Bathtub Shape Research Articles

<b>Ristić-Balakrishnan extended exponential distribution

In this paper, we introduce and study a new generalization of the extended exponential distribution, called the Ristić-Balakrishnan extended exponential distribution. The new model adds one parameter in the baseline model and its failure rate function can accommodate both inverted bathtub and bathtub shapes. Important distributions are obtained as a special case of our model, such as exponential and Lindley distributions. The main purpose is to define a new flexible distribution with great power adjustment to survival data sets. For this reason, we provide a comprehensive mathematical treatment of the new model. Furthermore, we use a real data set that proves empirically the power of adjustment of the new distribution compared to other competitive models in the literature.

Generalized Version of Complementary Lindley Power Series Distribution

In this paper, we shall introduce a new class of generalized complementary compound lifetime distributions which is obtained by compounding generalized Lindley distribution with power series distribution. This new family of continuous lifetime distributions so obtained will be called Complementary Generalized Lindley Power Series (CGLPS) distribution. The proposed class of distribution contains several lifetime distributions as its special cases that are very flexible to accommodate different types of data sets since the probability density function and hazard rate can take up different forms such as increasing, decreasing and upside down bathtub shapes which have been shown through graphs for some selected values of parameters and the potentiality proposed class has been tested statistically by using it to model some real life data set.

Posterior Properties of the Nakagami-m Distribution Using Noninformative Priors and Applications in Reliability

The Nakagami-m distribution plays an important role in problems related to communication engineering. This distribution can also be used for modeling reliability data as its hazard rate (mean residual life) function presents increasing (decreasing) or bathtub (unimodal) shapes. In this study, a Bayesian inference considering objective priors for the Nakagami-m distribution parameters is presented. We propose a theorem with sufficient and necessary conditions so that a general class of posteriors are proper. The impropriety of these posteriors can be observed by the behavior of the objective priors. This theorem is applied to different objective priors such as Jeffreys' rule, Jeffreys prior, maximal data information prior, and reference priors. Simulation studies were conducted to investigate the performance of the Bayes estimators. Finally, our methodology is illustrated using two real lifetime datasets showing that the Nakagami-m distribution can be used to describe lifetime data.

A New Family of Generalized Distributions Based on Alpha Power Transformation with Application to Cancer Data

In this paper, we propose a new method for generating distributions based on the idea of alpha power transformation introduced by Mahdavi and Kundu (Commun Stat Theory Methods 46(13):6543–6557, 2017). The new method can be applied to any distribution by inverting its quantile function as a function of alpha power transformation. We apply the proposed method to the Weibull distribution to obtain a three-parameter alpha power within Weibull quantile function. The new distribution possesses a very flexible density and hazard rate function shapes which are very useful in cancer research. The hazard rate function can be increasing, decreasing, bathtub or upside down bathtub shapes. We derive some general properties of the proposed distribution including moments, moment generating function, quantile and Shannon entropy. The maximum likelihood estimation method is used to estimate the parameters. We illustrate the applicability of the proposed distribution to complete and censored cancer data sets.

Weighted Weibull distribution: Bivariate and multivariate cases

Gupta and Kundu (Statistics 43 (2009) 621–643) introduced a new class of weighted exponential distribution and established its several properties. The probability density function of the proposed weighted exponential distribution is unimodal and it has an increasing hazard function. Following the same line Shahbaz, Shahbaz and Butt (Pak. J. Stat. Oper. Res. VI (2010) 53–59) introduced weighted Weibull distribution, and we derive several new properties of this weighted Weibull distribution. The main aim of this paper is to introduce bivariate and multivariate distributions with weighted Weibull marginals and establish their several properties. It is shown that the hazard function of the weighted Weibull distribution can have increasing, decreasing and inverted bathtub shapes. The proposed multivariate model has been obtained as a hidden truncation model similarly as the univariate weighted Weibull model. It is observed that to compute the maximum likelihood estimators of the unknown parameters for the proposed $p$-variate distribution, one needs to solve $(p+2)$ non-linear equations. We propose to use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters. We obtain the observed Fisher information matrix, which can be used for constructing asymptotic confidence intervals. One data analysis has been performed for illustrative purposes, and it is observed that the proposed EM algorithm is very easy to implement, and the performance is quite satisfactory.

A new lifetime distribution with applications in inventory and insurance

PurposeThe purpose of this paper is to introduce a new probability density function having both unbounded and bounded support with a wider applicability. While the distribution with bounded support on [0, 1] has applications in insurance and inventory management with ability to fit risk management data on proportions better than existing bounded distributions, the same with unbounded support is used as a lifetime model and is considered as an attractive alternative to some existing models in the reliability literature.Design/methodology/approachThe new density function, called modified exponential-geometric distribution is derived from the exponential-geometric distribution introduced by Adamidis and Loukas (1998). The support of the density function is shown to be both unbounded and bounded depending on the values of one of the shape parameters. Various properties of the density function are studied in detail and the parameters are estimated through maximum likelihood method of estimation. A number of applications related to reliability, insurance and inventory management are exhibited along with some useful data analysis.FindingsA single probability distribution with both unbounded and bounded support, which does not seem to exist in the reliability literature, is introduced in this paper. The proposed density function exhibits varying shapes including U-shape, and the failure rate also shows increasing, decreasing and bathtub shapes. The Monte Carlo simulation shows that the estimates of the parameters are quite stable with low standard errors. The distribution with unbounded support is shown to have competitive features for lifetime modeling through analysis of two data sets. The distribution with bounded support on [0, 1] is shown to have application in insurance and inventory management and is found to t data on proportions related to risk management better than some existing bounded distributions.Originality/valueThe authors introduce an innovative probability distribution which contributes significantly in insurance and inventory management besides its remarkable statistical and reliability properties.

A generalization of discrete Weibull distribution

ABSTRACTIn this paper, we introduce a new family of discrete distributions and study its properties. It is shown that the new family is a generalization of discrete Marshall-Olkin family of distributions. In particular, we study generalized discrete Weibull distribution in detail. Discrete Marshall-Olkin Weibull distribution, exponentiated discrete Weibull distribution, discrete Weibull distribution, discrete Marshall-Olkin generalized exponential distribution, exponentiated geometric distribution, generalized discrete exponential distribution, discrete Marshall-Olkin Rayleigh distribution and exponentiated discrete Rayleigh distribution are sub-models of generalized discrete Weibull distribution. We derive some basic distributional properties such as probability generating function, moments, hazard rate and quantiles of the generalized discrete Weibull distribution. We can see that the hazard rate function can be decreasing, increasing, bathtub and upside-down bathtub shape. Estimation of the parameters are done using maximum likelihood method. A real data set is analyzed to illustrate the suitability of the proposed model.

A new extension of Weibull distribution: Properties and different methods of estimation

The Weibull distribution has been generalized by many authors in recent years. Here, we introduce a new generalization of the Weibull distribution, called Alpha logarithmic transformed Weibull distribution that provides better fits than some of its known generalizations. The proposed distribution contains Weibull, exponential, logarithmic transformed exponential and logarithmic transformed Weibull distributions as special cases. Our main focus is the estimation from frequentist point of view of the unknown parameters along with some mathematical properties of the new model. The proposed distribution accommodates monotonically increasing, decreasing, bathtub and unimodal and then bathtub shape hazard rates, so it turns out to be quite flexible for analyzing non-negative real life data. We briefly describe different frequentist approaches, namely, maximum likelihood estimators, percentile based estimators, least squares estimators, weighted least squares estimators, maximum product of spacings estimators and compare them using extensive numerical simulations. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples. The potentiality of the distribution is analyzed by means of two real data sets.

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.

The Extended Burr XII Distribution with Variable Shapes for the Hazard Rate

We define and study a new continuous distribution called the exponentiated Weibull Burr XII. Its density function can be expressed as a linear mixture of Burr XII. Its hazard rate is very flexibile in accomodating various shapes including constant, decreasing, increasing, J-shape, unimodal or bathtub shapes. Various of its structural properties are investigated including explicit expressions for the ordinary and incomplete moments, generating function, mean residual life, mean inactivity time and order statistics. We adopted the maximum likelihood method for estimating the model parameters. The flexibility of the new family is illustrated by means of a real data application.

Transmuted Kumaraswamy-G Family of Distributions for Modelling Reliability Data

Abstract This research introduced the new class of distributions, called the transmuted Kumaraswamy (TKw) G family for modelling life testing problems. The new extended family is obtained by using the quadratic rank transmutation map method, which possesses a bathtub shape for its hazard rate. Some special models in the new family are discussed. We studied some mathematical properties of the new family, including the ordinary moments, moment generating functions, probability weighted moment, quantile function, and formulated the pdf of rth order statistics. The TKw G-distribution parameters are estimated using the method of maximum likelihood. We illustrate the potentiality of the new family with two applications to the failure and service times of Aircraft windshield data.

Bayesian failure-rate modeling and preventive maintenance optimization

New results are derived for the optimal preventive maintenance schedule of a single item over a finite horizon, based on Bayesian models of a failure rate function. Two types of failure rate functions—increasing and bathtub shapes—are considered. For both cases, optimality conditions and efficient algorithms to find an optimal maintenance schedule are given. A Bayesian parametric model for bathtub-shaped failure rate functions is used, while the class of increasing failure rate functions are tackled by an extended gamma process. We illustrate both approaches using real failure time data from the South Texas Project Nuclear Operating Company in Bay City, Texas.

Exponentiated Weibull regression for time-to-event data.

The Weibull, log-logistic and log-normal distributions are extensively used to model time-to-event data. The Weibull family accommodates only monotone hazard rates, whereas the log-logistic and log-normal are widely used to model unimodal hazard functions. The increasing availability of lifetime data with a wide range of characteristics motivate us to develop more flexible models that accommodate both monotone and nonmonotone hazard functions. One such model is the exponentiated Weibull distribution which not only accommodates monotone hazard functions but also allows for unimodal and bathtub shape hazard rates. This distribution has demonstrated considerable potential in univariate analysis of time-to-event data. However, the primary focus of many studies is rather on understanding the relationship between the time to the occurrence of an event and one or more covariates. This leads to a consideration of regression models that can be formulated in different ways in survival analysis. One such strategy involves formulating models for the accelerated failure time family of distributions. The most commonly used distributions serving this purpose are the Weibull, log-logistic and log-normal distributions. In this study, we show that the exponentiated Weibull distribution is closed under the accelerated failure time family. We then formulate a regression model based on the exponentiated Weibull distribution, and develop large sample theory for statistical inference. We also describe a Bayesian approach for inference. Two comparative studies based on real and simulated data sets reveal that the exponentiated Weibull regression can be valuable in adequately describing different types of time-to-event data.

Estimation of P (Y < X) for Two-Parameter Bathtub Shape Distribution Using Records: Bayesian and Non-Bayesian Approaches

Estimation of <i>P</i> (<i>Y</i><i> &lt;</i><i> X</i>) for Two-Parameter Bathtub Shape Distribution Using Records: Bayesian and Non-Bayesian Approaches

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.

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

A fractal model for real gas transport in porous shale

A model for real gas flow in shale gas matrices is proposed and consists of two main steps: (a) developing a microscopic (single pore) model for a real gas flow by generalizing our previously reported Extended Navier‐Stokes Equations (ENSE) method and (b) by using fractal theory concepts, up‐scaling the single pore model to the macroscopic scale. A prominent feature of the up‐scaled model is a predictor for the apparent permeability (AP). Both models are successfully validated with experimental data. The impact of the deviation of the gas behavior from ideality (real gas effect) on the gas transport mechanisms is investigated. The effect of the structural parameters (porosity Ф, the maximum pore diameter Dmax, and the minimum pore diameter Dmin) of the shale matrix on the apparent permeability is studied and a sensitivity analysis is performed to evaluate the significance of the parameters for gas transport. We find that (1) the real gas transport models for a single pore and porous shale matrix are both reliable and reasonable; (2) the real gas effect affects the thermodynamic parameters of the free gas and the adsorption and transport capacity of the adsorbed gas; (3) the real gas effect decreases the effective permeability for convective flow and surface diffusion; i.e., the derivation degree of the effective permeability for bulk diffusion and Knudsen diffusion increases with increasing pressure but presents a bathtub shape when the pore diameter is smaller than 10 nm; and (4) the apparent permeability increases with Ф, Dmax, and Dmin. It is more sensitive to Dmax, followed by the porosity. Dmin has a minor impact. © 2016 American Institute of Chemical Engineers AIChE J, 63: 1430–1440, 2017

Erlang–Lindley distribution

ABSTRACTIn this paper, Erlang–Lindley distribution (ErLD) is proposed which offers a more flexible model for waiting time data. It has the property that it can accommodate increasing, bathtub, and inverted bathtub shapes. Several statistical and reliability properties are derived and studied. The moments, its associated measures, and the limiting distributions of order statistics are derived. The model parameters are estimated by maximum likelihood and method of moments. An application of the proposed distribution to some waiting time data shows that it can give a better fit than other important lifetime models.

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.

Change-point analysis with bathtub shape for the exponential distribution

ABSTRACTLikelihood ratio type test statistic and Schwarz information criterion statistics are proposed for detecting possible bathtub-shaped changes in the parameter in a sequence of exponential distributions. The asymptotic distribution of likelihood ratio type statistic under the null hypothesis and the testing procedure based on Schwarz information criterion are derived. Numerical critical values and powers of two methods are tabulated for certain selected values of the parameters. The tests are applied to detect the change points for the predator data and Stanford heart transplant data.