## 1 Introduction  
The inference of stress-strength reliability in statistics is an important topic of interest. It has many applications in practical areas. In the stress-strength modeling, \(R=P(Y<X)\) is a measure of component reliability. If \(X\leq Y\), the component fails or the component may malfunction, where \(X\) is subject to \(Y\). \(R\) can also be considered in electrical and electronic systems. Many authors have studied its properties for many statistical models including double exponential, Weibull, generalized Pareto and Lindley distribution (see [1; 2; 3; 4]). Classical and Bayesian estimation of reliability in a multicomponent stress-strength model under a general class of inverse exponentiated distributions was researched by [PERSON]. [5]. Ref. [6] studied classical and Bayesian estimation of reliability in multicomponent stress-strength model under Weibull distribution. Furthermore, Refs. [7; 8; 9] also considered the problem of the stress-strength reliability.  
The inverse Weibull (IW) distribution has attracted much attention recently. If \(T\) denotes the random variable from Weibull model, we define \(X\) as follows \(X=1/T\), then the random variable \(X\) is said to have inverse Weibull distribution. It is a lifetime probability distribution that can be used in the reliability engineering discipline. The inverse Weibull distribution has the ability to model failure rates, which is quite common in reliability and biological studies. The inverse Weibull model was referred to with many different names like \"Frechet-type\" ([10]) and \"Complementary Weibull\" ([11]). [PERSON]. [11] discussed a graphical plotting technique to settle the suitability of the model. [PERSON]. [12] presented the IW distribution for modeling reliability data, this model was further discussed by researching the failures of mechanical components subject to degradation. Ref. [13] proposed a discrete inverse Weibull distribution and its parameters were estimated. The mixture model of two IW distributions and its identifiability properties were studied by [14]. For the theoretical analysis of IW distribution, we can refer to [15]. Ref. [16] proposed the generalized IW distribution and several properties of this model. For more details on the inverse Weibull distribution, see [17].  
In this paper, we focus on the estimation of the stress-strength reliability \(R=P(Y<X)\), where \(X\) and \(Y\) follow the inverse Weibull distribution. As far as we know, this model has not been previously studied, although, we believe it plays an important role in reliability analysis.  
We obtain the maximum likelihood estimator (MLE), approximate maximum likelihood estimator (AMLE) and the asymptotic distribution of the estimator. The asymptotic distribution is used to construct an asymptotic confidence interval. We also present two bootstrap confidence intervals of \(R\). By using Gibbs sampling technique, we obtain Bayes estimator of \(R\) and its corresponding credible interval. Finally, we present a real data example to illustrate the performance of different methods.  
The layout of this paper is organized as follows: in Section 2, we introduce the distribution of the inverse Weibull. In Section 3, we obtain the MLE of \(R\). In Section 4, we derive the estimator of \(R\) by approximating maximum likelihood equations. Different confidence intervals are presented in Section 5. In Section 6, Bayesian solutions are introduced. In Section 7, we compare different proposed methods using Monte Carlo simulation. A numerical example is also provided. Finally, in Section 8, we conclude the paper.