In recent years, a new class of models has been proposed to exhibit the bathtub-shaped failure rate functions. The Weibull extension model is one of these models, which is asymptotically related to the ordinary Weibull model and is capable of modeling the bathtub-shaped and increasing failure rate lifetime data. This paper presents the conditional inference for constructing the confidence intervals for the Weibull extension parameters based on the generalized order statistics. For measuring the performances of this approach comparing to the Asymptotic maximum likelihood estimates, Simulation studies have been carried out, that indicated the conditional intervals possess a good statistical properties and they can perform quite well even when the sample size is extremly small. An illustrative examples based on real data are given to illustrate the confidence intervals developed in this paper.

This article proposes and studies a new three-parameter generalized model of the inverse Gompertz distribution, in the so-called Kumaraswamy inverse Gompertz distribution. The main advantage of the new model is that it has "an upside down bathtub-shaped curve hazard rate function" depending upon the shape parameters. Several of its statistical and mathematical properties including quantiles, median, mode, moments, probability weighted moment, entropy function, skewness and kurtosis are derived. Moreover, the reliability and hazard rate functions, mean time to failure, mean residual and inactive lifetimes are also concluded. The maximum likelihood approach is done here to estimate the new model parameters. A simulation study is conducted to examine the performance of the estimators of this model. Finally, the usefulness of the proposed distribution is illustrated with different engineering applications to complete, type-II right censored, and upper record data and it is found that this model is more flexible when it is compared to well-known models in the statistical literature.

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The performance of a repairable bridge network system is improved by using the availability equivalence factors. All components for the bridge system have constant failure and repair rates. The system is improved through the use of five methods: reduction, increase, hot duplication, warm duplication, and cold duplication methods. The availability of the original and improved systems is derived. Two types of availability equivalence factors of the system are obtained to compare different system designs. Numerical example to interpret how to utilize the obtained results is provided.

In this article, a new four-parameter lifetime model called the exponentiated generalized inverted Gompertz distribution is studied and proposed. The newly proposed distribution is able to model the lifetimes with upside-down bathtub-shaped hazard rates and is suitable for describing the negative and positive skewness. A detailed description of some various properties of this model, including the reliability function, hazard rate function, quantile function, and median, mode, moments, moment generating function, entropies, kurtosis, and skewness, mean waiting lifetime, and others are presented. The parameters of the studied model are appreciated using four various estimation methods, the maximum likelihood, least squares, weighted least squares, and Cramér-von Mises methods. A simulation study is carried out to examine the performance of the new model estimators based on the four estimation methods using the mean squared errors (MSEs) and the bias estimates. The flexibility of the proposed model is clarified by studying four different engineering applications to symmetric and asymmetric data, and it is found that this model is more flexible and works quite well for modeling these data.