Bathtub Failure Rate Research Articles

A new finite range increasing generalized failure rate distribution and it’s applications

Here we introduce a bounded distribution, bounded by 0 and 1, that has unimodal density function. The distribution function as well as the hazard rate function of the model have closed forms. Structural properties and asymptotic behavior of the distribution are studied and obtained a characterization. Moreover, the distribution is an increasing generalized failure rate distribution and is capable of modeling data with increasing and bathtub failure rates. Two real data sets are analyzed and it is shown that the proposed model fits better than some of the existing models.

A new bathtub and increasing failure rate model: An extension of the Mustapha type II distribution

This article introduces a new three-parameter lifetime model with an increasing and bathtub failure rate functions as an extension of the Mustapha type II distribution (MuII). The model can be very useful in statistical studies, reliability, computer sciences and engineering. Various mathematical and statistical properties of the distribution are discussed, such as moments, mean deviations, Bonferroni and Lorenz curves, entropy, order statistic, and extreme value distributions. Moreover, we consider the bivariate extension of the new model. Statistical inferences by the maximum likelihood method are discussed and assess by simulation studies. Applications of the proposed model to two right-skewed data are presented for illustration. The new model provides a better fit than some other existing distribution as measured by some model selection criteria and goodness of fits statistics.

A New Life-Time Model with Bathtub and Inverted Bathtub Failure Rate Function

In this paper, a three parameter model called Zubair- Kumaraswamy (Z-Kum) distribution is proposed. The extension was done using Zubair G-Family [1] of continuous probability distribution to extend the well knownKumaraswamy distribution to make it more flexible in modeling and predicting real world phenomenon. Some basic structural properties of the new distributions like cdf, pdf, quantile functions, moments, moment generating functions, characteristics functions and order statistics was obtained. Survival function, hazard function, reversed hazard rate function and a cumulative hazard rate function was also obtained. Behavior of the hazard rate plot exhibit increase, decrease, Bathtub and inverted Bathtub shape. Maximum likelihood estimate was used to estimate the Z-Kum distribution parameters. Monte Carlo simulation was carried out to evaluate the performance of MLE method adopted. Result of the simulation studies indicates that MLE is good for the estimation of our distribution parameters. To compare the proposed model with the other fitted existing models, analytical measure of goodness of fit of some information criterion was considered using three real life data sets. From the results obtained, it is evident that our proposed model gives better fit than the other competing models and therefore, our proposed model provide greater flexibility in modeling real world phenomenon.

A family of tests for trend change in failure rate function with right censored data

In this paper we extend the family of test proposed by Majumder and Mitra [Detecting trend change in failure functions-an L-statistic approach. Stat Pap. 2019;62:31–52. doi: 10.1007/s00362-018-01074-8] for testing exponentiality against bathtub (BT) failure rate (upside down bathtub (UBT)) to randomly right censored case. We derive the asymptotic distribution of the test statistic and using Monte Carlo simulation we obtain the empirical powers for certain alternative distributions. As a special case the asymptotic distribution of the test statistic proposed by Park [Testing whether failure rate changes its trend. IEEE Trans Reliab. 1988;37(4):375–378. doi: 10.1109/24.9842] is obtained. Finally some examples of real life data are given for illustrative purpose.

인공 신경망 예측기를 이용한 새로운 데이터 모델 기반 PV 어레이 고장 진단용 기준 신호 발생 방법

The power supply part of a PV system is called a PV array which is composed of many PV strings connected in parallel. A PV-string which is a set of PV modules is made by serial connection of many PV modules. PV modules presents irregularity of ouput power which is generated due to various factors such as average performance degradation of about 0.923% to 1.54% according to various researches and born power variations over production stages. Power degradation and born power variation are due to various factors such as material interactions(connection state of connectors between PV cells, corrosion, browning of encapsulation materials ...etc) and environment factors such as shading and soiling which refers to the accumulation of snow, dirt, leaves and bird droppings on PV modules. Such various minor factors can make side effects in safety side and economic one. Therefore several methodology for diagnosis of PV-array have been developed, which are classified into three types – image-based diagnosis approach, model-based approach and data-driven approach. In this paper, a new data-based approach(called data model) with good failure diagnosis reliability and economy is proposed, and verified by simulation using Python, published data and reasonable data generation. Based on the bathtub failure rate function, the fault diagnosis model requires a well-functioning predictor to generate a reference signal which evaluates the output characteristics of PV array under instantaneously varying environmental condition such as solar irradiance, temperature, etc. To implement a new data-based approach, an Artificial Neural Network (ANN) based predictor is applied as a reference generator for PV array’s fault diagnosis. r² value and the RMSE(Root Mean Square Error) are used to evaluate how far away individual learned predictions are from the actual measured values. Based on the reliability obtained from the learning result for a specific PV string, it is confirmed through pair comparison analysis(called t-test) that the learning result(called clustering possibility) between PV strings under different installation conditions is also reliable.

A robust bathtub-shaped failure time model for a two-component system with applications to complete and censored reliability data

ABSTRACT This article proposes a flexible additive model that adequately describes complex reliability and survival data. The proposed methodology, referred to as the flexible exponential power-Gompertz (FEPG4) distribution, is able to characterize the behavior of a complex system whose failure times have bathtub-shaped, with clear burn-in and wear-out change points and a low, yet lengthy, flat middle segment as its underlying failure rate distribution. We discuss some properties of the model. Parameter inferences are proposed under maximum likelihood and Bayesian techniques. We determine the Bayes estimators of the FEPG4 parameters and used Hamiltonian Monte Carlo for posterior simulations. Extensive simulation experiments are performed to validate the proposed estimators. For assessing the potential of the FEPG4, the model is compared with other recent bathtub distributions constructed via the same approach on devices’ failure and running times (censored and non-censored case) and failure times of some devices, each with the bathtub failure rate. Seven parametric and non-parametric selection criteria and other supporting plots are utilized for comparison purposes. Findings indicate that the FEPG4 model might be the best alternative for representing device failure times, particularly when the bathtub-shaped failure rate of the available data clearly illustrates its three phases.

A new failure times model for one and two failure modes system: A Bayesian study with Hamiltonian Monte Carlo simulation

This paper presents an additive Gompertz-Weibull (AGW) distribution, a four-parameter hybrid probability distribution, and its applications in reliability engineering. The failure rate (FR) function of the proposed model demonstrates an increasing trend and a variety of bathtub shapes with or without a low and yet long-stable segment, making it appropriate for modelling a wide variety of real-world problems. Some relationships between the AGW’s FR and its mean residual life functions are examined. For parameter estimation, maximum likelihood and Bayesian inferences are considered. For posterior simulations, we use Hamiltonian Monte Carlo to evaluate the Bayes estimators of the AGW parameters. We evaluate the performance of the proposed AGW model to that of other recent bathtub distributions constructed following the same approach on three failure time datasets. The first two datasets represent device failure times, while the third represents early cable-joint failure times, all with bathtub FR. For comparison, five parametric and nonparametric evaluation criteria and the fitted FR and mean residual life curves were employed. The results indicated that the AGW model would be the best choice for describing failure times, especially when the bathtub-shaped FR of the presented dataset exhibits its three segments.

On some properties of distributions possessing a bathtub-shaped failure rate average

AbstractThe life distribution of a device subject to shocks governed by a homogeneous Poisson process is shown to have a bathtub failure rate average (BFRA) when the probabilities $\bar{P}_k$ of surviving k shocks possess the corresponding discrete property. We prove closure under the formation of weak limits for BFRA distributions and explore related moment convergence issues within the BFRA family. Similar results for increasing and decreasing failure rate average distributions are obtained either independently or as consequences of our results. We also establish some results outlining the positions of various non-monotonic ageing classes such as bathtub failure rate, increasing initially then decreasing mean residual life, new worse then better than used in expectation, and increasing initially then decreasing mean time to failure in the hierarchy. Finally, an open problem is posed and a partial solution provided.

A new three-parameter model with support on a bounded domain: Properties and quantile regression model

In this article, we proposed a new three-parameter model with a bathtub failure rate. The main properties of the new model are derived, such as quantile function, moments, the moment of residual life, stress strength reliability parameter, order statistics, extreme value distribution, Shannon entropy, and Renyi entropy. Maximum likelihood estimation (MLE) is considered for the parameter estimation, and the information matrix is obtained. Simulation studies were used to assess the performances of the estimators by discussing their bias, mean square error, confidence interval, and coverage probability. In addition, we discussed the quantile regression model based on the proposed model; we examined the performance of their MLEs by simulation studies via the randomized quantile residuals. Two real data sets are used to illustrate the importance of the new model in practice.

Inferences and Engineering Applications of Alpha Power Weibull Distribution Using Progressive Type-II Censoring

As an extension of the standard Weibull distribution, a new crucial distribution termed alpha power Weibull distribution has been presented. It can model decreasing, increasing, bathtub, and upside-down bathtub failure rates. This research investigates the estimation of model parameters and some of its reliability characteristics using progressively Type-II censored data. To get estimates of unknown parameters, reliability, and hazard rate functions, the maximum likelihood, and Bayesian estimation approaches are studied. To acquire estimated confidence intervals for unknown parameters and reliability characteristics, the maximum likelihood asymptotic properties are used. The Markov chain Monte Carlo approach is used in Bayesian estimation to provide Bayesian estimates under squared error and LINEX loss functions. Furthermore, the highest posterior density credible intervals of the parameters and reliability characteristics are determined. A Monte Carlo simulation study is used to investigate the accuracy of various point and interval estimators. In addition, various optimality criteria are used to choose the best progressive censoring schemes. Two real data from the engineering field are analyzed to demonstrate the applicability and significance of the proposed approaches. Based on numerical results, the Bayesian procedure for estimating the parameters and reliability characteristics of alpha power Weibull distribution is recommended. The analysis of two real data sets showed that the alpha power Weibull distribution is a good model to investigate engineering data in the presence of progressive Type-II censoring.

A reliability and survival model for one and two failure modes system with applications to complete and censored datasets

Lifetime distributions that have bathtub-shaped failure rates with low and long constant region at the mid-time period are more realistic for modelling failure time datasets with similar failure rates. In this paper, we introduce a new model named flexible additive Chen–Gompertz distribution. The model has increasing and bathtub-shaped failure rate with or without low and long constant region, and can be adequately used in reliability context. Some distributional properties of the model are obtained. We present a complete Bayesian analysis of the proposed model for censored and non-censored datasets and provide the Bayes estimators of the model parameters using the Metropolis–Hastings algorithm by utilizing the square error loss function. We have also given the maximum likelihood estimators of the model parameters for complete and right-censored datasets and performed simulations on the two methods for different settings of the parameters and sample sizes. The applicability of the proposed distribution is demonstrated by fitting three different failure time datasets each with bathtub failure rate from reliability and survival analysis. The results of the applications illustrate that the model provides a good fit to all the three datasets compared to the other well-known Chen and Weibull extended distributions as evidenced by some goodness-of-fit tests.

The Marshall-Olkin Odd Burr III-G Family: Theory, Estimation, and Engineering Applications

We propose a new flexible class called the Marshall-Olkin odd Burr III family for generating continuous distributions and derive some of its statistical properties. We provide three special models which accommodate symmetrical, right-skewed and left-skewed shaped densities as well as bathtub, decreasing, increasing, reversed-J shaped and upside-down bathtub failure rate functions. The parameters are estimated by maximum likelihood, least squares and a percentile method. Some simulations investigate the accuracy of the three methods. We illustrate the utility of a special model through three applications to engineering field.

The Exponentiated Poisson-Exponential Distribution: A Distribution with Increasing, Decreasing and Bathtub Failure Rate

In this paper we propose a new three-parameters lifetime distribution with increasing, decreasing and bathtub failure rate depending on its parameters. The properties of the proposed distribution are discussed, including explicit algebraic formulae for its reliability and failure rate functions, quantiles and moments. A simulation study is performed in order to verify the behavior of the maximum likelihood estimates. The methodology is illustrated on a real data set.

A New Lifetime Distribution with Decreasing and Upside-Down Bathtub-Shaped Hazard Rate Function

We introduce a new lifetime distribution, called the alpha-power transformed Lomax (APTL) distribution which generalizes the Lomax distribution to provide better fits than the Lomax distribution and some of its known generalizations. Various properties of the proposed distribution, including explicit expressions for the quantiles, mode, moments, conditional moments, mean residual lifetime, stochastic ordering, Bonferroni and Lorenz curve, stress-strength reliability and order statistics are derived. The new distribution can have a decreasing and upside-down bathtub failure rate function depending on its parameters. The maximum likelihood estimators of the three unknown parameters of APTL are obtained. A simulation study is carried out to examine the performances of the maximum likelihood estimates in terms of their mean squared error using simulated samples. Finally, the potentiality of the distribution is analyzed by means of two real data sets. For the real data sets, this distribution is found to be superior in its ability to sufficiently model both the data sets as compared to the Lomax (L) distribution, exponentiated-Lomax (EL) distribution, gamma-Lomax (GL) distribution, beta-Lomax (BL) distribution and Kumaraswamy-Lomax (KuL) distribution.

Location Based Generalized Akash Distribution: Properties and Applications

In recent years, several statistical finite mixture models have been proposed to model the lifetime data with heterogeneity. The Lindley distribution has been highlighted by many authors for these types of lifetime data analysis. This paper introduces a new Lindley family distribution called location-based generalized Akash distribution (NGAD) with monotonic increasing and bathtub failure rates. The density function of NGAD is flexible to cover the left-skewed, right-skewed and symmetrical shapes with different tail-weights. Its fundamental structural properties and its ability to provide a suitable statistical model for various types of data sets are studied. The maximum likelihood (ML) method is used to estimate its unknown parameters and the performance of ML estimates are examined by a simulation study. Finally, several real-data sets with different characteristics are used to illustrate its flexibility. It is observed that NGAD provides a better fit than some other existing modified Lindley distributions.

Stochastic comparisons of series, parallel and [formula omitted]-out-of-[formula omitted] systems with heterogeneous bathtub failure rate type components

In this paper, we explore stochastic comparisons of order statistics arising from a two parameter distribution with bathtub shaped failure rate. We obtain detailed results in the contexts of the usual stochastic order, the hazard rate order, the reversed hazard rate order and the likelihood ratio order for series, parallel and k-out-of-n systems. Results involving multivariate chain majorization are also established.

A New Extension of Lindley Geometric Distribution and its Applications

A new four-parameter distribution called the beta Lindley-geometric distribution is proposed. The hazard rate function of the new model can be constant, decreasing, increasing, upside down bathtub or bathtub failure rate shapes. Various structural properties including of the new distribution are derived. The estimation of the model parameters is performed by maximum likelihood method. The usefulness of the new distribution is illustrated using a real data set.

A new uniform distribution with bathtub-shaped failure rate with simulation and application

Bathtub failure rate shape is widely used in industrial and medical applications. In this paper, a three-parameter lifetime distribution, so-called the generalized Weibull uniform distribution that extends the Weibull distribution, is proposed and studied. This distribution has bathtub-shaped or decreasing failure rate function which enables it to fit real lifetime data sets. Various structural properties of the new distribution are derived, including explicit expressions for the quantile function, moments, moment-generating function and order statistics. Parameter estimations are provided by a maximum likelihood estimation, and the performance of the maximum likelihood estimation is evaluated using a simulation study. An application to real-life data demonstrates that the proposed distribution can be very useful in fitting real data.

A New Extension of the Generalized Half Logistic Distribution with Applications to Real Data.

In this paper, we introduced a new three-parameter probability model called Poisson generalized half logistic (PoiGHL). The new model possesses an increasing, decreasing, unimodal and bathtub failure rates depending on the parameters. The relationship of PoiGHL with the exponentiated Weibull Poisson (EWP), Poisson exponentiated Erlang-truncated exponential (PEETE), and Poisson generalized Gompertz (PGG) model is discussed. We also characterized the PoiGHL sub model, i.e the half logistic Poisson (HLP), based on certain functions of a random variable by truncated moments. Several mathematical and statistical properties of the PoiGHL are investigated such as moments, mean deviations, Bonferroni and Lorenz curves, order statistics, Shannon and Renyi entropy, Kullback-Leibler divergence, moments of residual life, and probability weighted moments. Estimation of the model parameters was achieved by maximum likelihood technique and assessed by simulation studies. The stress-strength analysis was discussed in detail based on maximum likelihood estimation (MLE), we derived the asymptotic confidence interval of based on the MLEs, and examine by simulation studies. In three applications to real data set PoiGHL provided better fit and outperform some other popular distributions. In the stress-strength parameter estimation PoiGHL model illustrated as a reliable choice in reliability analysis as shown using two real data set.

Logarithm Transformed Fr´echet Distribution : Properties and Estimation

In this paper, a new three-parameter distribution called the Alpha Logarithm Transformed Fr\'{e}chet (ALTF) distribution is introduced which offers a more flexible distribution for modeling lifetime data. Various properties of the proposed distribution, including explicit expressions for the quantiles, moments, incomplete moments, conditional moments, moment generating function R\'{e}nyi and $\delta$-entropies, stochastic ordering, stress-strength reliability and order statistics are derived. The new distribution can have decreasing, reversed J-shaped and upside-down bathtub failure rate functions depending on its parameter values. The maximum likelihood method is used to estimate the distribution parameters. A simulation study is conducted to evaluate the performance of the maximum likelihood estimates. Finally, the proposed extended model is applied on real data sets and the results are given which illustrate the superior performance of the ALTF distribution compared to some other well-known distributions.