The Bivariate Transmuted Family of Distributions: Theory and Applications

The bivariate distributions are useful in simultaneous modeling of two random variables. These distributions provide a way to model models. The bivariate families of distributions are not much widely explored and in this article a new family of bivariate distributions is proposed. The new family will extend the univariate transmuted family of distributions and will be helpful in modeling complex joint phenomenon. Statistical properties of the new family of distributions are explored which include marginal and conditional distributions, conditional moments, product and ratio moments, bivariate reliability and bivariate hazard rate functions. The maximum likelihood estimation (MLE) for parameters of the family is also carried out. The proposed bivariate family of distributions is studied for the Weibull baseline distributions giving rise to bivariate transmuted Weibull (BTW) distribution. The new bivariate transmuted Weibull distribution is explored in detail. Statistical properties of the new BTW distribution are studied which include the marginal and conditional distributions, product, ratio and conditional momenst. The hazard rate function of the BTW distribution is obtained. Parameter estimation of the BTW distribution is also done. Finally, real data application of the BTW distribution is given. It is observed that the proposed BTW distribution is a suitable fit for the data used.     

A New Generalized Weibull Model: Classical and Bayesian Estimation

Statistical distributions play a prominent role in applied sciences, particularly in biomedical sciences. The medical data sets are generally skewed to the right, and skewed distributions can be used quite effectively to model such kind of data sets. In the present study, therefore, we propose a new family of distributions suitable for modeling right-skewed medical data sets. The proposed family may be called a new generalized-X family. A special sub-model of the proposed family called a new generalized-Weibull distribution is discussed in detail. 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. Finally, the proposed model is applied to the remission times of the stomach cancer patient’s data. The comparison of the goodness of fit results of the proposed model is made with the other competing models such as Weibull, Kumaraswamy Weibull, and exponentiated Weibull distributions. Certain analytical measures such as Akaike information criterion, Bayesian information criterion, Anderson Darling statistic, and Kolmogorov–Smirnov test statistic are considered to show which distribution provides the best fit to data. Based on these measures, it is showed that the proposed distribution is a reasonable candidate for modeling data in medical sciences and other related fields.     

A log-extended Weibull regression model

A bathtub-shaped failure rate function is very useful in survival analysis and reliability studies. The well-known lifetime distributions do not have this property. For the first time, we propose a location-scale regression model based on the logarithm of an extended Weibull distribution which has the ability to deal with bathtub-shaped failure rate functions. We use the method of maximum likelihood to estimate the model parameters and some inferential procedures are presented. We reanalyze a real data set under the new model and the log-modified Weibull regression model. We perform a model check based on martingale-type residuals and generated envelopes and the statistics AIC and BIC to select appropriate models.     

Inferences on the lifetime performance index for Weibull distribution based on censored observations using the max p-value method

In the service (or manufacturing) industries, process capability indices (PCIs) are utilised to assess whether product quality meets the required level. And the lifetime performance index (or larger-the-better PCI) C_L is frequently used as a means of measuring product performance, where L is the lower specification limit. Hence, this study first uses the max p-value method to select the optimum value of the shape parameter β of the Weibull distribution and β is given. Second, we also construct the maximum likelihood estimator (MLE) of C_L based on the type II right‐censored sample from the Weibull distribution. The MLE of C_L is then utilised to develop a novel hypothesis testing procedure provided that L is known. Finally, we give one practical example to illustrate the use of the testing procedure under given significance level α.     

Detection-recognition algorithm based on the Gabor transform for unknown signals embedded in unknown noise

In this paper we present the CFAR (Constant False Alarm Rate) two-step detection-recognition algorithm for unknown, non-stationary signals embedded in unknown noise, based on the discrete Gabor transform. In the detection step, the decision about the absence or the presence of a signal of interest in a background of noise should be taken. The term ‘recognition’ means recovering the signal waveform from a noisy signal after the detection step. The recognition can be reformulated as the non-stationary, time-varying filtering problem in a time–frequency domain. In this paper the Gabor time–frequency domain is taken into account and the Gabor transform is used both in the detection and the in the filtering step. The discrete Gabor transform (DGT) is under intensive study of mathematicians, what results in a number of new, efficient computational algorithms for long time series. The Gabor frame approach is used for computation analysis and synthesis windows. Data-driven approach to develop the detection-recognition algorithm is based on the assumption, that disturbing noise signal after the Gabor transform, can be successfully approximated by the Weibull distribution regardless noise distribution before the transformation. It is shown by intensive simulations, that a two-parameter model like the Weibull distribution is really appropriate. Scale and shape parameters of the Weibull distribution are easily estimated and the CFAR threshold used in detection, based on estimated parameters, can be computed. The case of a low SNR ratio, with additional assumption about a signal, is also considered. It is shown that the iterative form of the time-varying filtering, significantly improves the quality of the whole detection-recognition CFAR algorithm. This approach is successfully investigated on a real-life radar signal.     

RELIABILITY PREDICTION OF IMPERFECT SWITCHING SYSTEMS SUBJECT TO WEIBULL FAILURES

Reliability prediction plays a very important role in system design and evaluation. In order to accurately predict the system reliability, one should consider the system configuration and the failure distribution of its components. This paper discusses the imperfect switching system with one component in an active state and n spares in a standby state. When the operating component breaks down, the switch detects the failure via the sensor and the defective component is replaced with a functional spare, so the system can resume operation. The Weibull distribution is one of the most flexible failure distributions which is widely used because it can adequately describe the reliability behavior during the lifetime of present day components/systems. This paper assumes the operating components follow Weibull failures, but the spares, sensor and switch failures follow an exponential distribution. In addition, three assumptions are made with regard to its switch failures: (i) under the energized condition, (ii) under the failing-open condition, and (iii) under the failing-closed condition. Due to the intractability of the Weibull distribution in imperfect switching models, it is difficult to solve the multiple integration involved analytically. Therefore, a numerical integration method using Simpson's rule was selected as a tool to address the problem of multiple integration for the Weibull distribution. A recursive algorithm is developed for the reliability prediction of a series system with m imperfect switching sub-systems subject to Weibull failures. Finally, a sensitivity analysis is performed on the two parameters of the Weibull distribution, on the effect of spare addition, as well as different failing conditions (switch and sensor) on system reliability. A numerical example is also given to explain and demonstrate the practical application of the developed reliability prediction models.     

无失效数据机载多余度EWIS可靠性研究

针对高可靠度机载多余度EWIS各组成部分寿命服从指数分布但参数未知的情况，提出采用无失效数据可靠度分析方法评估EWIS的可靠度水平。通过Monte-Carlo仿真方法对连接形式为“先并联、后串联”EWIS各组成部分寿命进行抽样，利用“最小最大值”方法获得系统寿命的抽样值，用概率纸检验法初步判断EWIS寿命是否服从威布尔分布，再用Pearson拟合优度检验法判断EWIS寿命是否服从威布尔分布。结合无故障飞行时间的样本值与EWIS寿命服从威布尔分布的假设，采用无失效数据分析方法评估EWIS的可靠度水平。研究方法对机载多余度EWIS无失效数据可靠度分析有一定的贡献。     

Estimating the parameters of the Marshall-Olkin bivariate Weibull distribution by EM algorithm

In this paper we consider the Marshall-Olkin bivariate Weibull distribution. The Marshall-Olkin bivariate Weibull distribution is a singular distribution, whose both the marginals are univariate Weibull distributions. This is a generalization of the Marshall-Olkin bivariate exponential distribution. The cumulative joint distribution of the Marshall-Olkin bivariate Weibull distribution is a mixture of an absolute continuous distribution function and a singular distribution function. This distribution has four unknown parameters and it is observed that the maximum likelihood estimators of the unknown parameters cannot be obtained in explicit forms. In this paper we discuss about the computation of the maximum likelihood estimators of the unknown parameters using EM algorithm. We perform some simulations to see the performances of the EM algorithm and re-analyze one data set for illustrative purpose.     

Weibull分布在软件最优交付时间估算中的应用

软件最优交付时间在很大程度上取决于软件可靠性估算。Weibull分布较有效地描述了软件测试过程中错误发现的统计特征,是一种有效的软件可靠性分析模型。在分析Weibull分布的基础上,提出了Weibull分布的累计函数拟合模型,解决了错误发现的不连续问题,推导出两种利用Weibull累计函数模型进行软件最优交付时间的估算公式,并结合具体实例验证了模型的有效性。     

A generalized modified Weibull distribution for lifetime modeling

A four parameter generalization of the Weibull distribution capable of modeling a bathtub-shaped hazard rate function is defined and studied. The beauty and importance of this distribution lies in its ability to model monotone as well as non-monotone failure rates, which are quite common in lifetime problems and reliability. The new distribution has a number of well-known lifetime special sub-models, such as the Weibull, extreme value, exponentiated Weibull, generalized Rayleigh and modified Weibull distributions, among others. We derive two infinite sum representations for its moments. The density of the order statistics is obtained. The method of maximum likelihood is used for estimating the model parameters. Also, the observed information matrix is obtained. Two applications are presented to illustrate the proposed distribution.     

A generalized modified Weibull distribution for lifetime modeling

A four parameter generalization of the Weibull distribution capable of modeling a bathtub-shaped hazard rate function is defined and studied. The beauty and importance of this distribution lies in its ability to model monotone as well as non-monotone failure rates, which are quite common in lifetime problems and reliability. The new distribution has a number of well-known lifetime special sub-models, such as the Weibull, extreme value, exponentiated Weibull, generalized Rayleigh and modified Weibull distributions, among others. We derive two infinite sum representations for its moments. The density of the order statistics is obtained. The method of maximum likelihood is used for estimating the model parameters. Also, the observed information matrix is obtained. Two applications are presented to illustrate the proposed distribution.     

Full band watermarking in DCT domain with Weibull model

In the framework of maximum-likelihood detection for image watermarking schemes, the conventional Generalized Gaussian Distribution (GGD), Cauchy and Student’s t distributions often fail to model the pulse-like distributions, such as Discrete Cosine Transform (DCT) coefficient distribution. Meanwhile DCT DC coefficients are often neglected in the image watermarking schemes. In this paper an improved full band image watermarking algorithm with utilization of Weibull distribution modeling the DCT AC and DC coefficients is proposed. Experiments indicate that compared with other popluar distributions such as the GGD, the Weibull model gives a closer fit on the distribution of AC coefficients in absolute domain with a smaller Kullback-Leibler (KL) divergence and lower Mean Square Error (MSE). The watermarking scheme with Weibull modeling the DCT AC coefficients (Weibull-AC) exhibits strong robustness under the attack of scaling and median filtering. The watermarking scheme with Weibull modeling the DCT DC coefficients (Weibull-DC) yields a better detection accuracy for bright and more detailed images. Combining the above two advantages, the proposed Weibull based full band watermarking in DCT domain (Weibull-FB) further improves its robustness under the attack of JPEG compression and achieves 10.47 % overall increment in the detection accuracy compared with the baseline system while maintaining good invisibility in the view of structural similarity (SSIM).     

Comment on “A general model for long-tailed network traffic approximation”

The recent paper by Wang et al. (J. Supercomput. 38:155–172, 2006) proposed a Hyper Erlang model for long-tailed network traffic approximation. The paper argued that traditional models such as the Pareto, Weibull and log normal distributions are difficult to apply because of “their complex representations and theoretical properties”. The paper went on to say that the Pareto distribution “does not have analytic Laplace transform, and many other heavy-tailed distributions, such as Weibull and log normal also do not have closed-form Laplace transforms”. In the following, we would like to show that one can actually derive explicit expressions for Laplace transforms of heavy-tailed distributions. The next three sections provide explicit expressions for the Laplace transforms of the Pareto, Weibull and the log-normal distributions. To the best of our knowledge, these are the first known results on Laplace transforms of heavy-tailed distributions.

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A hybrid variable neighborhood search and simulated annealing algorithm to estimate the three parameters of the Weibull distribution

The Weibull distribution plays an important role in failure distribution modeling of reliability research. While there are three parameters in the general form of this distribution, for simplicity, one of its parameters is usually omitted and as a result, the others are estimated easily. However, due to its more flexibility, when the general form of the Weibull distribution is of interest, the estimation procedure is not an easy task anymore. For example, in the maximum likelihood estimation method, the likelihood function that is formed for a three-parameter Weibull distribution is very hard to maximize. In this paper, a new hybrid methodology based on a variable neighborhood search and a simulated annealing approach is proposed to maximize the likelihood function of a three-parameter Weibull distribution. The performance of the proposed methodology in terms of both the estimation accuracy and the required CPU time is then evaluated and compared to the ones of an existing current method through a wide range of numerical examples in which a sensitivity analysis is performed on the sample size. The results of the comparison study show that while the proposed method provides accurate estimates as well as those of the existing method, it requires significantly less CPU time.     

Log-modified Weibull regression models with censored data: Sensitivity and residual analysis

This paper proposes a regression model considering the modified Weibull distribution. This distribution can be used to model bathtub-shaped failure rate functions. Assuming censored data, we consider maximum likelihood and Jackknife estimators for the parameters of the model. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and we also present some ways to perform global influence. Besides, for different parameter settings, sample sizes and censoring percentages, various simulations are performed and the empirical distribution of the modified deviance residual is displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended for a martingale-type residual in log-modified Weibull regression models with censored data. Finally, we analyze a real data set under log-modified Weibull regression models. A diagnostic analysis and a model checking based on the modified deviance residual are performed to select appropriate models.     

恒定应力加速寿命试验模型及应用——威布尔分布

加速寿命试验是可靠性试验技术的基本方法之一。检验加速寿命试验中的数据是否服从威布尔分布,进而得出加速系数和加速寿命方程,用在不同应力水平下试验得到的数据,运用威布尔分布模型,并对其中的待估参数进行点估计和区间估计,最终得出产品在正常应力水平下的各项可靠性特征量的估计。     

Size distribution and waiting times for the avalanches of the Cell Network Model of Fracture

The Cell Network Model is a fracture model recently introduced that resembles the microscopical structure and drying process of the parenchymatous tissue of the Bamboo Guadua angustifolia. The model exhibits a power-law distribution of avalanche sizes, with exponent −3.0 when the breaking thresholds are randomly distributed with uniform probability density. Hereby we show that the same exponent also holds when the breaking thresholds obey a broad set of Weibull distributions, and that the humidity decrements between successive avalanches (the equivalent to waiting times for this model) follow in all cases an exponential distribution. Moreover, the fraction of remaining junctures shows an exponential decay in time. In addition, introducing partial breakings and cumulative damages induces a crossover behavior between two power-laws in the histogram of avalanche sizes. This results support the idea that the Cell Network Model may be in the same universality class as the Random Fuse Model.     

The log-exponentiated Weibull regression model for interval-censored data

In interval-censored survival data, the event of interest is not observed exactly but is only known to occur within some time interval. Such data appear very frequently. In this paper, we are concerned only with parametric forms, and so a location-scale regression model based on the exponentiated Weibull distribution is proposed for modeling interval-censored data. We show that the proposed log-exponentiated Weibull regression model for interval-censored data represents a parametric family of models that include other regression models that are broadly used in lifetime data analysis. Assuming the use of interval-censored data, we employ a frequentist analysis, a jackknife estimator, a parametric bootstrap and a Bayesian analysis for the parameters of the proposed model. We derive the appropriate matrices for assessing local influences on the parameter estimates under different perturbation schemes and present some ways to assess global influences. Furthermore, for different parameter settings, sample sizes and censoring percentages, various simulations are performed; in addition, the empirical distribution of some modified residuals are displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended to a modified deviance residual in log-exponentiated Weibull regression models for interval-censored data.     

Effects of light and heavy workload on air traffic tactical operations: a hazard rate model

This paper introduces a class of event history analysis used to examine how the operations of an air traffic controller change under light and heavy traffic workload. The analysis begins by assessing the hazard rate, h(t), of a transition (or spell) between the controller's communication and flight progress activities. h(t) is the instantaneous rate of going from one state (i.e. an activity of communication or flight progress) to another in a unit of time, given that the controller has been in the first state until time t. Results indicated that the spell distribution closely followed a Weibull distribution, a prerequisite for this analysis. The results also indicated that h(t) was more likely regulated by time in heavy than in light workload conditions, and that under heavy workload, indirect speech from the planner controller would decrease the h(t) for communication to flight progress spells. The results suggest that a dynamic model for the analysis of air traffic control may be necessary, and that the implications of using modular automation may not be straightforward. This technique may be of general use to examine temporal regularities in operating real-time control tasks.