Failure Rate Function Research Articles

The Gamma–Gompertz distribution: Theory and applications

In this article, we explore a new model with three parameters called Gamma–Gompertz (GGo) distribution which is generated by the gamma-X family. The GGo distribution provides better fits than some competing distributions such as three-parameter Gompertz, Gompertz–Lindley, Gompertz–geometric, Gompertz–Poisson, and inverse Gompertz distributions. Some well-known distributions are included in the GGo distribution as special sub-models. The density function of the distribution can be decreasing, unimodal and decreasing–increasing–decreasing shaped while the failure rate function can be increasing and unimodal shaped. Various properties of the GGo distribution are obtained. An extensive simulation study is carried out to assess the effectiveness of some classical estimation approaches which are discussed to estimate the model parameters. To demonstrate the potentiality of the GGo distribution, two real data sets are used and the bootstrap percentile confidence intervals are also obtained by bootstrap resampling. In addition, a new regression model is proposed via re-parametrization of the GGo distribution, called the log-GGo distribution. The maximum likelihood method is considered to estimate the unknown parameters of re-parametrized log-GGo distribution. The potentiality of log-GGo regression model is analyzed for HIV+ censored data set.

The Lactate-to-Platelet Ratio: A Novel Predictor for Short-Term Early Allograft Failure After Liver Transplantation

Early allograft dysfunction (EAD) is a criterion to evaluate initial graft dysfunction associated with inferior graft survival and postoperative complications after liver transplantation (LT). This study defined the lactate-to-platelet ratio (LPR) as lactate level immediately post-LT/platelet count on postoperative day 1 and evaluated its association with EAD and short-term graft failure. This study reviewed 434 deceased-donor LTs from individuals with confirmed brain death between January 2008 and December 2014. The area under the curve (AUC) was used to compare the predictive capacity for 3-month graft survival between EAD and the LPR. Along with LPR, the risk factors for 3-month graft failure were analyzed by multivariate analysis. EAD was reported in 127 patients (31%). The LPR in patients with EAD was significantly higher than that in patients without EAD (9.8 vs 5.9, P < .001). In the multivariate analysis, both the LPR (per 1.0 increase) and EAD were independent risk factors for 3-month graft failure (hazard ratio [HR] =1.03, P=.03; and HR=9.14, P=.001). The comparison of the AUCs between the LPR and EAD showed no significant difference (0.79 vs 0.78, P=.84), whereas the combination of EAD and LPR had a better predictive capacity than EAD alone (0.86 vs 0.78, P < .001). The LPR showed an inverse relationship for predicting 3-month graft survival. The LPR is a continuous parameter that enables prediction of initial graft function and estimation of the 3-month graft failure rate with the advantages of early availability and simple calculations.

Advanced Approach for Estimating Failure Rate Using Saddlepoint Approximation

In the present study paper, a failure (hazard) rate function approximates the probability distribution for the linear combination of a random variable considered a highly complex model. The saddlepoint approximation approach is used to approximate the probability mass function and the cumulative distribution function to derive the approximation of the failure (hazard) rate with a high level of accuracy. The superior performance of this method is shown by numerical simulations and comparison with the performance of other approximation methods.

A New Scale-Invariant Lindley Extension Distribution and Its Applications

A new scale-invariant extension of the Lindley distribution and its power generalization has been introduced. The moments and the moment-generating functions of the proposed models have closed forms. The failure rate, the mean residual life, and the α -quantile residual life functions have been explored. The failure rate function of these models accommodates increasing, bathtub-shaped, and increasing then bathtub-shaped forms. The parameters of the models have been estimated by the maximum likelihood method for the complete and right-censored data. In a simulation study, the efficiency and consistency of the maximum likelihood estimator have been investigated. Then, the proposed models were fitted to four data sets to show their flexibility and applicability.

Integration of Production Planning, Maintenance Scheduling and Noncentral Chi-square Chart Parameters with Random Failures and Multiple Assignable Causes

This study presents an integrated model of three dependent concepts including production planning, maintenance scheduling, and statistical process monitoring (SPM) in order to improve both economic and statistical features of the production systems. To bring the proposed model closer to real applications, the possibility of occurring several types of assignable causes during the production cycle is taken into account. Besides, a noncentral chi-square chart is used for simultaneous monitoring of the process mean and variability parameters. Moreover, it is assumed that both time-to-failure and time-to-shift are Weibull distributed random variables. In other words, the system may suddenly fail and goes to an out-of-control condition with two different increasing rates including the failure rate function and the shift occurrence rate function. Hence, to improve the system reliability, a nonuniform sampling scheme is developed in which a same integrated shift occurrence rate is obtained for all intervals. The goal of the model is to minimize the expected total cost per time unit, subject to some statistical constraints. Ultimately, three comparative studies are given to demonstrate the efficiency of the proposed model. The first one indicates that a nonuniform sampling scheme reduces the expected total cost and the length of out-of-control period. The second one confirms that considering the system failure not only increases the length of in-control period but also improves the economic feature of the production system. The third one shows that using chi-square chart decreases the quality loss cost which in turn decreases the expected total cost.

Bayesian Testing Procedure on the Lifetime Performance Index of Products Following Chen Lifetime Distribution Based on the Progressive Type-II Censored Sample

With the high demands on the quality of high-tech products for consumers, assuring the lifetime performance is a very important task for competitive manufacturing industries. The lifetime performance index CL is frequently used to monitor the larger-the-better lifetime performance of products. This research is related to the topic of asymmetrical probability distributions and applications across disciplines. Chen lifetime distribution with a bathtub shape or increasing failure rate function has many applications in the lifetime data analysis. We derived the uniformly minimum variance unbiased estimator (UMVUE) for CL, and we used this estimator to develop a hypothesis testing procedure of CL under a lower specification limit based on the progressive type-II censored sample. The Bayesian estimator for CL is also derived, and it is used to develop another hypothesis testing procedure. A simulation study is conducted to compare the average confidence levels for two procedures. Finally, one practical example is given to illustrate the implementation of our proposed non-Bayesian and Bayesian testing procedure.

Exponentiality Test Based on Progressively Type-II Censored Data Via Extension of Cumulative Tsallis Divergence

In this article, first new divergences are defined by using Tsallis divergence and a measure of discrepancy between equilibriums associated with two distributions is proposed. Then utilizing the progressively Type-II censored sample, we construct goodness of fit tests for exponentiality based on the estimation of proposed divergences. To investigate the performance of the mentioned tests, Monte Carlo simulations are performed. In order to study the power, the alternatives are considered according to the failure rate function. The powers of the proposed tests are then compared with other existing tests. As regards the last step of the study, in order to explain the use of the proposed tests, three examples are presented.

Study of the modified Weibull function to analyze reliability in engineering components that fail and are repairable

An engineering component is physically inoperative when it fails or has conditions that prevent it from operating and has a functional failure. The useful or operating times and the non-operating or downtimes are collected to determine the reliability or the probability of operation of the component during a period. The use of distributions is the most accepted method for determining component reliability. Among the main distributions used are the exponential, Log Normal, Weibull, etc. The Weibull distribution is widely used to estimate the reliability of a component, understood as the physical capacity of the element to function according to operating parameters. A study of three models based on Weibull modifications was conducted to determine the reliability of repairable components subject to maintenance activities. Two models represent the modified Weibull function with two parameters and an additional model that represents the Weibull function transmuted with failure rate function that fits the Davies curve. The characteristics, the physical interpretation, the applications, and the feasibility of using each model in analysis of reliability in elements and repairable in engineering are presented.

The exponentiated Akash distribution with properties and applications

In this study, a new two-parameter lifetime distribution named the exponentiated Akash distribution is introduced and studied. Akash distribution is considered an extension of the lindly distribution and can be easily expressed as a mixture of an exponential distribution with scale parameter (θ) and a gamma distribution with shape parameter equal 3 and scale parameter θ . The reliability (survival) function, hazard rate function (also known as the failure rate function), the reversed hazard rate and the cumulative hazard rate function of the new distribution are obtained. Some statistical properties of the exponentiated Akash distribution such as the moments, moment generating function, order statistics, Renyi entropy and quantile function are derived. The method of maximum likelihood is used to estimate the unknown parameters of this distribution. Finally, the performance of this new distribution is examined using some real life data sets to show its flexibility and better goodness of fit as compared with other lifetime distributions such as Akash and exponential based on the Akaike information criterion.

Prognostic Analysis of High-Speed Cylindrical Roller Bearing Using Weibull Distribution and k-Nearest Neighbor

Abstract Bearing remnant operational life can be determined by implementing a data-driven prognostics method. In this work, the bearing run-to-failure data from experimentation on test rig is used to extract time-domain features. The sudden change in time-domain information signifies the fault inception which led to failure stage promptly. The monotonicity metric is utilized to select the optimal feature set that best represents bearing degradation. Principal component analysis (PCA) is used for dimension reduction and fusion, and a unidimensional health indicator (HI) is constructed. Fluctuations of HI are smoothed by fitting it with a Weibull failure rate function (WFRF) and the corresponding parameters are estimated using nonlinear least-squares method. By inverting the model, the predicted time values are calculated, and hence remnant operational life of bearing is evaluated and compared with the actual life from experimental data. The performance assessment metrics utilized are mean absolute percentage error (MAPE), mean-square error (MSE), root-mean-square error (RMSE), and bias. Besides this, an online degradation state classification method using the k-nearest neighbor (KNN) classifier is implemented. The KNN model performance is assessed by constructing receiver operating characteristics (ROC) curve, which indicates the value of area under the curve (AUC) equal to 0.94, representing high accuracy of the KNN. The remaining useful life (RUL) is predicted within 95% confidence limits, and the predicted RUL almost follows the actual one with some fluctuations. The model performance is found promising and can be implemented to evaluate the remaining useful life of bearing.

Estimation and prediction for the Rayleigh distribution based on double Type-I hybrid censored data

Based on a new lifetime test scheme proposed in this article called double Type-I hybrid censoring scheme, the point estimates and confidence intervals of the unknown parameter, reliability function and failure rate function of the Rayleigh distributed lifetime are obtained. According to the posterior predictive distribution, the predictive estimates are given under three loss functions. Monte-Carlo method is used to simulate double Type-I hybrid censored data. The means and relative errors of various estimators are calculated. The influence of sample size on the estimated accuracy is explained. Finally, a numerical example is given to calculate the estimated values.

Bounded Odd Inverse Pareto Exponential Distribution: Properties, Estimation, and Regression

In this paper, we introduce a new three-parameter distribution defined on the unit interval. The density function of the distribution exhibits different kinds of shapes such as decreasing, increasing, left skewed, right skewed, and approximately symmetric. The failure rate function shows increasing, bathtub, and modified upside-down bathtub shapes. Six different frequentist estimation procedures were proposed for estimating the parameters of the distribution and their performance assessed via Monte Carlo simulations. Applications of the distribution were illustrated by analyzing two datasets and its fit compared to that of other distributions defined on the unit interval. Finally, we developed a regression model for a response variable that follows the new distribution.

A discrete analogue of odd Weibull-G family of distributions: properties, classical and Bayesian estimation with applications to count data

In the statistical literature, several discrete distributions have been developed so far. However, in this progressive technological era, the data generated from different fields is getting complicated day by day, making it difficult to analyze this real data through the various discrete distributions available in the existing literature. In this context, we have proposed a new flexible family of discrete models named discrete odd Weibull-G (DOW-G) family. Its several impressive distributional characteristics are derived. A key feature of the proposed family is its failure rate function that can take a variety of shapes for distinct values of the unknown parameters, like decreasing, increasing, constant, J-, and bathtub-shaped. Furthermore, the presented family not only adequately captures the skewed and symmetric data sets, but it can also provide a better fit to equi-, over-, under-dispersed data. After producing the general class, two particular distributions of the DOW-G family are extensively studied. The parameters estimation of the proposed family, are explored by the method of maximum likelihood and Bayesian approach. A compact Monte Carlo simulation study is performed to assess the behavior of the estimation methods. Finally, we have explained the usefulness of the proposed family by using two different real data sets.

Model for ancient Greek and Roman coinage production

Coinage production of ancient powers such as Athens and Rome is usually inferred from die statistics of monetary issues. The present work applies a Kaplan-Meier analysis of resistance to failure to 29 sets of well-documented monetary issues. The failure rate function assumes a U-shaped form known in reliability engineering literature as the ‘bathtub curve’. With the geometric distribution of die failure being demonstrably violated for a large fraction of the data sets, the die distribution of each data set was instead fitted by a mixture of two Weibull distributions corresponding to two failure regimes. Dies can be divided into bad dies, failing early for various reasons, and good dies, failing late by fatigue. The dual populations reflect the efforts of the smiths at the time to produce bronze dies that would meet two conflicting needs: the reduction of premature die failure (= infant mortality) and the limitation of ductile deformation during minting. The variable proportions of the two populations suggest that not all workshops had fully mastered die technology. Because of the dichotomy induced by contrasting mechanical properties, corrections for missing dies based on singletons and causes of die failure must be carefully assessed for each data set.

New Method for Generating New Families of Distributions

This article presents a new method for generating distributions. This method combines two techniques—the transformed—transformer and alpha power transformation approaches—allowing for tremendous flexibility in the resulting distributions. The new approach is applied to introduce the alpha power Weibull—exponential distribution. The density of this distribution can take asymmetric and near-symmetric shapes. Various asymmetric shapes, such as decreasing, increasing, L-shaped, near-symmetrical, and right-skewed shapes, are observed for the related failure rate function, making it more tractable for many modeling applications. Some significant mathematical features of the suggested distribution are determined. Estimates of the unknown parameters of the proposed distribution are obtained using the maximum likelihood method. Furthermore, some numerical studies were carried out, in order to evaluate the estimation performance. Three practical datasets are considered to analyze the usefulness and flexibility of the introduced distribution. The proposed alpha power Weibull–exponential distribution can outperform other well-known distributions, showing its great adaptability in the context of real data analysis.

On Some Properties of $$\alpha $$-Mixtures

The $$\alpha $$ -mixtures is a new, flexible family of distributions that includes many mixture models as special cases. This paper is mainly focused on relevant stochastic comparisons and ageing properties of $$\alpha $$ -mixtures of survival functions. In particular, we prove that ageing properties of $$\alpha $$ -mixtures for additive and multiplicative hazards models depend on the properties of the baseline failure rate functions and the corresponding conditional moments of mixing distributions. Partial orderings of the finite $$\alpha $$ -mixtures in the sense of the usual stochastic order and the hazard rate order are discussed. Finally, we extend some results on the shape of the mixture failure rate obtained in the literature for usual mixtures to the case of $$\alpha $$ -mixtures.

Statistical Properties of a New Bathtub Shaped Failure Rate Model With Applications in Survival and Failure Rate Data

In this study, we proposed a flexible lifetime model identified as the modified exponentiated Kumaraswamy (MEK) distribution. Some distributional and reliability properties were derived and discussed, including explicit expressions for the moments, quantile function, and order statistics. We discussed all the possible shapes of the density and the failure rate functions. We utilized the method of maximum likelihood to estimate the unknown parameters of the MEK distribution and executed a simulation study to assess the asymptotic behavior of the MLEs. Four suitable lifetime data sets we engaged and modeled, to disclose the usefulness and the dominance of the MEK distribution over its participant models.

A new Weibull Exponentiated Inverted Weibull Distribution for modelling positively-skewed data

An Exponentiated Inverted Weibull Distribution (EIWD) has a hazard rate (failure rate) function that is unimodal, thus making it less efficient for modeling data with an increasing failure rate (IFR). Hence, the need to generalize the EIWD in order to obtain a distribution that will be proficient in modeling these types of dataset (data with an increasing failure rate). This paper therefore, extends the EIWD in order to obtain Weibull Exponentiated Inverted Weibull (WEIW) distribution using the Weibull-Generator technique. Some of the properties investigated include the mean, variance, median, moments, quantile and moment generating functions. The explicit expressions were derived for the order statistics and hazard/failure rate function. The estimation of parameters was derived using the maximum likelihood method. The developed model was applied to a real-life dataset and compared with some existing competing lifetime distributions. The result revealed that the (WEIW) distribution provided a better fit to the real life dataset than the existing Weibull/Exponential family distributions.

A New Lifetime Model: Copulas, Properties and Real Lifetime Data Applications

A new four parameter lifetime model called the Weibullgeneralized Lomax is proposed and studied. The new density function can be "right skewed", "symmetric" and "left skewed" and its corresponding failure rate function can be "monotonically decreasing", " monotonically increasing" and "constant". The skewness of the new distribution can negative and positive. The maximum likelihood method is employed and used for estimating the model parameters. Using the "biases" and "mean squared errors", we performed simulation experiments for assessing the finite sample behavior of the maximum likelihood estimators. The new model deserved to be chosen as the best model among many well-known Lomax extension such as exponentiated Lomax, gamma Lomax, Kumaraswamy Lomax, odd log-logistic Lomax, Macdonald Lomax, beta Lomax, reduced odd log-logistic Lomax, reduced Burr-Hatke Lomax, reduced WG-Lx, special generalized mixture Lomax and the standard Lomax distributions in modeling the "failure times" and the "service times" data sets.

Properties of the Transmuted Burr XII Distribution, Regression and its Applications

In this paper, we define and study a four-parameter model called the transmuted Burr XII distribution. We obtain some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, order statistics, probability weighted moments and entropies. We formulate and develop a log-linear model using the new distribution so-called the log-transmuted Burr XII distribution for modeling data with a unimodal failure rate function, as an alternative to the log-McDonald Burr XII, log-beta Burr XII, log-Kumaraswamy Burr XII, log-Burr XII and logistic regression models. The flexibility of the proposed models is illustrated by means of three applications to real data sets.