General results for the beta-modified Weibull distribution

We study in detail the so-called beta-modified Weibull distribution, motivated by the wide use of the Weibull distribution in practice, and also for the fact that the generalization provides a continuous crossover towards cases with different shapes. The new distribution is important since it contains as special sub-models some widely-known distributions, such as the generalized modified Weibull, beta Weibull, exponentiated Weibull, beta exponential, modified Weibull and Weibull distributions, among several others. It also provides more flexibility to analyse complex real data. Various mathematical properties of this distribution are derived, including its moments and moment generating function. We examine the asymptotic distributions of the extreme values. Explicit expressions are also derived for the chf, mean deviations, Bonferroni and Lorenz curves, reliability and entropies. The estimation of parameters is approached by two methods: moments and maximum likelihood. We compare by simulation the performances of the estimates from these methods. We obtain the expected information matrix. Two applications are presented to illustrate the proposed distribution.

The exponential COM-Poisson distribution

The Conway-Maxwell Poisson (COMP) distribution as an extension of the Poisson distribution is a popular model for analyzing counting data. For the first time, we introduce a new three parameter distribution, so-called the exponential-Conway-Maxwell Poisson (ECOMP) distribution, that contains as sub-models the exponential-geometric and exponential-Poisson distributions proposed by Adamidis and Loukas (Stat Probab Lett 39:35-42, 1998) and KuAY (Comput Stat Data Anal 51:4497-4509, 2007), respectively. The new density function can be expressed as a mixture of exponential density functions. Expansions for moments, moment generating function and some statistical measures are provided. The density function of the order statistics can also be expressed as a mixture of exponential densities. We derive two formulae for the moments of order statistics. The elements of the observed information matrix are provided. Two applications illustrate the usefulness of the new distribution to analyze positive data.

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-Gdensity 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, Renyi 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.

Reliability studies of the skew normal distribution

It has been observed in various practical applications that data do not conform to the normal distribution, which is symmetric with no skewness. The skew normal distribution proposed by Azzalini(1985) is appropriate for the analysis of data which is unimodal but exhibits some skewness. The skew normal distribution includes the normal distribution as a special case where the skewness parameter is zero. In this thesis, we study the structural properties of the skew normal distribution, with an emphasis on the reliability properties of the model. More specifically, we obtain the failure rate, the mean residual life function, and the reliability function of a skew normal random variable. We also compare it with the normal distribution with respect to certain stochastic orderings. Appropriate machinery is developed to obtain the reliability of a component when the strength and stress follow the skew normal distribution. Finally, IQ score data from Roberts (1988) is analyzed to illustrate the procedure.

The generalized inverse Weibull distribution

The inverse Weibull distribution has the ability to model failure rates which are quite common in reliability and biological studies. A three-parameter generalized inverse Weibull distribution with decreasing and unimodal failure rate is introduced and studied. We provide a comprehensive treatment of the mathematical properties of the new distribution including expressions for the moment generating function and the rth generalized moment. The mixture model of two generalized inverse Weibull distributions is investigated. The identifiability property of the mixture model is demonstrated. For the first time, we propose a location-scale regression model based on the log-generalized inverse Weibull distribution for modeling lifetime data. In addition, we develop some diagnostic tools for sensitivity analysis. Two applications of real data are given to illustrate the potentiality of the proposed regression model.

The beta modified Weibull distribution

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

Weibull model selection for reliability modelling

A large number of models have been derived from the two-parameter Weibull distribution and are referred to as Weibull models. They exhibit a wide range of shapes for the density and hazard functions, which makes them suitable for modelling complex failure data sets. The WPP and IWPP plot allows one to determine in a systematic manner if one or more of these models are suitable for modelling a given data set. This paper deals with this topic.

The Effect of Operational Disturbances on Reliability and Operation Time Distribution of NCmachine Tools in FMS

This thesis concentrates on studying the operational disturbance behavior of machine tools integrated into FMS. Operational disturbances are short term failures of machine tools which are especially disruptive to unattended or unmanned operation of FMS. The main objective was to examine the effect of operational disturbances on reliability and operation time distribution for machine tools. The theoretical part of the thesis covers the fimdamentals of FMS relating to the subject of this study. The concept of FMS, its benefits and operator's role in FMS operation are reviewed. The importance of reliability is presented. The terms describing the operation time of machine tools are formed by adopting standards and references. The concept of failure and indicators describing reliability and operational performance for machine tools in FMSs are presented. The empirical part of the thesis describes the research methodology which is a combination of automated (ADC) and manual data collection. By using this methodology it is possible to have a complete view of the operation time distribution for studied machine tools. Data collection was carried out in four FMSs consisting of a total of 17 machine tools. Each FMS's basic features and the signals of ADC are described. The indicators describing the reliability and operation time distribution of machine tools were calculated according to collected data. The results showed that operational disturbances have a significant influence on machine tool reliability and operational performance. On average, an operational disturbance occurs every 8,6 hours of operation time and has a down time of 0,53 hours. Operational disturbances cause a 9,4% loss in operation time which is twice the amount of losses caused by technical failures (4,3%). Operational disturbances have a decreasing influence on the utilization rate. A poor operational disturbance behavior decreases the utilization rate. It was found that the features of a part family to be machined and the method technology related to it are defining the operational disturbance behavior of the machine tool. Main causes for operational disturbances were related to material quality variations, tool maintenance, NC program errors, ATC and machine tool control. Operator's role was emphasized. It was found that failure recording activity of the operators correlates with the utilization rate. The more precisely the operators record the failure, the higher is the utilization rate. Also the FMS organizations which record failures more precisely have fewer operational disturbances.

Characterization of Probability Distributions Using the Residual Entropy Function

The present study on the characterization of probability distributions using the residual entropy function. The concept of entropy is extensively used in literature as a quantitative measure of uncertainty associated with a random phenomenon. The commonly used life time models in reliability Theory are exponential distribution, Pareto distribution, Beta distribution, Weibull distribution and gamma distribution. Several characterization theorems are obtained for the above models using reliability concepts such as failure rate, mean residual life function, vitality function, variance residual life function etc. Most of the works on characterization of distributions in the reliability context centers around the failure rate or the residual life function. The important aspect of interest in the study of entropy is that of locating distributions for which the shannon’s entropy is maximum subject to certain restrictions on the underlying random variable. The geometric vitality function and examine its properties. It is established that the geometric vitality function determines the distribution uniquely. The problem of averaging the residual entropy function is examined, and also the truncated form version of entropies of higher order are defined. In this study it is established that the residual entropy function determines the distribution uniquely and that the constancy of the same is characteristics to the geometric distribution

A bayesian analysis for the parameters of the exponential-logarithmic distribution

The exponential-logarithmic is a new lifetime distribution with decreasing failure rate and interesting applications in the biological and engineering sciences. Thus, a Bayesian analysis of the parameters would be desirable. Bayesian estimation requires the selection of prior distributions for all parameters of the model. In this case, researchers usually seek to choose a prior that has little information on the parameters, allowing the data to be very informative relative to the prior information. Assuming some noninformative prior distributions, we present a Bayesian analysis using Markov Chain Monte Carlo (MCMC) methods. Jeffreys prior is derived for the parameters of exponential-logarithmic distribution and compared with other common priors such as beta, gamma, and uniform distributions. In this article, we show through a simulation study that the maximum likelihood estimate may not exist except under restrictive conditions. In addition, the posterior density is sometimes bimodal when an improper prior density is used. © 2013 Copyright Taylor and Francis Group, LLC.

New Bivariate Lifetime Distributions Based on Bath-Tub Shaped Failure Rate

A class of lifetime distributions which has received considerable attention in modelling and analysis of lifetime data is the class of lifetime distributions with bath-tub shaped failure rate functions because of their extensive applications. The purpose of this thesis was to introduce a new class of bivariate lifetime distributions with bath-tub shaped failure rates (BTFRFs). In this research, first we reviewed univariate lifetime distributions with bath-tub shaped failure rates, and several multivariate extensions of a univariate failure rate function. Then we introduced a new class of bivariate distributions with bath-tub shaped failure rates (hazard gradients). Specifically, the new class of bivariate lifetime distributions were developed using the method of Morgenstern’s method of defining bivariate class of distributions with given marginals. The computer simulations and numerical computations were used to investigate the properties of these distributions.

Resting heart rate and incident heart failure and cardiovascular mortality in older adults: role of inflammation and endothelial dysfunction: the PROSPER study.

AIMS: Resting heart rate is a promising modifiable cardiovascular risk marker in older adults, but the mechanisms linking heart rate to cardiovascular disease are not fully understood. We aimed to assess the association between resting heart rate and incident heart failure (HF) and cardiovascular mortality, and to examine whether these associations might be attributable to systemic inflammation and endothelial dysfunction. METHODS AND RESULTS: We studied 4084 older adults aged 70-82 years with known cardiovascular risk factors or previous cardiovascular disease, without pre-existing HF or beta-blockers in the PROSPER study. Over a 3.2-year follow-up period, we examined incident HF hospitalization and cardiovascular mortality according to resting heart rate, along with C-reactive protein (CRP), interleukin-6 (IL-6), tissue plasminogen activator (tPA), and von Willebrand factor (vWf). Mean heart rate was 67 b.p.m. for men and 70 b.p.m. for women. CRP, IL-6, tPA, and vWf levels were all positively correlated with heart rate. After multivariate adjustment, heart rate was associated with HF hospitalization [hazard ratio (HR) 1.78 for highest vs. lowest distribution third, 95% confidence interval (CI) 1.21-2.63, P= 0.003] and cardiovascular mortality (HR 1.74, 95% CI 1.23-2.47, P= 0.002). Further adjustment for both IL-6 and vWf levels decreased HR to 1.60 (95% CI 1.08-2.38, P= 0.020) for HF and to 1.50 (95% CI 1.04-2.15, P= 0.028) for cardiovascular mortality. CONCLUSION: Increased heart rate is associated with increased systemic inflammation and endothelial dysfunction. These factors are likely to contribute to, but do not fully explain, the risk of HF and cardiovascular death associated with increased heart rate in older age.

Wave-height hazard analysis in Eastern Coast of Spain : Bayesian approach using generalized Pareto distribution

Standard practice of wave-height hazard analysis often pays little attention to the uncertainty of assessed return periods and occurrence probabilities. This fact favors the opinion that, when large events happen, the hazard assessment should change accordingly. However, uncertainty of the hazard estimates is normally able to hide the effect of those large events. This is illustrated using data from the Mediterranean coast of Spain, where the last years have been extremely disastrous. Thus, it is possible to compare the hazard assessment based on data previous to those years with the analysis including them. With our approach, no significant change is detected when the statistical uncertainty is taken into account. The hazard analysis is carried out with a standard model. Time-occurrence of events is assumed Poisson distributed. The wave-height of each event is modelled as a random variable which upper tail follows a Generalized Pareto Distribution (GPD). Moreover, wave-heights are assumed independent from event to event and also independent of their occurrence in time. A threshold for excesses is assessed empirically. The other three parameters (Poisson rate, shape and scale parameters of GPD) are jointly estimated using Bayes' theorem. Prior distribution accounts for physical features of ocean waves in the Mediterranean sea and experience with these phenomena. Posterior distribution of the parameters allows to obtain posterior distributions of other derived parameters like occurrence probabilities and return periods. Predictives are also available. Computations are carried out using the program BGPE v2.0

Abrasion, transport and distribution of sediment in selected streams of Southern Ontario and Western New York /

The rate of decrease in mean sediment size and weight per square metre along a 54 km reach of the Credit River was found to depend on variations in the channel geometry. The distribution of a specific sediment size consist of: (1) a transport zone; (2) an accumulation zone; and (3) a depletion zone. These zones shift downstream in response to downcurrent decreases in stream competence. Along a .285 km man-made pond, within the Credit River study area, the sediment is also characterized by downstream shifting accumulation zones for each finer clast size. The discharge required to initiate movement of 8 cm and 6 cm blocks in Cazenovia Creek is closely approximated by Baker and Ritter's equation. Incipient motion of blocks in Twenty Mile Creek is best predicted by Yalin's relation which is more efficient in deeper flows. The transport distance of blocks in both streams depends on channel roughness and geometry. Natural abrasion and distribution of clasts may depend on the size of the surrounding sediment and variations in flow competence. The cumulative percent weight loss with distance of laboratory abraded dolostone is defined by a power function. The decrease in weight of dolostone follows a negative exponential. In the abrasion mill, chipping causes the high initial weight loss of dolostone; crushing and grinding produce most of the subsequent weight loss. Clast size was found to have little effect on the abrasion of dolostone within the diameter range considered. Increasing the speed of the mill increased the initial amount of weight loss but decreased the rate of abrasion. The abrasion mill was found to produce more weight loss than stream action. The maximum percent weight loss determined from laboratory and field abrasion data is approximately 40 percent of the weight loss observed along the Credit River. Selective sorting of sediment explains the remaining percentage, not accounted for by abrasion.

A simple relation between the Leimkuhler curve and the mean residual life

In this paper, a simple relation between the Leimkuhler curve and the mean residual life is established. The result is illustrated with several models commonly used in informetrics, such as exponential, Pareto and lognormal. Finally, relationships with some other reliability concepts are also presented. (C) 2010 Elsevier Ltd. All rights reserved.