Additive Weibull Research Articles

On the Transmuted AdditiveWeibull Distribution

In this article a continuous distribution, the so-called transmuted additive Weibull distribution, that extends the additive Weibull distribution and some other distributions is proposed and studied. We will use the quadratic rank transmutation map proposed by Shaw and Buckley (2009) in order to generate the transmuted additiveWeibull distribution. Various structural properties of the new distribution including explicit expressions for the moments, random number generation and order statistics are derived. Maximum likelihood estimation of the unknown parameters of the new model for completesample is also discussed. It will be shown that the analytical results are applicable to model real world data.

Kumaraswamy Transmuted Exponentiated Additive Weibull Distribution

This paper introduces a new lifetime model which is a generalization of the transmuted exponentiated additive Weibull distribution by using the Kumaraswamy generalized (Kw-G) distribution. With the particular case no less than \textbf{seventy nine} sub models as special cases, the so-called Kumaraswamy transmuted exponentiated additive Weibull distribution, introduced by Cordeiro and de Castro (2011) is one of this particular cases. Further, expressions for several probabilistic measures are provided, such as probability density function, hazard function, moments, quantile function, mean, variance and median, moment generation function, R\'{e}nyi and q entropies, order estatistics, etc. Inference is maximum likelihood based and the usefulness of the model is showed by using a real dataset.

The Additive Weibull-Geometric Distribution: Theory and Applications

In this paper, we introduce a new class of lifetime distributions which is called the additive Weibull geometric (AWG) distribution. This distribution obtained by compounding the additive Weibull and geometric distributions. The new distribution has a number of well-known lifetime special sub-models such as modified Weibull geometric, Weibull geometric, exponential geometric, among several others. Some structural properties of the proposed new distribution are discussed. 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.

A New Generalization of the Additive Weibull Distribution: Theory and Applications

The unified modeling language (UML) is widely used to analyze and design different software development artifacts in an object oriented development. Domain model is a significant artifact that models the problem domain and visually represents real world objects and relationships among them. It facilitates the comprehension process by identifying the vocabulary and key concepts of the business world. Category list technique identifies concepts and associations with the help of pre defined categories, which are important to business information systems. Whereas noun phrasing technique performs grammatical analysis of use case description to recognize concepts and associations. Both of these techniques are used for the construction of domain model, however, no empirical evidence exists that evaluates the quality of the resultant domain model constructed via these two basic techniques. A controlled experiment was performed to investigate the impact of category list and noun phrasing technique on quality of the domain model. The constructed domain model is evaluated for completeness, correctness and effort required for its design. The obtained results show that category list technique is better than noun phrasing technique for the identification of concepts as it avoids generating unnecessary elements i.e. extra concepts, associations and attributes in the domain model. The noun phrasing technique produces a comprehensive domain model and requires less effort as compared to category list. There is no statistically significant difference between both techniques in case of correctness.

THE WEIBULL-POWER FUNCTION DISTRIBUTION WITH APPLICATIONS

Recently, several attempts have been made to define new models that extend well-known distributions and at the same time provide great flexibility in modelling real data. We propose a new four-parameter model named the Weibull-power function (WPF) distribution which exhibits bathtub-shaped hazard rate. Some of its statistical properties are obtained including ordinary and incomplete moments, quantile and generating functions, Renyi and Shannon entropies, reliability and or- der statistics. The model parameters are estimated by the method of maximum likelihood. A bivariate extension is also proposed. The new distribution can be implemented easily using statistical software pack- ages. We investigate the potential usefulness of the proposed model by means of two real data sets. In fact, the new model provides a better fit to these data than the additive Weibull, modified Weibull, Sarahan-Zaindin modified Weibull and beta-modified Weibull distribu- tions, suggesting that it is a reasonable candidate for modeling survival data.

On the Additive Weibull Distribution

We derive explicit expressions for the moments, incomplete moments, quantile function and generating function of the additive Weibull model pioneered by Xie and Lai (1995), which is a quite flexible distribution for fitting lifetime data with bathtub-shaped failure rate function. In addition, we estimate the model parameters by maximum likelihood and determine the observed information matrix. The flexibility of the additive Weibull distribution is illustrated by means of one application to real data.

Parameter Estimations for Some Modifications of the Weibull Distribution

Proposed by the Swedish engineer and mathematician Ernst Hjalmar Waloddi Weibull (1887-1979), the Weibull distribution is a probability distribution that is widely used to model lifetime data. Because of its flexibility, some modifications of the Weibull distribution have been made from several researches in order to best adjust the non-monotonic shapes. This paper gives a study on the performance of two specific modifications of the Weibull distribution which are the exponentiated Weibull distribution and the additive Weibull distribution.

The discrete additive Weibull distribution: A bathtub-shaped hazard for discontinuous failure data

Although failure data are usually treated as being continuous, they may have been collected in a discrete manner, or in fact be discrete in nature. Reliability models with bathtub-shaped hazard rate are fundamental to the concepts of burn-in and maintenance, but how well do they incorporate discrete data? We explore discrete versions of the additive Weibull distribution, which has the twin virtues of mathematical tractability and the ability to produce bathtub-shaped hazard rate functions. We derive conditions on the parameters for the hazard rate function to be increasing, decreasing, or bathtub shaped. While discrete versions may have the same shaped hazard rate for the same parameter values, we find that when fitted to data the fitted hazard rate shapes can vary between versions. Our results are illustrated using several real-life data sets, and the implications of using continuous models for discrete data discussed.

Sample size dependence of flaw distributions for the prediction of brittle solids strength using additive Weibull bimodal distributions

A method to evaluate the mean strength of brittle solids at short gauge length from experiments performed at higher gauge length is proposed for bimodal fracture behavior. The method based on additive Weibull bimodal distributions takes the evolution of the relative proportion of flaws along the gauge length into account via a gauge length dependent mixing parameter. A linear dependence of this parameter vs. the gauge length is proposed. The approach is assessed using experimental results on carbon and E-glass fibers. The new method provides values for the average tensile strength of the fibers at 100 μm up to 27% higher than those calculated using the classical approach. The underestimation of the classical approach can be attributed to the weight of the severe category of flaws at short length that is considered to be the same as that determined at the experimental gauge length, which can be several orders higher. A simplified approach taking into account solely the more severe category of flaw is shown to be applicable for the prediction of the strength at short gauge length independently of the nature of the fiber.

Additive Weibull Model for Reliability Analysis

Additive Weibull Model for Reliability Analysis

Useful Periods for Lifetime Distributions With Bathtub Shaped HazardRate Functions

We propose computationally tractable formal mathematical definitions for the 'useful period' of lifetime distributions with bathtub shaped hazard rate functions. Detailed analysis of the reduced additive Weibull hazard rate function illustrates its utility for identifying such useful periods. Examples of several other bathtub shaped hazard rate functions are also presented with applications to lifetime data. The suggestion is made of defining and considering analogous 'stable periods' in the case of the corresponding upside-down bathtub shaped mean residual life functions.

Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function

Lifetime distributions for many components usually have a bathtub-shaped failure rate in practice. However, there are very few practical models to model this type of failure rate function. In this paper we study a simple model based on adding two Weibull survival functions. Some simplifications of the model are also presented. The graphical estimation technique based on the conventional Weibull plot is demonstrated to be useful in this case.

Use of a mixed-weibull distribution for the identification of PD phenomena

The interaction between partial discharge (PD) phenomena occurring in insulating systems is investigated in this paper. In particular, the recognition of two PD phenomena simultaneously active is approached by means of a 5-parameter additive Weibull distribution. Various shapes of the PD height distribution, obtained from measurements performed on specimens of stator bars and windings of ac rotating machines, are considered. It is shown that the proposed probability function fits well the partial discharge height distributions. By this way the probability distribution relevant to each concurring PD phenomenon can be derived, analyzed and identified. Moreover, the standard average quantities are estimated for each phenomenon.

Use of a mixed-Weibull distribution for the identification of PD phenomena [rotating machines

The interaction between partial discharge (PD) phenomena occurring in insulating systems is investigated in this paper. In particular, the recognition of two PD phenomena simultaneously active is approached by means of a five-parameter additive Weibull distribution. Various shapes of the PD height distribution, obtained from measurements performed on specimens of stator bars and windings of ac rotating machines, are considered. It is shown that the proposed probability function fits well the partial discharge height distributions. The probability distribution relevant to each concurring PD phenomenon can be derived, analysed and identified. Moreover, the standard average quantities are estimated for each phenomenon.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The computation of yield and drift reliability of electronic circuits : F. Jenson. Microelectron. & Reliab.11 (1972), p. 139

Although failure data are usually treated as being continuous, they may have been collected in a discrete manner, or in fact be discrete in nature. Reliability models with bathtub-shaped hazard rate are fundamental to the concepts of burn-in and maintenance, but how well do they incorporate discrete data? We explore discrete versions of the additive Weibull distribution, which has the twin virtues of mathematical tractability and the ability to produce bathtub-shaped hazard rate functions. We derive conditions on the parameters for the hazard rate function to be increasing, decreasing, or bathtub shaped. While discrete versions may have the same shaped hazard rate for the same parameter values, we find that when fitted to data the fitted hazard rate shapes can vary between versions. Our results are illustrated using several real-life data sets, and the implications of using continuous models for discrete data discussed.