Bathtub Shape Research Articles

Developments of Efficient Trigonometric Quantile Regression Models for Bounded Response Data

The choice of an appropriate regression model for econometric modeling minimizes information loss and also leads to sound inferences. In this study, we develop four quantile regression models based on trigonometric extensions of the unit generalized half-normal distributions for the modeling of a bounded response variable defined on the unit interval. The desirable shapes of these distributions, such as left-skewed, right-skewed, reversed-J, approximately symmetric, and bathtub shapes, make them competitive models for bounded responses with such traits. The maximum likelihood method is used to estimate the parameters of the regression models, and Monte Carlo simulation results confirm the efficiency of the method. We demonstrate the utility of our models by investigating the relationship between OECD countries’ educational attainment levels, labor market insecurity, and homicide rates. The diagnostics reveal that all our models provide a good fit to the data because the residuals are well behaved. A comparative analysis of the trigonometric quantile regression models with the unit generalized half-normal quantile regression model shows that the trigonometric models are the best. However, the sine unit generalized half-normal (SUGHN) quantile regression model is the best overall. It is observed that labor market insecurity and the homicide rate have significant negative effects on the educational attainment values of the OECD countries.

A New Exponentiated Beta Burr Type X Distribution: Model, Theory, and Applications

In recent years, many attempts have been carried out to develop the Burr type X distribution, which is widely used in fitting lifetime data. These extended Burr type X distributions can model the hazard function in decreasing, increasing and bathtub shapes, except for unimodal. Hence, this paper aims to introduce a new continuous distribution, namely exponentiated beta Burr type X distribution, which provides greater flexibility in order to overcome the deficiency of the existing extended Burr type X distributions. We first present its density and cumulative function expressions. It is then followed by the mathematical properties of this new distribution, which include its limit behaviour, quantile function, moment, moment generating function, and order statistics. We use maximum likelihood approach to estimate the parameters and their performance is assessed via a simulation study with varying parameter values and sample sizes. Lastly, we use two real data sets to illustrate the performance and flexibility of the proposed distribution. The results show that the proposed distribution gives better fits in modelling lifetime data compared to its sub-models and some extended Burr type X distributions. Besides, it is very competitive and can be used as an alternative model to some nonnested models. In summary, the proposed distribution is very flexible and able to model various shaped hazard functions, including the increasing, decreasing, bathtub, and unimodal.

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

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

New Lifetime Distribution for Modeling Data on the Unit Interval: Properties, Applications and Quantile Regression

Probability distributions are very useful in modeling lifetime datasets. However, no specific distribution is suitable for all kinds of datasets. In this study, the bounded truncated Cauchy power exponential distribution is proposed for modeling datasets on the unit interval. The probability density function exhibits desirable shapes, such as left-skewed, right-skewed, reversed J, and bathtub shapes, whereas the hazard rate function displays J and bathtub shapes. For the purpose of modeling dependence between measures in a dataset, a bivariate extension of the proposed distribution is developed. The bivariate probability density function displays monotonic and non-monotonic shapes, making it suitable for modeling complex bivariate relations. Subsequently, the applications of the distribution are illustrated using COVID-19 data. The results revealed that the new distribution provides a better fit to the datasets compared to other existing distributions. Finally, a new quantile regression model is developed and its application demonstrated. The generated quantile regression model offers a decent fit to the data, according to the residual analysis.

SciSpot: Scientific Computing On Temporally Constrained Cloud Preemptible VMs

Scientific computing applications are being increasingly deployed on cloud computing platforms. Transient servers such as EC2 spot instances and Google Preemptible VMs, can be used to lower the costs of running applications on the cloud by up to <inline-formula><tex-math notation="LaTeX">$10\times$</tex-math></inline-formula> . However, the frequent preemptions and resource heterogeneity of these transient servers introduces many challenges in their effective and efficient use. In this paper, we develop techniques for modeling and mitigating preemptions of transient servers, and present SciSpot, a software framework that enables low-cost scientific computing on the cloud. SciSpot deploys applications on Google Cloud Preemptible Virtual Machines that exhibit temporally constrained preemptions: VMs are always preempted in a 24 hour interval. Our empirical analysis shows that the preemption rate is generally bathtub shaped, which raises multiple fundamental challenges in performance modeling and policy design. We develop a new reliability model for temporally constrained preemptions, and use statistical mechanics to show why the bathtub shape is generally exhibited. SciSpot’s design is guided by our observation that many emerging scientific computing applications that integrate machine learning with simulations, can be deployed as “bags” of jobs, which represent multiple instantiations of the same computation with different physical model parameters. For a bag of jobs, SciSpot finds the optimal transient server on-the-fly, by taking into account the price, performance, and preemption rates of different servers. SciSpot reduces costs by <inline-formula><tex-math notation="LaTeX">$5\times$</tex-math></inline-formula> compared to conventional cloud deployments, and reduces makespans by up to <inline-formula><tex-math notation="LaTeX">$10\times$</tex-math></inline-formula> compared to conventional high performance computing clusters.

On the Coronavirus Disease Death Rate Modeling Utilizing Generalized Exponential Kumaraswamy

This paper aims to model COVID-19 in different countries, including Iran, Canada, Italy, and Mexico. A novel five-parameter lifetime distribution termed the Odd generalized exponential Kumaraswamy-inverse exponential distribution (OEKIE) is presented by combining the Kumaraswamy-inverse exponential distribution with the odd generalized exponential generator. The theoretical features of the new distribution, as well as its reliability functions, moments, and order statistics are investigated. The novel distribution has numerous advantages, since its density has a variety of symmetric and asymmetric forms. Furthermore, The graphs of the hazard rate function exhibit various asymmetrical shapes such as decreasing, increasing, upside-down bathtub shapes, and inverted J-shapes making OEKIE suitable for modeling hazards behaviors more likely to be observed in practical settings like human mortality, and biological applications. The OEKIE’s parameters are estimated using the maximum likelihood approach. The effectiveness of OEKIE is demonstrated through both numerical study and applications to four COVID-19 mortality rate data sets. The OEKIE provides best fits to COVID-19 data compared to other extended forms of the Kumaraswamy and inverse exponential distributions which may attract wider applications in different fields.

Effects of piston shape and nozzle specifications on part-load operation of natural gas–diesel dual-fuel RCCI engine and its application to high load extension

• Effects of piston shape and multi-angle nozzle were evaluated for DF RCCI operation. • Enhanced air utilization w/multi-angle nozzle reduced combustion and HT losses. • The highest GIE of RCCI improved as 51.8 % by hardware modifications. • Intake air amount could be increased by lowering CR (from 17 to 15) at the maximum load. • The maximum load of RCCI could be extended from 81 (reference) to 101 kW of BP. In this study, the effects of modifications of both the piston shape and injector nozzle are experimentally investigated for high load extension of a natural gas–diesel dual-fuel reactivity-controlled compression ignition (RCCI) engine. First, a bathtub shape with a reduced compression ratio is applied to the modified piston which was reported to improve the flame propagation in squish volume, the cooling loss, and the air induction rate under the same limit of in-cylinder pressure from previous studies. In addition, a multiangle two-layer injector is used to extend the maximum load, which has been proven to be effective for unburned hydrocarbon emissions and gross indicated efficiency in part-load operation in our previous study. The experiments are conducted under both 1200 rpm part-load operation (approximately 1950 J/cylinder per cycle of fuel input energy) and 1800 rpm maximum load operation. The objectives are to comprehensively evaluate the effects of the modifications on natural gas–diesel RCCI operation and verify the effects of all load conditions. The experimental results reveal that the changes in the piston—reduction in the compression ratio and a bathtub piston shape—are effective for both NOx and smoke emissions as the diesel injection timing is advanced. Although the multi-hole double layer (combined narrow and wide injection) has less effect on emissions compared to that of the change in piston, it affects the mixing process inside the cylinder, thus retarding the combustion phasing, prolonging the combustion duration, and eventually reducing the cooling loss. Both modifications are highly effective for advanced diesel injection timing corresponding to RCCI operation, which extends high-load operation from 80.9 to 101 kW of brake power, relatively 24.8 % increase, by adopting both the piston and injector modifications. Based on the modifications, the mass fraction of natural gas is suggested to be approximately 80 %. A higher fraction causes the variation in the maximum in-cylinder pressure to be extremely sensitive to allow robust operation. In contrast, if it is lower, the amounts of both NOx and smoke emissions exceed the emission standard levels (0.4 g/kWh and 15 mg/kWh for NOx and smoke emissions, respectively, corresponding to Stage-V of non-road engine operation).

Bayesian Life Analysis of Generalized Chen's Population Under Progressive Censoring

Chen's model with bathtub shape provides an appropriate conceptual for the hazard rate of various industrial products and clinical cases. This article deals with the problem of estimating the model parameters, reliability and hazard functions of a three-parameter Chen distribution based on progressively Type-II censored sample have been obtained. Based on the s-normal approximation to the asymptotic distribution of the maximum likelihood estimates and log-transformed maximum likelihood estimates, the approximate confidence intervals for the unknown parameters, and any function of them, are constructed. Using independent gamma conjugate priors, the Bayes estimators of the unknown parameters and reliability characteristics are derived under different versions of a symmetric squared error loss functions. However, the Bayes estimators are obtained in a complex form, so we have been used Metropolis-Hastings sampler procedure to carry out the Bayes estimates and also to construct the corresponding credible intervals. To assess the performance of the proposed estimators, numerical results using Monte Carlo simulation study were reported. To determine the optimum censoring scheme among different competing censoring plans, some optimality criteria have been considered. A practical example using real-life data set, representing the survival times of head and neck cancer patients, is discussed to demonstrate how the applicability of the proposed methods in real phenomenon.

Alpha Power Odd Generalized Exponential Family of Distributions: Model, Properties and Applications

In this paper, we exhibit a general family of distributions called alpha power odd generalized exponential family of distributions. The new family is very flexible with increasing, decreasing, J, reversed-J, bathtub shapes. Statistical properties of the family such as quantile, expansion of density function, moments, incomplete moments, mean deviation, Bonferroni and Lorenz curves are proposed. The method of maximum likelihood to estimate the model parameters is used. Three applications based on real data sets are the importance and flexibility of the three proposed models.

The gamma power half-logistic distribution: theory and applications

In this paper, we propose a new three-parameter distribution defined on the positive real line, called the gamma power half-logistic distribution. It constitutes an extension of the power half-logistic distribution using the gamma generated mechanism. The capabilities of the parent distribution are thus improved in several aspects. In particular, the hazard rate function now presents increasing failure, decreasing failure, and bathtub shapes, which are demanded characteristics in the context of statistical modelling. Other features related to the quantiles, skewness, kurtosis, moments, incomplete moments, mean deviations, Bonferroni and Lorenz curves, stochastic ordering, reliability parameter, and distributions of order statistics are also discussed. Subsequently, the gamma power half-logistic model is investigated using real-world results. We use the classical maximum likelihood method for estimating the model parameters, with a simulation trial demonstrating the effectiveness of the method for large enough sample sizes. Then, four real-life data sets of different sizes are used for the concrete application of the model, demonstrating its superiority in fitting compared to similar models.

Analysis of Wilson‐Hilferty distribution under progressive Type‐II censoring

AbstractA new re‐parameterize form of the Wilson‐Hilferty distribution for data modelling with increasing, decreasing and bathtub shape hazard rates has been considered. This paper takes into account the estimation for the unknown model parameters, reliability function and hazard function based on two frequentist methods and Bayesian method of estimation using Type‐II progressively censored data. In frequentist method, besides conventional likelihood based estimation, another competitive method, known as maximum product of spacing (MPS) method is proposed to estimate the model parameters, reliability function and hazard function as an alternative approach to the common likelihood method. In Bayesian paradigm, we have also considered the MPS function as an alternative to the traditional likelihood function and both are also discussed under the Bayesian set up for unknown parameters, reliability function and hazard function. Moreover, for all considered unknown quantities, the approximate confidence intervals under the proposed frequentist approaches as well as the Bayes credible intervals are constructed. Extensive Monte‐Carlo simulation studies are conducted to evaluate the performance of the proposed estimates with respect to various criteria quantities. Furthermore, we discuss an optimal progressive censoring plan among different competing censoring plans using three optimality criteria. Finally, to show the applicability of the proposed methodologies in a real‐life scenario, an engineering dataset is analysed.

Modeling COVID-19 mortality data in four countries using odd generalized exponential Kumaraswamy-Inverse exponential distribution

This study aims to introduce an optimum model to assess the COVID-19 death rate in Saudi Arabia, Canada, Italy, and Mexico. A novel five-parameter lifetime distribution termed the Odd generalized exponential Kumaraswamy-inverse exponential distribution is presented by combining the Kumaraswamy-inverse exponential distribution with the odd generalized exponential generator. The theoretical features of the new distribution, as well as its reliability functions, moments, and order statistics are investigated. The odd generalized exponential Kumaraswamy-inverse exponential distribution is of special importance since its density has a variety of symmetric and asymmetric forms. Furthermore, the graphs of the hazard rate function exhibit various asymmetrical shapes such as decreasing, increasing, and upside-down bathtub shapes, and inverted J-shapes making The Odd generalized exponential Kumaraswamy-inverse exponential distribution suitable for modeling hazards behaviors more likely to be observed in practical settings like human mortality, and biological applications. The proposed distribution parameters are estimated using the maximum likelihood approach and its effectiveness is demonstrated through both numerical study and applications to four COVID-19 mortality rate data sets. The Odd generalized exponential Kumaraswamy-inverse exponential distribution provides the best fit to COVID-19 data compared to other extended forms of the Kumaraswamy and inverse exponential distributions which may attract wider applications in different fields.

Inverted Power Ishita Distribution and Its Application to Lifetime Data

In this paper, a new distribution, named ‘the Inverted Power Ishita distribution’, was introduced. It is an extension of the Ishita distribution and its capable of modelling real life data with upside down bathtub shape and heavy tails was introduced. Mathematical and statistical characteristics such as the quantile function, mode, moments and moment generating function, entropy measure, stochastic ordering and distribution of order statistics have been derived. Furthermore, reliability measures like survival function, hazard function and odds function have been derived. The method of maximum likelihood was used for estimating the parameters of the distribution. To demonstrate the applicability of the distribution, a numerical example was given. Based on the results, the proposed distribution performed better than the competing distributions, except for the inverse power rama. As the criterion values, the AIC, BIC, AICc values, for both the proposed distribution and the inverse power rama distribution are approximately the same, stating that both may be interchangeably used.

A novel transmuted lomax exponential distribution: Properties and applications

Statistical lifetime distributions play a very important role in modeling data sets in various fields. Extending the existing distributions is of great interest in statistical research. The modification of the distributions provides more flexible model as compared to existing one. In this article, we propose a new probability model using Quadratic Rank Transmutation Map technique, named as Transmuted Lomax Exponential Distribution (TLED). The new distribution can model data sets with increasing, decreasing and bathtub shape hazard rates. Various statistical properties of the proposed distribution such as moments, order statistics, quantile function, mean residual life function and characteristic function are derived. Further, the parameter estimates are obtained through Maximum Likelihood method along with asymptotic confidence intervals. The utility of the new model is evaluated by analyzing two real data sets. In order to access the performance of the new model, several goodness of fit measures is used. The results indicate that the new model best fits the data as compared to the other extensions of the Lomax distribution.

Marshall olkin extended exponentiated Gamma distribution and its applications

Different methods for obtaining new probability distributions have been introduced in the literature in recent years, for example, (Gupta et al., 1998) proposed an interesting uni-parametric lifetime distribution, Exponentiated Gamma (EG), which hazard function has increasing and bathtub shapes. In this paper, we build a new two-parameters distribution, the Marshall Olkin Extended Exponentiated Gamma (MOEEG) distribution, which is derived from the Marshall-Olkin method and the EG distribution. The hazard function of this new distribution can accommodate monotonic, non-monotonic and unimodal shapes, allowing a better fit to greater data variability. In addition to the great flexibility of fitting the data, it contains only two parameters providing a simple parameter estimation procedure, unlike other distributions proposed in the literature that have three or more parameters. Some properties of the new distribution considered in this paper are presented such as n-th time, r-th moment of residual life, r-thmoment of residual life inverted, stochastic ordering, entropy, mean deviation, Bonferroni and Lorenz curve, skewness, kurtosis, order statistics, and stress-strength parameter. We also apply two different estimation methods, maximum likelihood and Bayesian approach. Real data applications are presented to illustrate the usefulness of this new distribution.

Gompertz Ampadu Class of Distributions: Properties and Applications

This paper introduces a new generator family of distributions called the Gompertz Ampadu-G family. Based on the generator, the Lomax distribution was modified into Gompertz Ampadu Lomax. The new distribution has a flexible hazard rate function that has upside-down and bathtub shapes, including increasing and decreasing hazard rate functions. The distribution comes with some desirable statistical properties. The distribution is applied to real-life data. Parameter estimates and test statistics show a better fit for the competitive models.

Statistical Inference of Chen Distribution Based on Type I Progressive Hybrid Censored Samples

In this paper we study the problem of estimating unknown parameters of a two-parameter distribution with bathtub shape under the assumption that data are type I progressive hybrid censored. We derive maximum likelihood estimators and then obtain the observed Fisher information matrix. Bayes estimators are also obtained under the squared error loss function and highest posterior density intervals are constructed as well. We perform a simulation study to compare proposed methods and analyzed a real data set for illustration purposes. Finally we establish optimal plans with respect to cost constraints and obtain comments based on a numerical study.

Poisson modified weibull distribution with inferences on stress‐strength reliability model

AbstractThis paper is devoted to study a new four‐parameter model called a Poisson Modified Weibull Distribution (PMW). The motivation for the new distribution is to obtain the hazard rate function with multiple shapes, the most important of which is the bathtub shape which is very useful in the reliability analysis. Some of its statistical properties are discussed. Maximum likelihood, method of moments and Bayesian techniques are used for estimating the parameters and the Fisher's information matrix is presented. In specific, we focus on the study of Stress‐ Strength reliability parameter, , when and have a PMW distribution. The maximum likelihood and Bayesian estimators for are derived. A simulation study is performed to compare the performance of each method of estimation. Moreover, the goodness of fit for PMW distribution comparison with other models are carried out by means of two applications to real data sets to illustrate the flexibility of the proposed model. Finally, the Stress‐Strength model for PMW distribution was applied to a real data representing the failure times for two types of electrical insulators.

The U Family of Distributions: Properties and Applications

Abstract In this article, we develop a new general family of distributions aimed at unifying some well-established lifetime distributions and offering new work perspectives. A special family member based on the so-called modified Weibull distribution is highlighted and studied. It differs from the competition with a very flexible hazard rate function exhibiting increasing, decreasing, constant, upside-down bathtub and bathtub shapes. This panel of shapes remains rare and particularly desirable for modeling purposes. We provide the main mathematical properties of the special distribution, such as a tractable infinite series expansion of the probability density function, moments of several kinds (raw, incomplete, probability weighted…) with discussions on the skewness and kurtosis. The stochastic ordering structure and stress-strength parameter are also considered, as well as the basics of the order statistics. Then, an emphasis is put on the inferential features of the related model. In particular, the estimation of the model parameters is employed by the maximum likelihood method, with a simulation study to confirm the suitability of the approach. Three practical data sets are then analyzed. It is observed that the proposed model gives better fits than other well-known lifetime models derived from the Weibull model.

A Flexible Probability Model for Reliability Data Analysis: The Extended Half-Normal Distribution with Further Results

Recent years have shown growth in the potential applications of extensions of skew and half-normal distributions. In this paper, we provide and study, in detail, an extended class of such distributions. The presented distribution has a scale parameter and two shape parameters. The motivations behind the development of this distribution are: 1- the probability distribution function has skewed, unimodal and bathtub shapes with different styles and the hazard rate function is increasing with various shapes, which makes this distribution flexible enough to analyze reliability data; 2- it can be expressed as a mixture of the Generalized Half-Normal distribution and the new weighted Generalized Half-Normal distribution. A number of statistical results are derived. Monte Carlo simulation analysis is carried out with Maximum Likelihood Estimation to assess the performance of this technique for a variety of the distribution parameter values. The power of the distribution is demonstrated with real applications using two reliability datasets from the industry. The results reveal that the proposed distribution outperformed the Generalized Lindley, Exponentiated Exponential, Gamma, Alpha-Skew-Normal, the Half-Alpha- Skew-Normal, and the Modified Generalized Half-Normal distributions.