A New Extended-F Family: Properties and Applications to Lifetime Data
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Abstract:
In this article, a new approach is used to introduce an additional parameter to a continuous class of distributions. The new class is referred to as a new extended-F family of distributions. The new extended-Weibull distribution, as a special submodel of this family, is discussed. General expressions for some mathematical properties of the proposed family are derived, and maximum likelihood estimators of the model parameters are obtained. Furthermore, a simulation study is provided to evaluate the validity of the maximum likelihood estimators. Finally, the flexibility of the proposed method is illustrated via two applications to real data, and the comparison is made with the Weibull and some of its well-known extensions such as Marshall–Olkin Weibull, alpha power-transformed Weibull, and Kumaraswamy Weibull distributions.Keywords:
Exponentiated Weibull distribution
Weibull modulus
We introduce a new five-parameter model related to Weibull distribution, the so called. exponentiated Weibull Weibull (EWW) distribution. It incluides some new and earlier distributions. Fundamental properties are deduced. We deal with maximum likelihood (ML) method to obtain parameter estimators. The interest of the recommended distribution is confirmed through real data sets .
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In reliability, a Bi-Weibull distribution can be represented by a combination of two
decreasing, constant or increasing failure rate functions. Several versions of Bi-Weibull
distributions can be found but these versions differ in the way in which the two Weibull
distributions were combined and in the number of parameters specified. In this paper we
determine the Bi-Weibull distribution from a combination of two Weibull distributions each
having 2 or 3-parameters. We also determine the probability density, cumulative distribution,
survival and hazard functions of the 4, 5, and 6-parameter Bi-Weibull distributions. The result
is that by increasing the number of parameters would give the distribution more flexibility. An
application of the 4-parameter Bi-Weibull distribution is also included using real data
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This chapter describes two enhanced versions of Weibull distribution. These are Nukiyama-Tanasawa distribution, and q-Weibull distribution. The Nukiyama-Tanasawa distribution is an extension of the Weibull-II distribution. Employing a method similar to that used to develop the q-Gaussian distribution from the normal distribution, the q-Weibull distribution is developed from the Weibull distribution. It is another in the family of Tsallis distributions. If q is set to 1, the distribution reverts to the Weibull-II distribution. The chapter shows the effect of adjusting q. Using the stock level example, the chapter also hows that q-Weibull fits the data much better than Weibull. The parameter q is fitted as -2.75, with δ as 2.10 and β as 3802. As an example, the Weibull distribution predicts the probability of the stock level falling below 300 tonnes is 0.8%, equivalent to three occasions per year. The q-Weibull puts it much higher at 2.3%, or eight occasions per year.
Exponentiated Weibull distribution
Weibull modulus
Log-logistic distribution
Log-Cauchy distribution
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Weibull modulus
Exponentiated Weibull distribution
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Goodness of fit
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We derive a common linear representation for the densities of four generalizations of the two-parameter Weibull distribution in terms of Weibull densities. The four generalized Weibull distributions briefly studied are: the Marshall-Olkin-Weibull, beta-Weibull, gamma-Weibull and Kumaraswamy-Weibull distributions. We demonstrate that several mathematical properties of these generalizations can be obtained simultaneously from those of the Weibull properties. We present two applications to real data sets by comparing these generalized distributions. It is hoped that this paper encourage developments of further generalizations of the Weibull based on the same linear representation.
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A Weibull space is defined by three coordinates of failure probability, effective volume and strength. The two-parameter Weibull distribution function is presented by a plane (Weibull plane) in the Weibull space. The Weibull modulus can be obtained from the gradient of the Weibull plane. Therefore, the Weibull modulus can be evaluated by the method of least squares from the strength data obtained by various tests of different effective volume specimens. The Weibull modulus estimated by this method is more accurate than that by the conventional Weibull plot method. The Weibull modulus estimated by this method is expected to be twice as accurate as that by the conventional method, particularly when the strength data are obtained from two different volumes whose ratio is more than 100.
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In this article, we introduce a new three-parameter lifetime model called the Burr X exponentiated Weibull model. The major justification for the practicality of the new lifetime model is based on the wider use of the exponentiated Weibull and Weibull models. We are motivated to propose this new lifetime model because it exhibits increasing, decreasing, bathtub, J shaped and constant hazard rates. The new lifetime model can be viewed as a mixture of the exponentiated Weibull distribution. It can also be viewed as a suitable model for fitting the right skewed, symmetric, left skewed and unimodal data. We provide a comprehensive account of some of its statistical properties. Some useful characterization results are presented. The maximum likelihood method is used to estimate the model parameters. We prove empirically the importance and flexibility of the new model in modeling two types of lifetime data. The proposed model is a better fit than the Poisson Topp Leone-Weibull, the Marshall Olkin extended-Weibull, gamma-Weibull , Kumaraswamy-Weibull , Weibull-Fréchet, beta-Weibull, transmuted modified-Weibull, Kumaraswamy transmuted- Weibull, modified beta-Weibull, Mcdonald-Weibull and transmuted exponentiated generalized-Weibull models so it is a good alternative to these models in modeling aircraft windshield data as well as the new lifetime model is much better than the Weibull-Weibull, odd Weibull-Weibull, Weibull Log-Weibull, the gamma exponentiated-exponential and exponential exponential-geometric models so it is a good alternative to these models in modeling the survival times of Guinea pigs. We hope that the new distribution will attract wider applications in reliability, engineering and other areas of research.
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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.
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The Weibull distribution is used to characterize the parameters of survival data. However, the two-parameter version of the Weibull model does not include a shift parameter. As the first breakages are sometimes occurring far from 0, the shape of the 2-parameter Weibull distribution must be extreme to accommodate such data. Here we show that Weibull distributions with extreme shapes are degenerate distributions which do not contain information about shape anymore. Because of this, any 2- parameter Weibull distribution with large shape parameter can be mimicked by another large-shaped distribution. The argument is further illustrated with simulated results. We present simple solutions to detect and avoid such situations.
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This chapter contains sections titled: The Weibull Cumulative Distribution Function (CDF), Percentiles, Moments, and Hazard Function The Minima of Weibull Samples Transformations The Conditional Weibull Distribution Quantiles for Order Statistics of a Weibull Sample Simulating Weibull Samples References Exercises
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