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

Abstract


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 nstandby 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) 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)

A. 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

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:

B. Bayesian estimation using informative priors based on Weighted Squared Error loss function
The prior distribution of the parameters is gamma distribution with hyperparameters 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 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: ) 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: 1000 replicates and the simulation study calculated by (R Studio ).And to compute the execution of the (S-S.R.) estimator as in steps: A. Generate random values for by the inverse function according to: