Point and Interval Estimation of Stress-Strength Model for Exponentiated Inverse Rayleigh distribution

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 nun-informative 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 nun-informative 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.

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 stress-strength (S-S.R.), it has two types (classical and modern) stress-strength, the classical stress-strength 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 stress-strength 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 Monte-Carlo simulation (MCS) for n-standby system when both of stress and strength follows a particular continuous distribution (5). And the stress-strength reliability estimated using Maximum Likelihood, Method of Moment, Least Square Method, and Weighted Least Square Method when X, T, Z are followed New Weibull-Pareto Distribution with unknown shape parameter (6).

Reliability formula

Deriving The formula of the reliability of stress-strength  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 ).

Maximum likelihood 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 (S-S.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 non-informative priors based on Weighted Squared Error loss function (W.S.E.L.F)

1. Bayesian estimation using Non-informative Jeffrey's prior based on Weighted Squared Error loss function

 The non-informative Jeffrey's prior for the shape parameter  is (8):

     , where  is the Fisher information for the parameter

The non-informative 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 (S-S.R.) based on Weighted Squared Error loss function we solved the following equation:

The expectation in the denominator using   Non-informative 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:

1. Bayesian estimation using informative priors  based on Weighted Squared 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 stress-strength 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 stress-strength 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 stress-strength reliability function of the model P(T<X<Z), the asymptotic variance of the stress-strength reliability function  will be:

Where  represent variance-covariance matrix for  , And  is a row vector with a dimension of 1*k and it represents the derivative of the stress-strength reliability function with respect to  .

To find the interval estimation of the stress-strength 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 variance-covariance 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 stress-strength 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 stress-strength 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 stress-strength 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 (S-S.R.) of Exponentiated Inverse Rayleigh distribution from three estimators which are ( Maximum likelihood estimator  , Bayesian estimator using Non-informative 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 (S-S.R.) estimator as in steps:

1. Generate random values for by the inverse function according to:

    where u is generated from the uniform distribution.

1. Calculate the mean of the estimators by
2. Evaluate the mean square error (MSE) for the estimators

And the estimator with smallest Mean square error (MSE) considered the best estimator under that size.

Table 1: Simulation results when  

Discussion

The results of simulation showed that the reliability value is  , 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 nun-informative 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.

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

Conclusion