Stress-strength Model Research Articles

Generalized Gompertz - Generalized Gompertz Distribution

In the present paper, we introduce a new generated family of continuous distributions based on generalized Gompertz distribution. Then generalized Gompertz - generalized Gompertz distribution is proposed as a special case of this new family. The probability density, cumulative distribution, reliability and hazard rate functions are introduced. Additionally, the most essential statistical properties of this new distribution such as the mean, variance, coefficient of skewness, coefficient of kurtosis, characteristic function, quantiel, median, Shannon and relative entropies along with stress strength model are obtained.

The mean remaining strength of parallel systems in a stress-strength model based on exponential distribution

The mean remaining strength of any coherent system is one of the important characteristics in stress-strength reliability. It shows that the system on the average how long can be safe under the stress. In this paper, we consider the mean remaining strength of the parallel systems in the stress-strength model. We assume that the strength and stress components constitute parallel systems separately. The mean remaining strength and its estimations are obtained when the all components follow the exponential distribution. The likelihood ratio order between the remaining strength of the parallel systems is presented for two-component case. The simulation study is performed to compare the derived estimates and their results are presented.

Bayesian inference on reliability in a multicomponent stress-strength bathtub-shaped model based on record values

In the literature, there are a well-developed estimation techniques for the reliability assessment in multicomponent stress-strength models when the information about all the experimental units are available. However, in real applications, only observations that exceed (or fall below) the current value may be recorded. In this paper, assuming that the components of the system follow bathtub-shaped distribution, we investigate Bayesian estimation of the reliability of a multicomponent stress-strength system when the available data are reported in terms of record values. Considering squared error, linex and entropy loss functions, various Bayes estimates of the reliability are derived. Because there are not closed forms for the Bayes estimates, we will use Lindley’s method to calculate the approximate Bayes estimates. Further, for comparison purposes, the maximum likelihood estimate of the reliability parameter is obtained. Finally, simulation studies are conducted in order to evaluate the performances of the proposed procedures and analysis of real data sets is provided.

English

The stress-strength model is proposed based on the m-generalized order statistics and the corresponding concomitant. For the dependency between m-generalized order statistics and its concomitant, a bivariate copula expansion is considered and the stressstrength model is obtained for two special cases of order statistics and upper record values. In the particular case of copula function, the generalized Farlie-Gumbel-Morgenstern bivariate distribution function is considered with proportional reversed hazard functions as marginal functions. Based on the order statistics and record values, two estimators of stress-strength are presented using a procedure similar to the inference functions for margins. Finally, a simulation study is carried out which shows the good performance of the proposed estimators for a finite sample.

Reliability estimation in a multicomponent stress–strength model under generalized half-normal distribution based on progressive type-II censoring

ABSTRACTIn this paper, based on progressively Type-II censored samples, the problem of estimation of multicomponent stress–strength reliability under generalized half-normal (GHN) distribution is considered. The reliability of a k-component stress-strength system is estimated when both stress and strength variates are assumed to have a GHN distribution with various cases of same and different shape and scale parameters. Different methods such as the maximum likelihood estimates (MLEs) and Bayes estimation are discussed. The expectation maximization algorithm and approximate maximum likelihood methods are proposed to compute the MLE of reliability. The Lindley's approximation method, as well as Metropolis–Hastings algorithm, are applied to compute Bayes estimates. The performance of the proposed procedures is also demonstrated via a Monte Carlo simulation study and an illustrative example.

The Fréchet Topp Leone-G Family of Distributions: Properties, Characterizations and Applications

A new family of continuous distributions which ensure model flexiblity, is introduced based on the Frechet distribution and Topp Leone-G family. Two special sub-models of the new family are discussed. We provide some distributional properties of this family in the general setting such as the series expansions of density, moments, generating function, stress strength model, Renyi and Shannon entropies, probability weighted moments and order statistics. Certain characterizations of the proposed family are presented. The maximum likelihood estimates and the observed information matrix are obtained for the model parameters. We assess the performance of the maximum likelihood estimators by means of a graphical simulation study. The potentiality of the new class is shown via two applications to real data sets.

The Topp Leone Generalized Inverted Kumaraswamy Distribution: Properties and Applications

We propose a new four parameters continuous model called the Topp Leone generalized inverted Kumaraswamy distribution which extends the generalized inverted Kumaraswamy distribution. Basic mathematical properties of the new distribution are invstigated such as; quantile function, raw and incomplete moments, generating functions, probability weighted moments, order statistics, Rényi entropy, stochastic ordering and stress strength model. The method of maximum likelihood is used to estimate the model parameters of the new distribution. A Monte Carlo simulation is conducted to examine the behavior of the parameter estimates. The applicability of the new model is demonstrated by means of two real applications.

Estimation of reliability of multicomponent stress-strength inverted exponentiated Rayleigh model

ABSTRACTIn this paper, we consider the estimation of the multicomponent reliability by assuming the inverted exponentiated Rayleigh distribution. Both stress and strength are assumed to have an inverted exponentiated Rayleigh distribution with common scale parameter. The random variable representing the stress experienced by the system and representing the strength of system available to overcome the stress. The system works flawlessly only if at least out of strength variables exceed the random stress. The multicomponent reliability of the system is given by . We estimate by using frequentist and Bayesian approaches. Bayes estimates of have been obtained by using Markov Chain Monte Carlo methods since joint posteriors of the parameters does not have the explicit forms. We also construct asymptotic and highest probability density credible intervals for . The behavior of the proposed estimators is studied on the basis of estimated risks through Monte Carlo simulations. Finally, a data set is analyzed for illustrative purposes.Abbreviations: PDF: Probability Density Function; CDF: Cumulative Density Function; IER: Inverted Exponentiated Rayleigh; MLE: Maximum likelihood estimators; HPD: Highest Posterior Density; UBT: Upside down Bathtub; HNC: Head and Neck Cancer data; MCMC: Monte-Carlo Markov Chains; CP: Coverage Probability; KS: Kolmogorov - Smirnov.

The Stress-Strength Model: R=P(Y<X) Gamma Case by Mathematica

In this paper, we considered the estimation of R=P(Y<X), dubbed as Stress-Strength Model (SSM), in a framework of gamma distribution with unknown scale and shape parameters. This is not an easy computing problem in conventional mathematics. For such problems, a very important option is using a Mathematica software, which has been employed to obtain solutions to this problem.

The Exponentiated Generalized Topp Leone-G Family of Distributions: Properties and Applications

In this paper, we propose a new class of continuous distributions called the exponentiated generalized Topp Leone-G family that extends the Topp Leone-G family introduced by Al-Shomrani et al. (2016). We derive explicit expressions for certain mathematical properties of the new family such as; ordinary and incomplete moments, generating functions, reliability analysis, Lorenz and Bonferroni curves, Rényi entropy, stress strength model, moment of residual and reversed residual life, order statistics and extreme values. We discuss the maximum likelihood estimates and the observed information matrix for the model parameters. Two real data sets are used to illustrate the flexibility of the new family.

EXPONENTIATED HALF-LOGISTIC LOMAX DISTRIBUTION WITH PROPERTIES AND APPLICATION

A new three-parameter continuous model called the exponentiated half-logistic Lomax distribution is introduced in this paper. Basic mathematical properties for the proposed model were investigated which include raw and incomplete moments, skewness, kurtosis, generating functions, Rényi entropy, Lorenz, Bonferroni and Zenga curves, probability weighted moment, stress strength model, order statistics, and record statistics. The model parameters were estimated by using the maximum likelihood criterion and the behaviours of these estimates were examined by conducting a simulation study. The applicability of the new model is illustrated by applying it on a real data set.

On estimating the reliability in a multicomponent stress-strength model based on Chen distribution

In this article, we obtain point and interval estimates of multicomponent stress-strength reliability model of an s-out-of-j system using classical and Bayesian approaches by assuming both stress and strength variables follow a Chen distribution with a common shape parameter which may be known or unknown. The uniformly minimum variance unbiased estimator of reliability is obtained analytically when the common parameter is known. The behavior of proposed reliability estimates is studied using the estimated risks through Monte Carlo simulations and comments are obtained. Finally, a data set is analyzed for illustrative purposes.

Bayesian and classical estimation of reliability in a multicomponent stress-strength model under adaptive hybrid progressive censored data

The statistical inference of multicomponent stress-strength reliability under the adaptive Type-II hybrid progressive censored samples for the Weibull distribution is considered. It is assumed that both stress and strength are two Weibull independent random variables. We study the problem in three cases. First assuming that the stress and strength have the same shape parameter and different scale parameters, the maximum likelihood estimation (MLE), approximate maximum likelihood estimation (AMLE) and two Bayes approximations, due to the lack of explicit forms, are derived. Also, the asymptotic confidence intervals, two bootstrap confidence intervals and highest posterior density (HPD) credible intervals are obtained. In the second case, when the shape parameter is known, MLE, exact Bayes estimation, uniformly minimum variance unbiased estimator (UMVUE) and different confidence intervals (asymptotic and HPD) are studied. Finally, assuming that the stress and strength have the different shape and scale parameters, ML, AML and Bayesian estimations on multicomponent reliability have been considered. The performances of different methods are compared using the Monte Carlo simulations and for illustrative aims, one data set is investigated.

Interval estimation for the generalized inverted exponential distribution under progressive first failure censoring

ABSTRACTIn this paper, we consider the inferential procedures for the generalized inverted exponential distribution under progressive first failure censoring. The exact confidence interval for the scale parameter is derived. The generalized confidence intervals (GCIs) for the shape parameter and some commonly used reliability metrics such as the quantile and the reliability function are explored. Then the proposed procedure is extended to the prediction interval for the future measurement. The GCIs for the reliability of the stress-strength model are discussed under both equal scale and unequal scale scenarios. Extensive simulations are used to demonstrate the performance of the proposed GCIs and prediction interval. Finally, an example is used to illustrate the proposed methods.

On Shrinkage Estimation for R(s, k) in Case of Exponentiated Pareto Distribution

This paper concerns with deriving and estimating the reliability of the multicomponent system in stress-strength model R(s,k), when the stress and strength are identical independent distribution (iid), follows two parameters Exponentiated Pareto Distribution(EPD) with the unknown shape and known scale parameters. Shrinkage estimation method including Maximum likelihood estimator (MLE), has been considered. Comparisons among the proposed estimators were made depending on simulation based on mean squared error (MSE) criteria.

Some Results on the Stress–Strength Reliability under the Distortion Functions

In this paper, the system reliability in a stress–strength model, when the distributions of the stress and the strength random variables are replaced by two distorted distributions, is considered. The conditions to increase the reliability under several distortion functions are studied. Furthermore, the changes in the residual stress–strength reliability, the mean remaining strength and the number of working components of a system are investigated when the original distributions have been affected by distortion functions. Some of the previous results in the literatures can be achieved as special cases of our results.

Reliability Estimation in Multicomponent Stress-strength Model based on Generalized Pareto Distribution

The paper deals with the estimation of multicomponent system reliability where the system has k components with their strengths X1, X2, … Xk being independently and identically distributed random variables and each component is experiencing a random stress Y. The s-out-of-k system is said to function if atleast s out of k (1 ≤ s ≤ k) strength variables exceed the random stress. The reliability of such a system is derived when both strength and stress variables follow generalized Pareto distribution. The system reliability is estimated using maximum likelihood and Bayesian approaches. The maximum likelihood estimators are derived under both simple random sampling and ranked set sampling schemes. Lindley's approximation technique is used to get approximate Bayes estimators. The reliability estimators obtained from both the methods are compared by using mean squares error criteria and real data analysis is carried out to illustrate the procedure.

The Topp Leone odd Lindley-G family of distributions : Properties and applications

We introduce a new class of continuous distributions called the Topp Leone odd Lindley-G family. Some mathematical properties of the new family such as; the raw and incomplete moments, moment generating function, stress strength model, Rényi entropy, probability weighted moments and order statistics are investigated. We discuss the maximum likelihood estimates and the observed information matrix for the model parameters. The potentiality of the new family is illustrated by means of three applications to real data sets.

A Reliability-based Approach for Performance Optimization of Service Industries: An Application to Healthcare Systems

The performance of healthcare systems has always been a significant concern of policy makers. Performance in healthcare systems is assessed by multiple criteria, and it is a complex task to optimize the performance, considering the fact that outputs are mostly interrelated and some external factors affect performance of the system. This study aims to apply the Multi-Response Optimization (MRO) method for reliability-based performance optimization of healthcare systems. In this study, we consider all types of response variables so that strength features of the system are optimized against external stresses. The proposed approach in this research is based on the stress-strength model, and stresses are assumed to be normally distributed. Also, we present the implementation results of the proposed approach to a hospital in Iran for reliability-based optimization of the bed capacity planning. The results indicate that the proposed approach is powerful in tackling the inherent complexity of healthcare systems.

Reliability estimation of stress–strength model using finite mixture distributions under progressively interval censoring

This paper considers the reliability estimation of the stress–strength model based on progressively Type-I interval censored data, where the random stress variable follows a Lindley distribution and the random strength variable follows a finite mixture of exponential distributions. The maximum likelihood estimation and 95% confidence interval estimation of the stress–strength reliability are deduced by using EM algorithm and Bootstrap sampling, respectively. The Bayesian estimation and 95% highest posterior density credible interval of the stress–strength reliability under squared error loss function are obtained by using the Metropolis–Hastings within Gibbs algorithm. To test the homogeneity of the finite mixture distributions, the D -test statistic is introduced. Then, we use the D -test statistic to test the homogeneity of a real data, compare the finite mixture exponential distributions with a single exponential distribution by using the Akaike information criterion (AIC) values, and analyze this data using the proposed methodology. Finally, Monte Carlo simulations are performed for illustrative purpose. • The stress–strength model applied to the progressively Type-I interval censoring. • The finite mixture distributions are applied to describe the strength variables. • The reliability is estimated by EM algorithm and MH within Gibbs algorithm. • The homogeneity of the mixture distribution is tested by the D-test statistic.