation Study
In this section, we present some results based on Monte Carlo simulations to compare the performance of different methods: MLE, UMVUE and PTSSSE of Cascade Reliability Model(2+1) using different sample (20, 40, and 60). For this purpose the following steps of Monte Carlo simulation are used based on Mean Squared Errors criteria with 1000 replicates: Step1: Generate random samples which follow the continuous uniform distribution defined on the interval (0,1), as , , … , and , , … , ,for all 1,2, … , 1,2, … , . respectively., Step2: Applying an inverse transformation approach to generate random variables follows IED as follows: : :

/ ln
And, by the same method, we get:
Step4: find the cascade reliability R of the MLE using equation 21 .
Step5: find the uniformly minimum variance unbiased method of R using equation 27 .
Step7: Calculate the MSE based on (L=1000) trials as follows:

Results of Simulation
In this section, the simulation results are used to determine the best outcome of the conceder estimation methods ( , UMVUE, PTSSSE) of cascade reliability model (2+1) estimator based on one parameter IED. In the cascade reliability model (2+1) of estimate the system reliability , , , , , )).The following tables of mean square error show, at most the orders rank of the estimators as follows: , and respectively, that means is the best than the others estimators. The following tables (1-9) will present the simulation results. , , , based on inverted exponential distribution were used in this paper to verify the performance of different estimators which are ; Maximum likelihood estimation, Unbiased estimation method and Preliminary test single stage shrinkage (PTSSS) estimator using different samples 20, 40, and 60 . The Monte Carlo Simulation was exhibited to analyses and comparison between these methods. the results show that the performance of (PTSSS) estimator was appropriate behavior and it is efficient estimator than the others in the sense of at most. While had the second rank and followed by .