Abstract
This paper deals with finding a formula for the stressstrength reliability function for complete data when the strength (X) falls between the stress (T) and the stress (Z) ; where X,T,Z are independent random variables and follow the Exponentiated Inverse Rayleigh Distribution with unknown shape parameters and common known scale parameter , and estimate this formula with the Maximum Likelihood Estimate method (MLE) and the Bayesian method using Noninformative priors and informative priors under Weighted Square Error Loss Function ( WSELF ) ,Also the interval estimation had been done for the reliability function that based on the Maximum Likelihood Estimator .
Simulation study is used to determine the best estimator; the results showed that Bayesian estimation using informative priors based on Weighted Square Error Loss Function is the best estimator For the equal sizes , and Bayesian estimation using Noninformative priors based on Weighted Square Error Loss Function is the best estimator when the size of the stress sample (Z) larger than the size of (X,T) , and Maximum Likelihood Estimator is the best estimator For the rest sizes
Highlights
In this paper, point and interval estimation was presented to estimate the reliability function P (T< X< Z) when each of X, Z and T follows Exponentiated Inverse Rayleigh Distribution with different shape parameters for complete data, the point estimation included maximum likelihood method and Bayesian estimation using informative Gamma priors and nuninformative priors based on Weighted Square Error Loss Function ( WSELF ) for interval estimation confidence interval were estimated for the reliability function P (T< X< Z) based on maximum likelihood estimator of the reliability , Simulation results that appeared confirm that the value of the reliability and confidence intervals is between (0,1) which match the statistical theory and the Bayesian estimation using informative priors based on Weighted Square Error Loss Function is the best estimator for the equal sizes , and Bayesian estimation using nuninformative priors based on Weighted Square Error Loss Function is the best estimator when the size (w) of the stress sample (Z) larger than the sizes of (X,T) , and Maximum Likelihood Estimator is the best estimator For the rest cases.
Full Text
Introduction
Exponentiated Inverse Rayleigh distribution (EIR) is a life time distribution used in reliability estimation and statistical quality control techniques. it's a generalization of inverse Rayleigh distribution that developed by Nadarajah and Kotz (1). they suggested a method of generating new exponential type distribution by using reliability function:
Where R( ) is the reliability function of Inverse Rayleigh distribution.
The C.D.F of Exponentiated Inverse Rayleigh distribution is :
And the P.D.F of EIR distribution is:
where indicates the scale parameter and indicates the shape parameter.
Note that When the Exponentiated Inverse Rayleigh distribution (EIR) distribution turns into Inverse Rayleigh (IR) distribution.
As for the reliability of the stressstrength (SS.R.), it has two types (classical and modern) stressstrength, the classical stressstrength explained the life of the component and describe the ability (strength (x)) of the component to still functional when it subject to random stress (T). and interest to estimate the probability of the component's strength (X) exceed the stress (T);
And the component either fail or the system containing the component might malfunction when .
The second type is P(T<X<Z); which the current study concerned with evaluating and estimating, P(T<X<Z) represent that the strength of the component (X) should not be only greater than the component's stress(T) but also should be smaller than the other component's stress (Z).
For example, blood pressure which has two limits (systolic and diastolic) and the person's blood pressure should be between these limits (2).
In the past 45 years; a case of stressstrength reliability P(T<X<Z) considered when the cumulative functions of T and Z are known and pdf of X is unknown but its observation is available (3). The reliability estimated where X, T and Z are independent and follow a Weibull distribution with different unknown scale parameters and commonly known shape parameter, in presence of k outliers in the strength X, the moment estimator and maximum likelihood (MLE) estimators and mixture estimators of the reliability are derived (4). Then the reliability R = P(X<T<Z) was estimated using MonteCarlo simulation (MCS) for nstandby system when both of stress and strength follows a particular continuous distribution (5). And the stressstrength reliability estimated using Maximum Likelihood, Method of Moment, Least Square Method, and Weighted Least Square Method when X, T, Z are followed New WeibullPareto Distribution with unknown shape parameter (6).
Deriving The formula of the reliability of stressstrength function P(T<X<Z) under complete data for a component's strength (X) that falls in between the stresses T and Z respectively , will be as follows (7) :
,where
, where and are cumulative distribution functions .
Suppose that : ,
And suppose that T and Z are independently random stresses following Respectively,where is the scale parameter and are the shape parameters ,and (X) is random strength and independent from T and Z and follow EIR then :
Is a cum
Substituting the result of in R to get the formula of R
Point Estimation
Point estimation is a process of finding an approximate value of unknown parameters from statistics taken from one or several samples of the population . this section shall discuss two types of point estimation ( maximum likelihood estimation , Bayesian estimation ).
Let be a sample of random observations of strength taken from EIR With known scale parameter and unknown shape parameter , then the likelihood function of the sample is given by:
Where:
And let be a samples of random stresses observation taken from Respectively, that their scale parameter is known and equal ;and shape parameters are unknown, are independent from each other and from , then the likelihood functions of the samples are given by :
Where:
Where:
The maximum likelihood estimators of the parameters (
, , ,
The MLE for the (SS.R.) can be found by applying the invariance property on for the MLE of
Bayesian estimation
This part estimates the stress strength reliability using Bayesian estimation method and under consideration that it performed for complete data by using informative and noninformative priors based on Weighted Squared Error loss function (W.S.E.L.F)
The noninformative Jeffrey's prior for the shape parameter is (8):
, where is the Fisher information for the parameter
The noninformative prior for ( : , ,
The posterior distribution for is
Which is the kernel of gamma distribution Where D =
Then the complete posterior distribution Of is:
Similarly, the posterior distribution for are
Where
Where
Since The joint posterior can be found as follows:
The Weighted Squared Error loss function (9)takes the following form:
To find the Bayesian estimation ( ) for (SS.R.) based on Weighted Squared Error loss function we solved the following equation:
The expectation in the denominator using Noninformative Jeffrey's prior based on Weighted Squared Error loss function is:
=
By solving the integrations which is kernels of gamma distribution
Substituting equation above in to get the Bayesian estimation using non informative prior based on Weighted Square Error Loss Function:
The prior distribution of the parameters is gamma distribution with hyper – parameters with pdf's as follows (10) :
,
Then the posterior for will be as follows:
Since random variables The joint posterior distribution For can be found as:
the posterior distribution for each parameter is :
, where
, Where
, where
the estimated reliability function based on Weighted Square Error Loss Function when the priors are informative is defined as:
Where
By solving the integrations which is kernels of gamma distribution
Substituting equation above in to get the Bayesian estimation using informative prior based on Weighted Square Error Loss Function:
Interval Estimation
The confidence interval can be defined as a numerical range that is expected to contain the true value of an unknown parameter, As for interval estimation; it is the estimate of the unknown parameter within a certain range (period) of values with a certain probability. This probability is called the confidence level and is symbolized by the symbol (1 estimation error) .
To find the estimated confidence interval (interval estimation) of the stressstrength reliability function , the asymptotic variances of the estimated parameters must be found first; Then the interval estimation of the reliability is generated based on these variances, in this section interval estimation of the stressstrength reliability function of the model P(T<X<Z) will be found based on estimated reliability by the Maximum Likelihood method and Assuming to be for large samples.
And the formula for the asymptotic variances of reliability function will be found according to the following theorem :
Theorem : Let be statistics for the parameters such that as then the probability distribution of the difference between the statistics and the parameters is in the following form:
Where means "converges in distribution to" , and is a matrix with k*k dimension, which represent variance  covariance matrix for the estimated parameters and that 0 represents a zero vector with dimension k*1.
If is a function in terms of statistics such that all its first derivatives with respect to parameters exist ; and is a function in terms of the parameters when then the Asymptotic distribution of is :
Where represents the value of the asymptotic variance of function which can be found by the formula:
Where represent variance  covariance matrix for the statistics , And is a row vector with a dimension of 1*k and it represents the derivative of the function in terms of parameters with respect to its parameters:
.
By Applying this theorem to the stressstrength reliability function of the model P(T<X<Z), the asymptotic variance of the stressstrength reliability function will be:
Where represent variancecovariance matrix for , And is a row vector with a dimension of 1*k and it represents the derivative of the stressstrength reliability function with respect to .
To find the interval estimation of the stressstrength reliability function for the model P (T<X<Z) based on estimated reliability by the Maximum Likelihood method for large samples and it is necessary to find the variancecovariance matrix for the estimated parameters that have been estimated by Maximum Likelihood method and it can be found using C. R. lower bound note that Maximum likelihood estimators are unbiased for large samples (n→∞,m→∞,W→∞) ; As a result will be (11) :
is the inverse of and can be found as follows:
The asymptotic variance of the stressstrength reliability function of the model P(T<X<Z) based on estimated reliability by the Maximum Likelihood method will be :
Where :
The interval of the stressstrength reliability function for large samples using the reliability function estimated by the maximum likelihood method take the following form:
Then The interval estimation of the stressstrength reliability function will be :
By applying in the equation above ; the lower limit and the upper limit will be respectively as follows:
Simulation study
In this section , the simulation study was used to determine best estimator for the stress strength reliability (SS.R.) of Exponentiated Inverse Rayleigh distribution from three estimators which are ( Maximum likelihood estimator , Bayesian estimator using Noninformative Jeffrey's prior based on Weighted Square Error Loss Function ( ) , Bayesian estimation using informative gamma prior based on Weighted Square Error Loss Function ( ), and the mean square error for the estimators had been evaluated with different sample sizes (25,50,100) when ( )and for gamma priors ( ) for 1000 replicates and the simulation study calculated by (R Studio). And to compute the execution of the (SS.R.) estimator as in steps:
where u is generated from the uniform distribution.
And the estimator with smallest Mean square error (MSE) considered the best estimator under that size.
Table 1: Simulation results when
(n,m,w) 




Interval 

Lower 
APar 

(25,25,25) 
Mean 
0.184584317 
0.164417829 
0.176681509 
0.12042947 
0.24873915 

MSE 
0.001527285 
0.001048327 
0.001528945 


50,50,50) 
Mean 
0.186202242 
0.174131039 
0.182202162 
0.14083792 
0.23156656 

MSE 
0.000718744 
0.000557947 
0.000710385 


(100,100,100) 
Mean 
0.184534952 
0.178257293 
0.18254518 
0.15245753 
0.21661237 

MSE 
0.000388508 
0.000348124 
0.000390614 


(25,25,50) 
Mean 
0.185781972 
0.145876463 
0.178296983 
0.12884985 
0.24271409 

MSE 
0.001060386 
0.001939215 
0.001026011 


(25,25,100) 
Mean 
0.186850981 
0.1359154 
0.179556714 
0.13389838 
0.23980357 

MSE 
0.000767187 
0.002699673 
0.000757229 


(25,50,50) 
Mean 
0.184952622 
0.166039524 
0.179933962 
0.13945289 
0.23045235 

MSE 
0.00068232 
0.00079885 
0.00068633 


(25,100,50) 
Mean 
0.183886009 
0.178066435 
0.180117901 
0.14535414 
0.22241787 

MSE 
0.000552438 
0.000587057 
0.000578431 


(25,100,100) 
Mean 
0.183999333 
0.16549432 
0.180461586 
0.15163552 
0.21636314 

MSE 
0.000365037 
0.000689656 
0.000399329 


(50,25,25) 
Mean 
0.185441981 
0.169539427 
0.178535151 
0.12138310 
0.24950086 

MSE 
0.001311157 
0.00145187 
0.001322262 


(50,25,50) 
Mean 
0.186606875 
0.152011108 
0.180136913 
0.12978291 
0.24343083 

MSE 
0.001025344 
0.001494132 
0.001049792 


(50,25,100) 
Mean 
0.186869292 
0.1420354 
0.180626489 
0.13403299 
0.23970558 

MSE 
0.000860419 
0.002159555 
0.000833483 


(50,100,50) 
Mean 
0.185695982 
0.187800412 
0.182942888 
0.14732410 
0.22406786 

MSE 
0.000486642 
0.000552427 
0.000489908 


(50,100,100) 
Mean 
0.184454042 
0.174602358 
0.181948287 
0.15228087 
0.21662720 

MSE 
0.00034235 
0.000386029 
0.00034768 


(100,25,25) 
Mean 
0.184187804 
0.169683182 
0.177817778 
0.12017696 
0.24819864 

MSE 
0.001375285 
0.001444861 
0.00137899 


(100,25,50)

Mean 
0.186530954 
0.153787513 
0.180581522 
0.12976114 
0.24330076 

MSE 
0.001015693 
0.001376026 
0.001037458 


(100,25,100) 
Mean 
0.18677032 
0.144365563 
0.181045696 
0.13399227 
0.23954836 

MSE 
0.000828736 
0.001958911 
0.000835621 


(100,50,50) 
Mean 
0.185000338 
0.175929686 
0.181539631 
0.13970387 
0.23029681 

MSE 
0.000548374 
0.000579833 
0.000549518 


(100,50,100) 
Mean 
0.186065711 
0.165842967 
0.182830893 
0.14588510 
0.22624632 

MSE 
0.000481623 
0.000653763 
0.000488444 






The experiment was also applied on anther values and it showed the same results.