Bathtub-shaped Hazard Function Research Articles

Bayesian and empirical-Bayesian evaluations of the Topp–Leone lifetime model using lower record statistics

Record statistics are generally used to measure a process’ stochastic behavior at previously unheard-of values, either upper or lower. It can be obtained or calculated using order statistics from a sample whose size is dictated by the values and order of occurrence of observations. Due to its finite support and bathtub-shaped hazard function, the Topp–Leone (TL) distribution is appealing for reliability investigations. In today’s practical applications, estimating unknown parameters and predicting upcoming records is an essential issue. Thus, in this work, the TL distribution’s lower record values are taken into account. Against the symmetric squared-error loss, both Bayes and empirical-Bayes approaches are used to estimate the TL’s parameter and calculate prediction limits for upcoming lower record values. A Monte Carlo simulation experiment is performed to assess the accuracy of the estimation methodologies provided. To demonstrate the operability and usefulness of the various approaches, two useful applications are studied. The numerical evaluations show that the suggested approaches perform effectively and that the provided estimations are adequate in practice.

A NEWCLASS OF TAN-G FAMILY OF DISTRIBUTIONS WITH PROPERTIES AND APPLICATIONS

This research article is dedicated to the exploration of a novel family of distributions that is based on a tangent transformation. The new class of distributions, which we have named the new class tan-G (NCT-G) family, has been developed using the ratio of the cumulative distribution function (CDF) G(x) and 1+G(x) of a baseline distribution. We provide an overview of the general properties of this family of distributions. To demonstrate the applicability of the NCT-G family, we have utilized the inverse Weibull distribution as a baseline and introduced a new member of the suggested family. This new distribution exhibits a reverse-J, increasing, or inverted bathtub-shaped hazard function. We have also explored some statistical properties of this distribution, as well as its associated parameters estimated through maximum likelihood estimation (MLE). To evaluate the accuracy of the estimation procedure, we have conducted a Monte Carlo simulation. We found that even for small sample sizes, biases and mean square errors decrease as the sample size increases. Additionally, we have applied the NCT-IW distribution to two real data sets. By using model selection criteria and goodness of fit test statistics, we empirically proved that our suggested model outperforms other existing models, most of which have a greater number of parameters.

An improved modified Weibull distribution applied to predict the reliability evolution of an aircraft lock mechanism

A new improved modified Weibull distribution (IMW), which consists of two tandem failure models is proposed in this paper. The order statistics, moment estimate and maximum likelihood estimate of the new distribution are investigated. The distribution is flexible for modeling the three phases of the modified bathtub-shaped hazard function. The modified distribution allows for enhanced statistical simulation. The new distribution is analyzed and applied to two sets of known lifetime data to demonstrate the advantages of the model. To verify the practical engineering applications of the model, reliability evolution test data for an aircraft door lock mechanism are analyzed and fitted. Based on the obtained trends of the lock hook angle and lock hook distance, we predict the evolutionary data for numerous mechanism samples, thus reducing the test cost. By considering the subdistribution and removing one parameter, the fitting effect is improved over that of the traditional distribution model.

Inference for Local Parameters in Convexity Constrained Models

ABSTRACT In this article, we develop automated inference methods for “local” parameters in a collection of convexity constrained models based on the natural constrained tuning-free estimators. A canonical example is given by the univariate convex regression model, in which automated inference is drawn for the function value, the function derivative at a fixed interior point, and the anti-mode of the convex regression function, based on the widely used tuning-free, piecewise linear convex least squares estimator (LSE). The key to our inference proposal in this model is a pivotal joint limit distribution theory for the LS estimates of the local parameters, normalized appropriately by the length of certain data-driven linear piece of the convex LSE. Such a pivotal limiting distribution instantly gives rise to confidence intervals for these local parameters, whose construction requires almost no more effort than computing the convex LSE itself. This inference method in the convex regression model is a special case of a general inference machinery that covers a number of convexity constrained models in which a limit distribution theory is available for model-specific estimators. Concrete models include: (i) log-concave density estimation, (ii) s-concave density estimation, (iii) convex nonincreasing density estimation, (iv) concave bathtub-shaped hazard function estimation, and (v) concave distribution function estimation from corrupted data. The proposed confidence intervals for all these models are proved to have asymptotically exact coverage and oracle length, and require no further information than the estimator itself. We provide extensive simulation evidence that validates our theoretical results. Real data applications and comparisons with competing methods are given to illustrate the usefulness of our inference proposals. Supplementary materials for this article are available online.

The Kumaraswamy Lindley Regression Model with Application on the Egyptian Stock Exchange

We introduce and study the Kumaraswamy Lindely Distribution (KLD) model, which has increasing, decreasing, upside-down bathtub and bathtub shaped hazard functions.. We perform a Monte Carlo simulation study to assess the finite sample behavior of the maximum likelihood estimates of the parameters. We define a new regression model based on the new distribution. The new regression was applied to data from the Egyptian stock exchange in the period of (2015-2019). Finally, we study some properties of regression Residual analysis The martingale residual, Deviance component residual.

A New Lifetime Exponential-X Family of Distributions with Applications to Reliability Data

Modeling reliability data with nonmonotone hazards is a prominent research topic that is quite rich and still growing rapidly. Many studies have suggested introducing new families of distributions to modify the Weibull distribution to model the nonmonotone hazards. In the present study, we propose a new family of distributions called a new lifetime exponential-X family. A special submodel of the proposed family called a new lifetime exponential-Weibull distribution suitable for modeling reliability data with bathtub-shaped hazard rates is discussed. The maximum-likelihood estimators of the model parameters are obtained. A brief Monte Carlo simulation study is conducted to evaluate the performance of these estimators. For illustrative purposes, two real applications from reliability engineering with bathtub-shaped hazard functions are analyzed. The practical applications show that the proposed model provides better fits than the other nonnested models.

A New Flexible Bathtub-Shaped Modification of the Weibull Model: Properties and Applications

Many studies have suggested the modifications and generalizations of the Weibull distribution to model the nonmonotone hazards. In this paper, we combine the logarithms of two cumulative hazard rate functions and propose a new modified form of the Weibull distribution. The newly proposed distribution may be called a new flexible extended Weibull distribution. Corresponding hazard rate function of the proposed distribution shows flexible (monotone and nonmonotone) shapes. Three different characterizations along with some mathematical properties are provided. We also consider the maximum likelihood estimation procedure to estimate the model parameters. For the illustrative purposes, two real applications from reliability engineering with bathtub-shaped hazard functions are analyzed. The practical applications show that the proposed model provides better fits than the other nonnested models.

The beta exponentiated Nadarajah-Haghighi distribution: theory, regression model and application

Abstract We introduce and study the beta exponentiated Nadarajah-Haghighi model, which has increasing, decreasing, upside-down bathtub and bathtub shaped hazard functions. Some of its mathematical properties are determined including a power series for the quantile function. We perform a Monte Carlo simulation study to assess the finite sample behavior of the maximum likelihood estimates of the parameters. We define a new regression model based on the new distribution. The potentiality of this regression model is proved empirically by means of a real dataset related to diabetic retinopathy study.

Failure models driven by a self-correcting point process in earthquake occurrence modeling

The long-term recurrence of strong earthquakes is often modeled according to stationary Poisson processes for the sake of simplicity. However, renewal and self-correcting point processes (with nondecreasing hazard functions) are more appropriate. Short-term models mainly fit earthquake clusters due to the tendency of an earthquake to trigger other earthquakes. In this case, self-exciting point processes with nonincreasing hazard are especially suitable. To provide a unified framework for analysis of earthquake catalogs, Schoenberg and Bolt proposed the short-term exciting long-term correcting model in 2000, and in 2005, Varini used a state-space model to estimate the different phases of a seismic cycle. Both of these analyses are combinations of long-term and short-term models, and the results are not completely satisfactory, due to the different scales at which these models appear to operate. In this study, we propose alternative modeling. First, we split a seismic sequence into two groups: the leader events, non-secondary events the magnitudes of which exceed a fixed threshold; and the remaining events, which are considered as subordinate. The leader events are assumed to follow the well-known self-correcting point process known as the stress-release model. In the interval between two subsequent leader events, subordinate events are expected to cluster at the beginning (aftershocks) and at the end (foreshocks) of that interval; hence, they are modeled by a failure process that allows bathtub-shaped hazard functions. In particular, we examined generalized Weibull distributions, as a large family that contains distributions with different bathtub-shaped hazards, as well as the standard Weibull distribution. The model is fit to a dataset of Italian historical earthquakes, and the results of Bayesian inference based on the Metropolis–Hastings algorithm are shown.

Alpha power transformed inverse Lindley distribution: A distribution with an upside-down bathtub-shaped hazard function

The inverse Lindley distribution has been generalized by many authors in recent years. Here, we introduce a new generalization called alpha power transformed inverse Lindley (APTIL) distribution that provides better fits than the inverse Lindley distribution and some of its known generalizations. The new model includes the inverse Lindley distribution as a special case. Various properties of the proposed distribution, including explicit expressions for the mode, moments, conditional moments, mean residual lifetime, Bonferroni and Lorenz curves, entropies, stochastic ordering, stress–strength reliability and order statistics are derived. The new distribution can have an upside-down bathtub failure rate function depending on its parameters. The model parameters are obtained by the method of maximum likelihood estimation. The approximate confidence intervals of the model parameters are also obtained. A simulation study is carried out to examine the performance of the maximum likelihood estimators of the parameters. Finally, two data sets have been analyzed to show how the proposed model works in practice.

A new flexible hazard rate distribution: Application and estimation of its parameters

ABSTRACTWe introduce a new class of flexible hazard rate distributions which have constant, increasing, decreasing, and bathtub-shaped hazard function. This class of distributions obtained by compounding the power and exponential hazard rate functions, which is called the power-exponential hazard rate distribution and contains several important lifetime distributions. We obtain some distributional properties of the new family of distributions. The estimation of parameters is obtained by using the maximum likelihood and the Bayesian methods under squared error, linear-exponential, and Stein’s loss functions. Also, approximate confidence intervals and HPD credible intervals of parameters are presented. An application to real dataset is provided to show that the new hazard rate distribution has a better fit than the other existing hazard rate distributions and some four-parameter distributions. Finally , to compare the performance of proposed estimators and confidence intervals, an extensive Monte Carlo simulation study is conducted.

On modified Kies distribution and its applications

In this paper, we consider a class of bathtub-shaped hazard function distribution through modifying the Kies distribution and investigate some of its important properties by deriving expressions for its percentile function, raw moments, stress-strength reliability measure etc. The parameters of the distribution are estimated by the method of maximum likelihood and discussed some of its reliability applications with the help of certain real life data sets. In addition, the asymptotic behavior of the maximum likelihood estimators of the parameters of the distribution is examined by using simulated data sets.

Change points estimations of bathtub-shaped hazard functions

The life data analysis has been finding increasing importance for every industry, resulting in higher quality and cost reduction in current fierce competitive market. The hazard function for life models have three distinct phases of burn-in, useful life and wear-out shown in the bath-tub curves. Change points estimation is important for bathtub-shaped hazard function models in reliability and life data analysis for the product developer and designers to have a relatively accurate estimation of the burn-in, useful life and onset of the wear-out life phases. The applications are for determination and assisting to plan appropriate burn-in, guarantee, maintenance, repair and replacement strategies. In this research, life time interval is studied for bathtub shaped hazard function. Two change points are calculated for burn-in and useful life phases, as well as useful life and wear out phases. Two criteria are used in this study for determination of the change point including (1) minimum of hazard function and (2) maximum change in slope of hazard function. This research is structured as a parametric approach with Bayesian inference utilized as the parameter estimation. The modified Weibull distribution are determined as the suitable models for simulation of all three life phases. In this paper, failure data of an electronic system case is used for the method demonstration.

A two-parameter discrete distribution with a bathtub hazard shape

This paper introduces a two-parameter discrete distribution based on a continuous two-parameter bathtub distribution. It is the only two-parameter discrete distribution that shows a bathtub-shaped hazard function. Some statistical properties of the distribution are discussed. Three different methods are used to estimate its two unknown parameters. The point estimators of the parameters have no closed form. The bootstrap method is used to estimate the distributions of these point estimators. Different approximations of the interval estimations for the two-parameters are discussed. Real data sets are analyzed to show how this distribution works in practice. A simulation study is performed to investigate the properties of the estimations obtained and compare their performances.

Transmuted Singh-Maddala Distribution: A new Flexible and Upside-Down Bathtub Shaped Hazard Function Distribution

The Singh-Maddala distribution is very popular to analyze the data on income, expenditure, actuarial, environmental, and reliability related studies. To enhance its scope and application, we propose four parameters transmutedSingh-Maddala distribution, in this study. The proposed distribution is relatively more flexible than the parent distribution to model a variety of data sets. Its basic statistical properties, reliability function, and behaviors of the hazard function are derived. The hazard function showed the decreasing and an upside-down bathtub shape that is required in various survival analysis. The order statistics and generalized TL-moments with their special cases such as L-, TL-, LL-, and LH-moments are also explored. Furthermore, the maximum likelihood estimation is used to estimate the unknown parameters of the transmuted Singh-Maddala distribution. The real data sets are considered to illustrate the utility and potential of the proposed model. The results indicate that the transmuted Singh-Maddala distribution models the datasets better than its parent distribution.

Reliability analysis using an exponential power model with bathtub-shaped failure rate function: a Bayes study.

Models with bathtub-shaped hazard function have been widely accepted in the field of reliability and medicine and are particularly useful in reliability related decision making and cost analysis. In this paper, the exponential power model capable of assuming increasing as well as bathtub-shape, is studied. This article makes a Bayesian study of the same model and simultaneously shows how posterior simulations based on Markov chain Monte Carlo algorithms can be straightforward and routine in R. The study is carried out for complete as well as censored data, under the assumption of weakly-informative priors for the parameters. In addition to this, inference interest focuses on the posterior distribution of non-linear functions of the parameters. Also, the model has been extended to include continuous explanatory variables and R-codes are well illustrated. Two real data sets are considered for illustrative purposes.

An additive modified Weibull distribution

A new five-parameter lifetime distribution with increasing, decreasing and bathtub-shaped hazard function is proposed in this paper. An important property of the new distribution is that it is very flexible to model the bathtub-shaped hazard rate data. For a bathtub-shaped hazard rate with sharp change in the initial and wear-out phase, the new distribution can provide a better fit than many other modified Weibull distributions. The properties of the new distribution including moments, order statistics and the Weibull type probability plot (WPP) are studied. The estimation of parameters is discussed by using the method of maximum likelihood. Finally, the superiority of the proposed distribution and its four-parameter sub-model are tested by analyzing two well-known lifetime data sets.

An extension of the generalized exponential distribution

The two-parameter generalized exponential distribution has been used recently quite extensively to analyze lifetime data. In this paper the two-parameter generalized exponential distribution has been embedded in a larger class of distributions obtained by introducing another shape parameter. Because of the additional shape parameter, more flexibility has been introduced in the family. It is observed that the new family is positively skewed, and has increasing, decreasing, unimodal and bathtub shaped hazard functions. It can be observed as a proportional reversed hazard family of distributions. This new family of distributions is analytically quite tractable and it can be used quite effectively to analyze censored data also. Analyses of two data sets are performed and the results are quite satisfactory.

The beta modified Weibull distribution

A five-parameter distribution so-called the beta modified Weibull distribution is defined and studied. The new distribution contains, as special submodels, several important distributions discussed in the literature, such as the generalized modified Weibull, beta Weibull, exponentiated Weibull, beta exponential, modified Weibull and Weibull distributions, among others. The new distribution can be used effectively in the analysis of survival data since it accommodates monotone, unimodal and bathtub-shaped hazard functions. We derive the moments and examine the order statistics and their moments. We propose the method of maximum likelihood for estimating the model parameters and obtain the observed information matrix. A real data set is used to illustrate the importance and flexibility of the new distribution.

Bayesian reliability applications of a combined lifecycle failure distribution

This research seeks to better understand how to update sectional time-to-failure (TTF) distributions, such as the Sandia National Laboratories' developed CoMBined Lifecycle (CMBL) distribution, when new operational TTF data become available. With a bathtub-shaped hazard function, the CMBL probability density function provides an application-friendly method for characterizing a component's failure or lifecycle distribution. Its five parameters are chosen specifically to make it relatively easy to elicit the probability of failure distribution from subject matter experts and limited data. Once a characterization of the component?s lifecycle in terms of failure probability is established, a methodology for how to update that characterization based on the availability of new operational failure data is required. The updating process presented here uses a Bayesian changepoint methodology to return updated CMBL distribution parameters based on new operational TTF data modelled as a Poisson process. In this methodology, the changepoints are determined first, and when combined with the counts of the TTF data, provide enough information to estimate the remaining CMBL distribution parameters. The method developed in this effort for updating the CMBL distribution and other TTF distributions should prove valuable in optimizing large scale system-of-systems supply/repair chain models.