Classical and Bayesian Estimation of Stress-Strength Reliability from Generalized Inverted Exponential Distribution based on Upper Records

This paper deals with the problem of classical and Bayesian estimation of stress-strength reliability (R=P(X<Y)) based on upper record values from generalized inverted exponential distribution (GIED). Hassan {et al.} (2018) discussed the maximum likelihood estimator (MLE) and Bayes estimator of $R$ by considering that the scale parameter to be known for defined distribution while we consider the case when all the parameters of GIED are unknown. In the classical approach, we have discussed MLE and uniformly minimum variance estimator (UMVUE). In Bayesian approach, we have considered the Bays estimator of R by considering the squared error loss function. Further, based on upper records, we have considered the Asymptotic confidence interval based on MLE, Bayesian credible interval and bootstrap confidence interval for $R$. Finally, Monte Carlo simulations and real data applications are being carried out for comparing the performances of the estimators of R.


Introduction
Let X and Y be two independent random variables and represents the probability that X does not exceeds Y.In terms of stress-strength reliability, X denotes the stress applied to a device having strength Y.For proper functionning of the device, the strength of the device Y must exceeds to stress X. Stress-strength reliability model appears in many practical situations like, in medical sciences, structural engineering, natural phenomena like a flood, earthquakes, etc.In medical science, let X and Y stand for the effect of the control treatment and the new treatment respectively.When new treatment applied over a control treatment then the quantity express the effectiveness of the new treatment compared with the control treatment.In structural engineering, while building a bridge, X and Y shows the stress (load on the bridge) and strength (capacity) of the bridge respectively.Bridge survive only if the strength of the bridge is greater than the stress applied on it.In such a situation shows the survival probability of the bridge.In statistical literature, estimation of has been widely studied under the assumption that X and Y are independent random variables belonging to the same family of distributions.Several researchers have deduced the estimators of R. Awad et al. (1981)  when X and Y have a bivariate exponential distribution.For an extensive and lucid literature review regarding estimation and application of the stress-strength reliability, readers are referred to Johnson (1988) and Kotz et al. (2003).
In stress-strength reliability, the problem of estimating R is carried out for different data sets such as complete, censored and so on.However, many situations are appeared in practical life where observations are more extreme than the current extreme values.A natural example is industrial stress testing where only items are destroyed which are more weaker than other observed failed items, see Ahmadi and Arghami (2003 a, b).This type of data is called "Record Data" or "Record Values".Chandler (1952) developed the mathematical theory of record values and discussed its basic properties.Consequently, many researchers considered record data for their work of interest.A detailed treatment with extensive references are provided by Ahsanullah (1995)  be an infinite sequence of identically and independently distributed (iid) random variables.An observation for every j i  .We shall assume that j X occurs at time j, then the record time sequence is defined as

,
. The joint probability density function (pdf) of first n upper records is given by The exponential distribution played a very important role in reliability theory.Several researchers have considered its generalized and inverse form namely generalized exponential distribution (Gupta and Kundu, 1999) and inverted exponential distribution (Dey, 2007).Abouammoh and Alshingiti (2009) and its pdf is given by- are discussed in Section 4 and 5 respectively.In Section 6, Monte Carlo simulations are carried out to check the efficiency of aforesaid estimators of R. A real data example is presented in Section 7 for the purpose of illustration.

Maximum Likelihood Estimation
In this Section, we consider the problem of estimating based on upper record values from GIED.Here, we obtained MLE of R by assuming that all the parameters of GIED are unknown.
Let X~GIED(λ,α) and Y~GIED(λ,β) be independent random variables.Let be the stress-strength reliability and it can be seen that

=
. Then the likelihood function based on these upper records values is given by where f, F and g, G represent the pdf, cdf of GIED(λ,α) and GIED(λ,β) respectively.Also ) ( 1 ) ( Putting the values of f, F, g and G in (2.1), the likelihood function can be written as From the above expression, it is very difficult to find the exact distribution of ML R ˆ.Therefore, we use another method to construct the CI of R namely asymptotic distribution of MLE and parametric bootstrap method.

3.
CI for R In this Section, we discussed two different methods to obtain the confidence interval for R namely the method of asymptotic normality and parametric bootstrap method.

Asymptotic CI
Here, we deduced the expression for asymptotic CI when  is unknown by using the multivariate delta method (see Wasserman, 2003, p.99).To compute the asymptotic distribution of ML R ˆ we need to find an asymptotic variance of ML R ˆ.However, it is well known that the asymptotic variance is the inverse of the Fisher information matrix which is given as: However, to find the expectation of the above defined terms are very complicated so, under some regularity conditions, we have used observed information matrix define as; Using the multivariate delta method (Soliman et al. 2013) to find the approximate estimate of the asymptotic variance of ML R ˆ, R  ˆ as follows: , where The asymptotic distribution of N  and it can also be written as :

Parametric bootstrap CI
Here, we have constructed a bootstrap CI for R by using a parametric percentile bootstrap method (Efron. 1982).The following algorithm is used to generate the parametric bootstrap estimates of R .
Step-1.Simulate a random sample from Uniform (0,1).Using this simulated value compute random sample for Step-2.Generate an independent parametric bootstrap sample using Step-4.Repeat the step-2 and step-3 N times to obtained the parametric bootstrap estimates for a given x .The approximate is an unbiased estimator for R .From equation (4.1) and (4.2), it can be seen that .where . Before solving the above integral, it is required to find the conditional distribution of .By some algebraic simplification, we get.and simplifying simultaneously, we get Here, it can be seen that the integral value depends on the value of

Case-II: when
is the Gauss hyper-geometric function.

Bayesian Estimation
This section presents a study of Bayesian estimation of R .We know that in Bayesian inference, we need some prior distributions for unknown parameters of parent distribution.we have considered gamma distributions as a prior distribution for  ,  and  with pdf given as 0 , , 0 , ) ( In order to find the posterior distribution of R , we need to obtain joint posterior distribution of  ,  and  .Using Bayes' theorem (Wasserman, 2003).From (5.1), (5.2), (5.3) and (5.4) the joint posterior distribution of  ,  and  is given by From the above equation, the joint posterior is very complicated and hence it is not possible to obtain a closed form or explicit expression for Bayes estimator of R .Therefore, to simulate the samples from the posterior distribution, we have considered the MCMC approach to find a sample based inferences.Solimon et al. (2013) considered the MCMC approach for stress-strength reliability model for the complete sample using modified weibull distribution.

Simulation Study
carried out to compare the performances of MLE and UMVUE and observed that UMVUE perform better than MLE in the sense of MSE.Moreovr, it is noticed that asymptotic CI provided the smallest average width of CI for different sample sizes as compare to MLE and bootstrap CIs.
introduced the GIED in reliability estimation.Ghitany et al. (2013) discussed the likelihood estimation for a general class of inverted exponential distribution based on complete and censored samples.Further, several researchers considered the estimation of the parameters of GIED based on complete, censored samples and record values.For example: Dey and Dey (2014 a, 2014 b), Dey and Pradhan (2014), Dube et al. (2016), Dey et al. (2016), Panahi (2017) and Gunasekera (2018) and so on.The cumulative distribution function (cdf) of GIED(λ,α) is written as 0


for the Lomax distribution based on upper record values.In this paper, Mahmoud et al. (2016) described the MLE of stress-strength reliability in two cases, when all the parameters are unknown and when scale parameter is common and known.Amin (2017) has discussed the estimation of stress-strength reliability based on upper record values for Kumaraswamy Exponential distribution.Recently, Dhanya and Jeevavand (2018) have considered the Bayesian estimation of squared error loss function and linex loss function and MLE of stressstrength reliability for power function distribution with different shape and same scale parameter based on records.Inference for the two-parameter bathtub-shaped distribution based on record data has been considered by Raqab et al. (2018).Rasethuntsa and Nadar (2018) have discussed the MLE, its asymptotic distribution and Bayes estimator under symmetric squared error loss function of stress-strength reliability in a multi-component system with nonidentical component strengths from a family of Kumaraswamy generalized distribution based on upper records.Khan and Khatoon (2019) have obtained MLE, UMVUE and Bayesian estimaor of stress strength reliability for exponential distribution based on generalized order statistics.In this paper, we have derived the classical (MLE and UMVUE) and Bayesian estimators of stress-strength reliability based on upper record values from GIED by taking common scale parameter and different shape parameter.The rest of the paper is organized as follows: In Section 2, the MLE of 3 provided the asymptotic confidence interval and percentile bootstrap interval of stress-strength reliability.UMVUE and Bayesian inference of 2)Thus the log-likelihood function of the above expression is given by ] the MLE of λ is the solution of the non-linear equation MLE of R based on upper records becomes with mean a and variance b and the symbol ⎯→ ⎯ d denotes the convergence in distribution.Based on this asymptotic distribution, a of  ,  and  say ML of  ,  and  .then using these values, calculate ML R ˆ.Step-3.Calculate the maximum likelihood estimate of ML

R=
In this Section, we have derived the expression for UMVUE of R based on upper record values when the observations follow GIED with a common and known parameter  .The technique used for obtaining the UMVUE of R is similar to Khan and Arshad (be the induced upper records from this sequence of random variables, then the joint pdf of n upper records is given by

= 2 )
be the induced upper records from this sequence of random variables, then the joint pdf of m upper records is given by To obtain the UMVUE of R , we need an unbiased and complete sufficient statistics of  and  .Let ) sufficient statistics for  and  and have gamma distribution with parameters the application of Lehman -Scheff e  theorem (Lehmann and Casella (1998)), the UMVUE of R is given by

1
beta function with upper limit  .Using the relation of incomplete beta and Gauss hyper-geometric function ) 4.6) can also be written as


are hyper-parameters chosen to reflect prior knowledge about  ,  and  .From (2.2), the likelihood function can be re-written as ) considered the MLE of and Arnold et al. (1998).
We are interested to obtain the MLE of R based on upper records.However, to find the MLE of R , it is required to obtain the MLE of  and  say,