Increasing Failure Rate Function Research Articles

Statistical inference for a two-parameter distribution with a bathtub-shaped or increasing hazard rate function based on record values and inter-record times with an application to COVID-19 data

In this paper, we study the problem of estimation and prediction for a two-parameter distribution with a bathtub-shaped or increasing failure rate function based on lower records and inter-record times, and based on lower records without considering the inter-record times. The maximum likelihood and Bayesian approaches are employed to estimate the unknown parameters. As it seems that the Bayes estimates cannot be derived in a closed form, the Metropolis-Hastings within Gibbs algorithm is implemented to obtain the approximate Bayes point estimates. Bayesian prediction of a future record value is also discussed. A simulation study is conducted to evaluate the proposed point and interval estimators. A real data set consisting of COVID-19 data from Iran is analyzed to illustrate the application of the theoretical results of the paper. Moreover, a simulated data example is presented. Several concluding remarks end the paper.

Statistical inference of joint competing risks models from comparative bathtub shape distributions with hybrid censoring

In reliability engineering studies, the class of lifetime distributions with bathtub-shaped failure rate functions is particularly important since the lifetimes of electro-mechanical, electronic, and mechanical goods are frequently modelled with this characteristic. Comparative competing risk data are obtained in a variety of applications, including engineering, biological, medical, and other related fields. In this study, we adopted the statistical inference of joint distributions with a bathtub shape or increasing failure rate function (Chen distributions). The problem of determining the relative merits of products according to their lifetime duration is discussed under the hybrid type-I censoring scheme. Also, under consideration of independent causes of failure, a survival analysis and the assessment of one risk in the presence of other risks are discussed. The maximum likelihood (ML) method and the Bayes approach relative to the symmetric loss function are used to estimate the model parameters. Additionally, we build the classical confidence intervals (CIs) (asymptotic CI and bootstrap CI) to compare with the Bayes credible intervals. Moreover, theoretical results are evaluated and contrasted using a Monte Carlo simulation study. Finally, for illustration purposes, a real data set obtained from a laboratory experiment is evaluated using the suggested model with some brief comments.

Joint design of control chart and maintenance policy under multiple assignable causes and random failures by considering the statistical constraints

ABSTRACT In modern industries, statistical process monitoring (SPM) and maintenance management are extensively employed to increase the production rate of conforming items. Aiming at minimizing the expected total cost per time unit subject to some statistical constraints, this study proposes a hybrid quality-maintenance model for imperfect short-run process. To bring the proposed model closer to real short-run systems, it is considered that the process mean may shift to an out-of-control condition due to occurrence of several types of assignable causes. Furthermore, it is supposed that the time-to-failure follows a non-homogenous Poisson process implying that the system may suddenly fail with an increasing failure rate function. Moreover, a non-uniform sampling scheme is developed in order to improve the system reliability. Finally, the main advantages of the proposed model are highlighted by conducting two comparative studies. The first one illustrates the efficiency of the non-uniform sampling scheme in increasing the in-control time interval and decreasing the expected total cost. The second one confirms the importance of considering the system failure on both the expected total cost and in-control time interval.

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.

Comparison of estimators under different loss functions for two-parameter bathtub - shaped lifetime distribution

Chen is suggested a two-parameter distribution. This distribution can have increasing failure rate function or a bathtub-shaped that allows it to fit real lifetime data sets. The ML (Maximum Likelihood) and Bayes estimates of the parameters of Chen’s distribution are constituted in this paper. The approximate values of Bayesian estimates are obtained by using the Tierney-Kadane approach. Two-parameter bathtub-shaped distribution's estimations are derived using Jeffrey's extension prior under General entropy, Squared and Linex loss functions. Besides, performances of ML and Bayes estimates are compared concerning MSE's (Mean Square Error) by using Monte Carlo simulation. As a result, it has been seen that approximate Bayes estimates obtained under linex loss function are better than others. Moreover, real data analysis for his distribution is presented.

A New Modified Kies Family: Properties, Estimation Under Complete and Type-II Censored Samples, and Engineering Applications

In this paper, we introduce a new family of continuous distributions that is called the modified Kies family of distributions. The main mathematical properties of the new family are derived. A special case of the new family has been considered in more detail; namely, the two parameters modified Kies exponential distribution with bathtub shape, decreasing and increasing failure rate function. The importance of the new distribution comes from its ability in modeling positively and negatively skewed real data over some generalized distributions with more than two parameters. The shape behavior of the hazard rate and the mean residual life functions of the modified Kies exponential distribution are discussed. We use the method of maximum likelihood to estimate the distribution parameters based on complete and type-II censored samples. The approximate confidence intervals are also obtained under the two schemes. A simulation study is conducted and two real data sets from the engineering field are analyzed to show the flexibility of the new distribution in modeling real life data.

Estimation of reliability of multicomponent stress-strength of a bathtub shape or increasing failure rate function

Purpose The purpose of this paper is to deal with the Bayesian and non-Bayesian estimation methods of multicomponent stress-strength reliability by assuming the Chen distribution. Design/methodology/approach The reliability of a multicomponent stress-strength system is obtained by the maximum likelihood (MLE) and Bayesian methods and the results are compared by using MCMC technique for both small and large samples. Findings The simulation study shows that Bayes estimates based on γ prior with absence of prior information performs little better than the MLE with regard to both biases and mean squared errors. The Bayes credible intervals for reliability are also shorter length with competitive coverage percentages than the condence intervals. Further, the coverage probability is quite close to the nominal value in all sets of parameters when both sample sizes n and m increases. Originality/value The lifetime distributions used in reliability analysis as exponential, γ, lognormal and Weibull only exhibit monotonically increasing, decreasing or constant hazard rates. However, in many applications in reliability and survival analysis, the most realistic hazard rate is bathtub-shaped found in the Chen distribution. Therefore, the authors have studied the multicomponent stress-strength reliability under the Chen distribution by comparing the MLE and Bayes estimators.

The performance assessment on the lifetime performance index of products following Chen lifetime distribution based on the progressive type I interval censored sample

In the competitive manufacturing markets, quality engineers frequently are dedicated to evaluate whether the quality performance of products met the desire level. The lifetime performance index (LPI) C L is used to monitor the larger-the-better performance of lifetimes of products. Chen lifetime distribution with bathtub shape or increasing failure rate function has many applications in the lifetime data analysis. The maximum likelihood estimator for C L is derived and used to propose a hypothesis testing procedure under a lower specification limit based on the progressive type I interval censored sample. Finally, two practical examples are given to demonstrate the application of our proposed testing procedure to assess whether the lifetime performance reached the desire level.

ON THE PLANNING OF THE SCOPE OF NEW TECHNOLOGY TESTING

This paper is a follow-up to [1]. It examines the matters of planning of the scope of highly dependable objects testing. The process of new technology development and manufacture involves determining its dependability indicators. The most objective method of identifying dependability characteristics of products is field testing. One of the widely used testing plans is the [N,U,T] plan. This plan that involves testing N nonreparable samples within the time interval between 0 and a certain T. It is assumed that during the tests k objects fail, while N-k objects successfully pass the tests. Thus, at the outcome of the experiment we have a mixed sample that includes k times to failure and N-k right censored observation. If the tested object is highly dependable it is quite possible that within the time period [0,T] failures will not happen, i.e. k will be equal to 0, therefore the probability of failure within this time period is extremely low and the number of tested objects is limited. Nevertheless even in this situation it would be desirable to be able to be in control of the accuracy of the estimation obtained during such experiments. It is clear that the accuracy of such estimation will depend not only on the number of tested objects N, but also on the experiment duration. For a fixed N, as the observation time T grows the estimation accuracy increases due to the increasing proportion of complete times, while the proportion of censored ones goes down. It should be noted that when we talk about identifying the dependability characteristics of complex and costly objects we cannot test large batches of finished products. Therefore the problem consists in defining testing duration and size of the product batch to be tested subject to specified requirements for the accuracy of estimation of dependability characteristics obtained as the result of the tests. The scope planning is based on the manufacturer’s requirement to validate the lower bound of the probability of no failure P0 with a specified confidence level at a certain time point t0. The aim of the paper is to identify the test scope of a batch of finished products N(T) under the condition of fulfilment of the manufacturer’s requirement for compliance with the lower confidence bound of the probability of no failure with a specified confidence level 1 – α. Three failure distributions are under examination: exponential distribution law, Weibull distribution and distribution with linear rate function. The considered types of distribution law enable the research of objects with decreasing, constant and increasing failure rate function. Methods. In this paper the authors deduce formulas for calculation of the scope of experiment for a number of experiment durations. The estimates are obtained using the maximum likelihood method and methods of researching asymptotic properties of estimates through the Fisher information quantity. Conclusions. The findings allow for a substantiated approach to planning the scope of highly dependable objects testing. It is shown that the longer is the experiment duration the fewer products must be supplied for testing. The dependence is non-linear, close to hyperbolic and is conditioned by both the input parameters and the parametrization of the failure rate function.

Transmuted exponentiated Chen distribution with application to survival data

This article considers an extension of the exponentiated Chen distribution based on the quadratic rank transmutation map technique. Some structural properties of the transmuted exponentiated Chen distribution are discussed. The estimation procedure is performed using the method of maximum likelihood. Finally, the flexibility of the new distribution is illustrated using strengths of glass fibres data and nicotine in cigarettes data. References G. R. Aryal and C. P. Tsokos. Transmuted Weibull distribution: A generalization of the Weibull probability distribution. Euro. J. Pure Appl. Math. , 4(2):89–102, 2011. http://www.ejpam.com/index.php/ejpam/article/view/1170 Z. Chen. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Stats. Prob. Lett. , 49:155–161, 2000. doi:10.1016/S0167-7152(00)00044-4 Y. P. Chaubey and R. Zhang. An extension of Chen's family of survival distributions with bathtub shape or increasing hazard rate function. Commun. Stat. A: Theor. , 44(19):4049–4064, 2015. doi:10.1080/03610926.2014.997357 M. S. Khan and R. King. Transmuted modified Weibull distribution: A generalization of the modified Weibull probability distribution. Euro. J. Pure Appl. Math. , 6(1):66–88, 2013. http://www.ejpam.com/index.php/ejpam/article/view/1606 M. S. Khan, R. King and I. Hudson. Characterizations of the transmuted inverse Weibull distribution. ANZIAM J , 55:C197–C217, 2014. doi:10.21914/anziamj.v55i0.7785 M. S. Khan and R. King. A new class of transmuted inverse Weibull distribution for reliability analysis. Am. J. Math. Manage. Sci. . 33(4):261–286, 2014. doi:10.1080/01966324.2014.929989 M. S. Khan, R. King and I. Hudson. A new three parameter transmuted Chen lifetime distribution with application. J. Appl. Stat. Sci. , 21(3) 2015. https://www.novapublishers.com/catalog/product_info.php?products_id=54720 F. Merovci. Transmuted Rayleigh distribution. Austrian J. Stat. , 42(1):21–31, 2013. http://www.ajs.or.at/index.php/ajs/article/view/vol42,%20no1-2 R. L. Smith and J. C. Naylor. A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution. J. Roy. Stat. Soc. C–App. , 36:358–369, 1987. doi: 10.2307/2347795 W. T. Shaw and I. R. C. Buckley. The alchemy of probability distributions: Beyond Gram–Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map. Technical report, 2009. http://arxiv.org/abs/0901.0434 Federal Trade Commission (USA). Results of tar, nicotine and carbon monoxide testing for 1,249 varieties of domestic cigarettes sold in 1995. File No. 962 3099, Jan. 1998. https://www.ftc.gov/news-events/press-releases/1998/01/results-tar-nicotine-and-carbon-monoxide-testing-1249-varieties

Estimation and prediction for Chen distribution with bathtub shape under progressive censoring

ABSTRACTWe consider estimation of the unknown parameters of Chen distribution [Chen Z. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statist Probab Lett. 2000;49:155–161] with bathtub shape using progressive-censored samples. We obtain maximum likelihood estimates by making use of an expectation–maximization algorithm. Different Bayes estimates are derived under squared error and balanced squared error loss functions. It is observed that the associated posterior distribution appears in an intractable form. So we have used an approximation method to compute these estimates. A Metropolis–Hasting algorithm is also proposed and some more approximate Bayes estimates are obtained. Asymptotic confidence interval is constructed using observed Fisher information matrix. Bootstrap intervals are proposed as well. Sample generated from MH algorithm are further used in the construction of HPD intervals. Finally, we have obtained prediction intervals and estimates for future observations in one- and two-sample situations. A numerical study is conducted to compare the performance of proposed methods using simulations. Finally, we analyse real data sets for illustration purposes.

The lower-bound estimate of reliability based on the results of accelerated tests

The problem of the assessment of reliability characteristics is considered on the basis of the results of accelerated tests under a variable monotonically increasing load. A series of lower-bound estimates of the reliability function have been obtained for parametric and nonparametric models under the assumption that under constant loads the distribution of the failure-free time of a system “ages” with a monotonically increasing failure rate function.

Statistical Analysis of a Weibull Extension with Bathtub-Shaped Failure Rate Function

We consider the parameter inference for a two-parameter life distribution with bathtub-shaped or increasing failure rate function. We present the point and interval estimations for the parameter of interest based on type-II censored samples. Through intensive Monte-Carlo simulations, we assess the performance of the proposed estimation methods by a comparison of precision. Example applications are demonstrated for the efficiency of the methods.

A New Two-Parameter Lifetime Distribution with Bathtub, Up-Bathtub or Increasing Failure Rate Function

It is a common situation that the failure rate function has a bathtub shape for many mechanical and electronic components. A simple model based on the median of tree /or four identical independent random variables drown from the well known power distribution is presented for modeling this type of data. The failure rate can also be ups ide-down bathtub shaped or increasing. Most properties of the proposed distribution are investigated. Estimation procedures are intro duced as well as a graphical approach via probability plot. An application to a real data set is presented; and a simulation study is provid ed. Finally, some concluding remarks are presented.

A New Two-Parameter Lifetime Distribution with Bathtub, Up-Bathtub or Increasing Failure Rate Function

A New Two-Parameter Lifetime Distribution with Bathtub, Up-Bathtub or Increasing Failure Rate Function

General results for the Kumaraswamy-G distribution

For any continuous baseline G distribution [G.M. Cordeiro and M. de Castro, A new family of generalized distributions, J. Statist. Comput. Simul. 81 (2011), pp. 883–898], proposed a new generalized distribution (denoted here with the prefix ‘Kw-G’ (Kumaraswamy-G)) with two extra positive parameters. They studied some of its mathematical properties and presented special sub-models. We derive a simple representation for the Kw-G density function as a linear combination of exponentiated-G distributions. Some new distributions are proposed as sub-models of this family, for example, the Kw-Chen [Z.A. Chen, A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Statist. Probab. Lett. 49 (2000), pp. 155–161], Kw-XTG [M. Xie, Y. Tang, and T.N. Goh, A modified Weibull extension with bathtub failure rate function, Reliab. Eng. System Safety 76 (2002), pp. 279–285] and Kw-Flexible Weibull [M. Bebbington, C.D. Lai, and R. Zitikis, A flexible Weibull extension, Reliab. Eng. System Safety 92 (2007), pp. 719–726]. New properties of the Kw-G distribution are derived which include asymptotes, shapes, moments, moment generating function, mean deviations, Bonferroni and Lorenz curves, reliability, Rényi entropy and Shannon entropy. New properties of the order statistics are investigated. We discuss the estimation of the parameters by maximum likelihood. We provide two applications to real data sets and discuss a bivariate extension of the Kw-G distribution.

Parameter estimation for a two-parameter bathtub-shaped lifetime distribution

A two-parameter distribution was revisited by Chen (2000) [7]. This distribution can have a bathtub-shaped or increasing failure rate function which enables it to fit real lifetime data sets. Maximum likelihood and Bayes estimates of the two unknown parameters are discussed in this paper. It is assumed in the Bayes case that the unknown parameters have gamma priors. Explicit forms of Bayes estimators cannot be obtained. Different approximations are used to establish point estimates and two sided Bayesian probability intervals for the parameters. Monte Carlo simulations are applied to the comparison between the maximum likelihood estimates and the approximate Bayes estimates obtained under non-informative prior assumptions. Analysis of a real data set is also been presented for illustrative purposes.

Statistical inferences of a two-parameter distribution with the bathtub shape based on progressive censored sample

This study considers the hypothesis test and interval estimation of the shape parameter of a new two-parameter distribution with the bathtub shape or increasing failure rate function under type-II progressive censoring based on the proposed m pivotal quantities. The confidence region of two parameters of this distribution is also proposed. We give one numerical example to illustrate the proposed methods. Next, the Monte Carlo simulation is done to assess the behaviour of these pivotal quantities. Lastly, we find the optimal pivotal quantity based on the criteria of highest power, minimum confidence length and the minimum confidence region.

The exact hypothesis test for the shape parameter of a new two-parameter distribution with the bathtub shape or increasing failure rate function under progressive censoring with random removals

This study considers the exact hypothesis test for the shape parameter of a new two-parameter distribution with the shape of a bathtub or increasing failure rate function under type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial or a uniform distribution. Several test statistics are proposed and one numerical example is provided to illustrate the proposed hypothesis test for the shape parameter. Finally, a simulation study is performed to compare the power performances of all proposed test statistics. We concluded that the test statistic w 1 is more attractive than other methods as it has better performance than other test statistics for most cases based on the criteria of maximum power.

Optimal corrective maintenance contract planning for aging multi‐state system

AbstractThis paper considers an aging multi‐state system, where the system failure rate varies with time. After any failure, maintenance is performed by an external repair team. Repair rate and cost of each repair are determined by a corresponding corrective maintenance contract with a repair team. The service market can provide different kinds of maintenance contracts to the system owner, which also can be changed after each specified time period. The owner of the system would like to determine a series of repair contracts during the system life cycle in order to minimize the total expected cost while satisfying the system availability. Operating cost, repair cost and penalty cost for system failures should be taken into account. The paper proposes a method for determining such optimal series of maintenance contracts. The method is based on the piecewise constant approximation for an increasing failure rate function in order to assess lower and upper bounds of the total expected cost and system availability by using Markov models. The genetic algorithm is used as the optimization technique. Numerical example is presented to illustrate the approach. Copyright © 2009 John Wiley & Sons, Ltd.