Stress-strength Parameter Research Articles

Extended inverse Weibull distribution with reliability application

Abstract The aim of this paper is to introduce an extension of the inverse Weibull distribution which offers a more flexible distribution for modeling lifetime data. We extend the inverse Weibull distribution by Marshall–Olkin method (MOEIW). Some statistical properties of the MOEIW are explored, such as quantiles, moments and reliability. Moreover, the estimation of the MOEIW parameters is discussed by using Maximum Likelihood Estimation method. In addition, the estimation of the stress-strength parameter is discussed. Finally, the proposed extended model is applied on real data and the results are given which illustrate the superior performance of the MOEIW distribution compared to other models.

Statistical properties and different methods of estimation of transmuted Rayleigh distribution

This article addresses the various properties and different methods of estimation of the unknown parameters of the Transmuted Rayleigh (TR) distribution from the frequentist point of view. Although, our main focus is on estimation from frequentist point of view, yet, various mathematical and statistical properties of the TR distribution (such as quantiles, moments, moment generating function, conditional moments, hazard rate, mean residual lifetime, mean past lifetime, mean deviation about mean and median, the stochastic ordering, various entropies, stress-strength parameter and order statistics) are derived. We briefly describe different frequentist methods of estimation approaches, namely, maximum likelihood estimators, moments estimators, L-moment estimators, percentile based estimators, least squares estimators, method of maximum product of spacings, method of Cram\'er-von-Mises, methods of Anderson-Darling and right-tail Anderson-Darling and compare them using extensive numerical simulations. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples. Finally, the potentiality of the model is analyzed by means of two real data sets which is further illustrated by obtaining bias and standard error of the estimates and the bootstrap percentile confidence intervals using bootstrap resampling.

Bayesian inference for the Topp–Leone distribution based on lower k-record values

In this paper, the lower k-record values coming from the Topp–Leone distribution are used to construct Bayesian point and interval estimators for the shape parameter, the survival function and the reversed hazard rate function. The Bayes estimators are obtained under symmetric and asymmetric loss functions. We study the problem of predicting future k-records, and reconstructing past unobserved k-records. Bayesian estimation of the stress-strength parameter is also discussed. Finally, a simulation study and a real data example are presented for the purpose of illustration and comparison of the suggested point and interval estimators.

A complementary generalized linear failure rate-geometric distribution

ABSTRACTIn this article, we shall attempt to introduce a new class of lifetime distributions, which enfolds several known distributions such as the generalized linear failure rate distribution and covers both positive as well as negative skewed data. This new four-parameter distribution allows for flexible hazard rate behavior. Indeed, the hazard rate function here can be increasing, decreasing, bathtub-shaped, or upside-down bathtub-shaped. We shall first study some basic distributional properties of the new model such as the cumulative distribution function, the density of the order statistics, their moments, and Rényi entropy. Estimation of the stress-strength parameter as an important reliability property is also studied. The maximum likelihood estimation procedure for complete and censored data and Bayesian method are used for estimating the parameters involved. Finally, application of the new model to three real datasets is illustrated to show the flexibility and potential of the new model compared to rival models.

Alpha power Weibull distribution: Properties and applications

ABSTRACTIn this paper, a new lifetime distribution is defined and studied. We refer to the new distribution as alpha power Weibull distribution. The importance of the new distribution comes from its ability to model monotone and non monotone failure rate functions, which are quite common in reliability studies. Various properties of the proposed distribution are obtained including moments, quantiles, entropy, order statistics, mean residual life function, and stress-strength parameter. The maximum likelihood estimation method is used to estimate the parameters. Two real data sets are used to illustrate the importance of the proposed distribution.

Morgenstern type bivariate Lindley Distribution

In this paper, a bivariate Lindley distribution using Morgenstern approach is proposed which can be used for modeling bivariate life time data. Some characteristics of the distribution like moment generating function, joint moments, Pearson correlation coefficient, survival function, hazard rate function, mean residual life function, vitality function and stress-strength parameter R=Pr(Y<X), are derived. The conditions under which the proposed distribution is an increasing (decreasing) failure rate distribution and positive (negative) quadrant dependent is discussed. Also, the method of estimating model parameters and stress-strength parameter by maximum likelihood is elucidated. Numerical illustration using simulated data is carried out to access the estimates in terms of mean squared error and relative absolute bias.

The truncated distribution of the range for a Wiener process: Application to the stock price

In this paper, the exact truncated distribution of the stock price (truncated distributionfor the range of a Wiener process) is available among the established results in the field of mathematics (ProbabilityDistributions). Various statistical properties of the distribution are derived including reliability properties,moments, stress-strength parameter, order statistics, Bonferroni curve, Lorenz curve and Gini’s index. Areal data set is analyzed to clarify the effectiveness of this distribution.

Inferences on Stress-Strength Reliability from Weighted Lindley Distributions

This article deals with the estimation of the stress-strength parameter R = P(Y < X), when X and Y are two independent weighted Lindley random variables with a common shape parameter. The MLEs can be obtained by maximizing the profile log-likelihood function in one dimension. The asymptotic distribution of the MLEs are also obtained, and they have been used to construct the asymptotic confidence interval of R. Bootstrap confidence intervals are also proposed. Monte Carlo simulations are performed to verify the effectiveness of the different estimation methods, and data analysis has been performed for illustrative purposes.

A new lifetime model with decreasing, increasing, bathtub-shaped, and upside-down bathtub-shaped hazard rate function

A new three-parameter distribution with decreasing, increasing, bathtub-shaped and upside-down bathtub-shaped hazard rate function is proposed. The new distribution encompasses some previously known distributions as special cases. Basic mathematical properties of the new distribution (including the moment-generating function, moments, order statistics properties, Rényi entropy and stress–strength parameter) are derived. Its parameters are estimated by the method of maximum likelihood. An application is illustrated using a real data set.

Estimation of stress-strength reliability using record ranked set sampling scheme from the exponential distribution

In this paper, point and interval estimation of stress-strength reliability based on upper record ranked set sampling (RRSS) from one-parameter exponential distribution are considered. Maximum likelihood estimator (MLE) as well as the uniformly minimum variance unbiased estimator (UMVUE) of stress-strength parameter are derived and their performance are studied. Also, some confidence intervals for stress-strength parameter based on upper RRSS are constructed and then compared on the basis of a simulation study. Finally, a data set has been analyzed for illustrative purposes.

On an Extension of the Exponentiated Weibull Distribution

In this article, the exponentiated Weibull distribution is extended by the Marshall-Olkin family. Our new four-parameter family has a hazard rate function with various desired shapes depending on the choice of its parameters and, thus, it is very flexible in data modeling. It also contains two mixed distributions with applications to series and parallel systems in reliability and also contains several previously known lifetime distributions. We shall study some basic distributional properties of the new distribution. Some closed forms are derived for its moment generating function and moments as well as moments of its order statistics. The model parameters are estimated by the maximum likelihood method. The stress–strength parameter and its estimation are also investigated. Finally, an application of the new model is illustrated using two real datasets.

Bayesian and Non–Bayesian Estimation for Two Generalized Exponential Populations Under Joint Type II Censored Scheme

In this paper, Bayesian and non-Bayesian estimators have been obtained for two generalized exponential populations under joint type II censored scheme, which generalize results of Balakrishnan and Rasouli (2008) and Shafay et al. (2013) . The maximum likelihood estimators (MLEs) of the parameters and Bayes estimators have been developed under squared error loss function as well as under LINEX loss function. Moreover, approximate confidence region are also discussed and compared with two Bootstrap confidence regions. Also the MLE and three confidence intervals for the stress–strength parameter are explored. A numerical illustration for these new results is given.

Inferences on the Competing Risk Reliability Problem for Exponential Distribution Based on Fuzzy Data

The problem of estimating the reliability parameter originated in the context of reliability where X represents the strength subjected to a stress Y. But traditionally it is assumed that the available data from the stress and strength populations are performed in exact numbers. However, some collected data might be imprecise, and are represented in the form of fuzzy numbers. In this paper, we consider the estimation of the stress-strength parameter R, when X and Y are statistically independent exponential random variables, and the obtained data from both distributions are reported in the form of fuzzy numbers. We consider the classical and Bayesian approaches. In the Bayesian setting, we obtain the estimate of R by using the approximation forms of Lindley, and Tierney & Kadane, as well as a Markov Chain Monte Carlo method under the assumption of statistically independent gamma priors. The estimation procedures are discussed in detail, and compared via Monte Carlo simulations in terms of their average values and mean squared errors.

A Resilience Parameter Model Generated by a Compound Distribution

In this paper, we shall attempt to extend the generalized exponential geometric distribution of Silva et al. [1]. The new four-parameter distribution also generalizes the Weibull-geometric distribution of Barreto-Souza et al. [2], exponentiated Weibull, and several other lifetime distributions as special cases. A useful characteristic of the new distribution is that its failure rate function can have different shapes. We first study certain basic distributional properties of the new distribution and provide closed form expressions for its moment generating function and moments. General expressions are also obtained for the order statistics densities and stress-strength parameter. Our findings happen to enfold several known results as special cases. The model parameters are estimated by the maximum likelihood method and the Fisher information matrix is discussed. Finally, the model is applied to a real data set and its advantage over some rival models is illustrated.

Estimation of R=P[Y

Surles and Padgett [Inference for reliability and stress–strength for a scaled Burr type X distribution. Lifetime Data Anal. 2001;7:187–200] introduced a two-parameter Burr-type X distribution, which can be described as a generalized Rayleigh distribution. In this paper, we consider the estimation of the stress–strength parameter R=P[Y<X], when X and Y are both three-parameter generalized Rayleigh distributions with the same scale and locations parameters but different shape parameters. It is assumed that they are independently distributed. It is observed that the maximum-likelihood estimators (MLEs) do not exist, and we propose a modified MLE of R. We obtain the asymptotic distribution of the modified MLE of R, and it can be used to construct the asymptotic confidence interval of R. We also propose the Bayes estimate of R and the construction of the associated credible interval based on importance sampling technique. Analysis of two real data sets, (i) simulated and (ii) real, have been performed for illustrative purposes.

Pareto Poisson–Lindley distribution with applications

A new lifetime distribution is introduced based on compounding Pareto and Poisson–Lindley distributions. Several statistical properties of the distribution are established, including behavior of the probability density function and the failure rate function, heavy- and long-right tailedness, moments, the Laplace transform, quantiles, order statistics, moments of residual lifetime, conditional moments, conditional moment generating function, stress–strength parameter, Rényi entropy and Song's measure. We get maximum-likelihood estimators of the distribution parameters and investigate the asymptotic distribution of the estimators via Fisher's information matrix. Applications of the distribution using three real data sets are presented and it is shown that the distribution fits better than other related distributions in practical uses.

Reliability estimation in stress–strength models: an MCMC approach

In this paper, we consider the estimation of the stress–strength parameter R=P(Y<X) when X and Y are independent and both are modified Weibull distributions with the common two shape parameters but different scale parameters. The Markov Chain Monte Carlo sampling method is used for posterior inference of the reliability of the stress–strength model. The maximum-likelihood estimator of R and its asymptotic distribution are obtained. Based on the asymptotic distribution, the confidence interval of R can be obtained using the delta method. We also propose a bootstrap confidence interval of R. The Bayesian estimators with balanced loss function, using informative and non-informative priors, are derived. Different methods and the corresponding confidence intervals are compared using Monte Carlo simulations.

Estimation of the stress–strength reliability for the generalized logistic distribution

Ragab [A. Ragab, Estimation and predictive density for the generalized logistic distribution, Microelectronics and Reliability 31 (1991) 91–95] described the Bayesian and empirical Bayesian methods for estimation of the stress–strength parameter R=P(Y<X), when X and Y are independent random variables from two generalized logistic (GL) distributions having the same known scale but different shape parameters. In this current paper, we consider the estimation of R, when X and Y are both two-parameter GL distribution with the same unknown scale but different shape parameters or with the same unknown shape but different scale parameters. We also consider the general case when the shape and scale parameters are different. The maximum likelihood estimator of R and its asymptotic distribution are obtained and it is used to construct the asymptotic confidence interval of R. We also implement Gibbs and Metropolis samplings to provide a sample-based estimate of R and its associated credible interval. Finally, analyzes of real data set and Monte Carlo simulation are presented for illustrative purposes.

Inferences on Stress-Strength Reliability from Lindley Distributions

This article deals with the estimation of the stress-strength parameter R = P(Y < X) when X and Y are independent Lindley random variables with different shape parameters. The uniformly minimum variance unbiased estimator has explicit expression, however, its exact or asymptotic distribution is very difficult to obtain. The maximum likelihood estimator of the unknown parameter can also be obtained in explicit form. We obtain the asymptotic distribution of the maximum likelihood estimator and it can be used to construct confidence interval of R. Different parametric bootstrap confidence intervals are also proposed. Bayes estimator and the associated credible interval based on independent gamma priors on the unknown parameters are obtained using Monte Carlo methods. Different methods are compared using simulations and one data analysis has been performed for illustrative purposes.

A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function

A new three-parameter exponential-type family of distributions which can be used in modeling survival data, reliability problems and fatigue life studies is introduced. Its failure rate function can be constant, decreasing, increasing, upside-down bathtub or bathtub-shaped depending on its parameters. It includes as special sub-models the exponential distribution, the generalized exponential distribution [Gupta, R.D., Kundu, D., 1999. Generalized exponential distributions. Australian and New Zealand Journal of Statistics 41, 173–188] and the extended exponential distribution [Nadarajah, S., Haghighi, F., 2011. An extension of the exponential distribution. Statistics 45, 543–558]. A comprehensive account of the mathematical properties of the new family of distributions is provided. Maximum likelihood estimation of the unknown parameters of the new model for complete sample as well as for censored sample is discussed. Estimation of the stress–strength parameter is also considered. Two empirical applications of the new model to real data are presented for illustrative purposes.