Journal articles: 'Weibull distribution function'

1

Oluyede, Broderick. "The Gamma-Weibull-G Family of Distributions with Applications." Austrian Journal of Statistics 47, no. 1 (January 30, 2018): 45–76. http://dx.doi.org/10.17713/ajs.v47i1.155.

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Weibull distribution and its extended families has been widely studied in lifetime applications. Based on the Weibull-G family of distributions and the exponentiated Weibull distribution, we study in detail this new class of distributions, namely, Gamma-WeibullG family of distributions (GWG). Some special models in the new class are discussed. Statistical properties of the family of distributions, such as expansion of density function, hazard and reverse hazard functions, quantile function, moments, incomplete moments, generating functions, mean deviations, Bonferroni and Lorenz curves and order statistics are presented. We also present R´enyi entropy, estimation of parameters by using method of maximum likelihood, asymptotic confidence intervals and applications using real data

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Bokde, Neeraj, Andrés Feijóo, and Daniel Villanueva. "Wind Turbine Power Curves Based on the Weibull Cumulative Distribution Function." Applied Sciences 8, no. 10 (September 28, 2018): 1757. http://dx.doi.org/10.3390/app8101757.

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The representation of a wind turbine power curve by means of the cumulative distribution function of a Weibull distribution is investigated in this paper, after having observed the similarity between such a function and real WT power curves. The behavior of wind speed is generally accepted to be described by means of Weibull distributions, and this fact allows researchers to know the frequency of the different wind speeds. However, the proposal of this work consists of using these functions in a different way. The goal is to use Weibull functions for representing wind speed against wind power, and due to this, it must be clear that the interpretation is quite different. This way, the resulting functions cannot be considered as Weibull distributions, but only as Weibull functions used for the modeling of WT power curves. A comparison with simulations carried out by assuming logistic functions as power curves is presented. The reason for using logistic functions for this validation is that they are very good approximations, while the reasons for proposing the use of Weibull functions are that they are continuous, simpler than logistic functions and offer similar results. Additionally, an explanation about a software package has been discussed, which makes it easy to obtain Weibull functions for fitting WT power curves.

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Pogány, Tibor k., and Ram k. Saxena. "The gamma-Weibull distribution revisited." Anais da Academia Brasileira de Ciências 82, no. 2 (June 2010): 513–20. http://dx.doi.org/10.1590/s0001-37652010000200026.

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The five parameter gamma-Weibull distribution has been introduced by Leipnik and Pearce (2004). Nadarajah and Kotz (2007) have simplified it into four parameter form, using hypergeometric functions in some special cases. We show that the probability distribution function, all moments of positive order and the characteristic function of gamma-Weibull distribution of a random variable can be explicitely expressed in terms of the incomplete confluent Fox-Wright Psi-function, which is recently introduced by Srivastava and Pogány (2007). In the same time, we generalize certain results by Nadarajah and Kotz that follow as special cases of our findings.

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ALWAKEEL, ALI HUSEEN. "ON DISCRETE WEIBULL DISTRIBUTION." Journal of Economics and Administrative Sciences 20, no. 79 (October 1, 2014): 1–9. http://dx.doi.org/10.33095/jeas.v20i79.1969.

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Most of the Weibull models studied in the literature were appropriate for modelling a continuous random variable which assumes the variable takes on real values over the interval [0,∞]. One of the new studies in statistics is when the variables take on discrete values. The idea was first introduced by Nakagawa and Osaki, as they introduced discrete Weibull distribution with two shape parameters q and β where 0 < q < 1 and b > 0. Weibull models for modelling discrete random variables assume only non-negative integer values. Such models are useful for modelling for example; the number of cycles to failure when components are subjected to cyclical loading. Discrete Weibull models can be obtained as the discrete counterparts of either the distribution function or the failure rate function of the standard Weibull model. Which lead to different models. This paper discusses the discrete model which is the counterpart of the standard two-parameter Weibull distribution. It covers the determination of the probability mass function, cumulative distribution function, survivor function, hazard function, and the pseudo-hazard function.

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الوكيل, علي عبد الحسين. "ON DISCRETE WEIBULL DISTRIBUTION." Journal of Economics and Administrative Sciences 20, no. 79 (October 1, 2014): 1. http://dx.doi.org/10.33095/jeas.v20i79.807.

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Most of the Weibull models studied in the literature were appropriate for modelling a continuous random variable which assume the variable takes on real values over the interval [0,∞]. One of the new studies in statistics is when the variables takes on discrete values. The idea was first introduced by Nakagawa and Osaki, as they introduced discrete Weibull distribution with two shape parameters q and β where 0 < q < 1 and b > 0. Weibull models for modelling discrete random variables assume only non-negative integer values. Such models are useful for modelling for example; the number of cycles to failure when components are subjected to cyclical loading. Discrete Weibull models can be obtained as the discrete counter parts of either the distribution function or the failure rate function of the standard Weibull model. Which lead to different models. This paper discusses the discrete model which is the counter part of the standard two-parameter Weibull distribution. It covers the determination of the probability mass function, cumulative distribution function, survivor function, hazard function, and the pseudo-hazard function.

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Braimah, J. O., J. A. Adjekukor, N. Edike, and S. O. Elakhe. "A new Weibull Exponentiated Inverted Weibull Distribution for modelling positively-skewed data." Global Journal of Pure and Applied Sciences 27, no. 1 (March 5, 2021): 43–53. http://dx.doi.org/10.4314/gjpas.v27i1.6.

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An Exponentiated Inverted Weibull Distribution (EIWD) has a hazard rate (failure rate) function that is unimodal, thus making it less efficient for modeling data with an increasing failure rate (IFR). Hence, the need to generalize the EIWD in order to obtain a distribution that will be proficient in modeling these types of dataset (data with an increasing failure rate). This paper therefore, extends the EIWD in order to obtain Weibull Exponentiated Inverted Weibull (WEIW) distribution using the Weibull-Generator technique. Some of the properties investigated include the mean, variance, median, moments, quantile and moment generating functions. The explicit expressions were derived for the order statistics and hazard/failure rate function. The estimation of parameters was derived using the maximum likelihood method. The developed model was applied to a real-life dataset and compared with some existing competing lifetime distributions. The result revealed that the (WEIW) distribution provided a better fit to the real life dataset than the existing Weibull/Exponential family distributions.

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Ahmad, Zubair, and Zawar Hussain. "New Extended Weibull Distribution." Circulation in Computer Science 2, no. 6 (July 20, 2017): 14–19. http://dx.doi.org/10.22632/ccs-2017-252-31.

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This article considers a new function to propose a new lifetime model. The new model is introduced by utilizing the linear scheme of the two logarithms of cumulative hazard functions. The new model is named as new extended Weibull distribution, and is able to model data with unimodal or modified unimodal shaped failure rates. A brief explanation of the mathematical properties of the proposed model is provided. The model parameters will be estimated by deploying the maximum likelihood method. To illustrate the usefulness of the proposed model, an example will be discussed.

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Phani, K. K. "A New Modified Weibull Distribution Function." Journal of the American Ceramic Society 70, no. 8 (August 1987): C—182—C—184. http://dx.doi.org/10.1111/j.1151-2916.1987.tb05719.x.

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Hassan, Amal S., and Salwa M. Assar. "The Exponentiated Weibull-Power Function Distribution." Journal of Data Science 15, no. 4 (March 4, 2021): 589–614. http://dx.doi.org/10.6339/jds.201710_15(4).00002.

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Alizadeh, Morad, Muhammad Nauman Khan, Mahdi Rasekhi, and G.G Hamedani. "A New Generalized Modified Weibull Distribution." Statistics, Optimization & Information Computing 9, no. 1 (January 22, 2021): 17–34. http://dx.doi.org/10.19139/soic-2310-5070-1014.

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We introduce a new distribution, so called A new generalized modified Weibull (NGMW) distribution. Various structural properties of the distribution are obtained in terms of Meijer's $G$--function, such as moments, moment generating function, conditional moments, mean deviations, order statistics and maximum likelihood estimators. The distribution exhibits a wide range of shapes with varying skewness and assumes all possible forms of hazard rate function. The NGMW distribution along with other distributions are fitted to two sets of data, arising in hydrology and in reliability. It is shown that the proposed distribution has a superior performance among the compared distributions as evidenced via goodness--of--fit tests

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11

CHEN, ZHENMIN. "EXACT CONFIDENCE INTERVALS AND JOINT CONFIDENCE REGIONS FOR THE PARAMETERS OF THE WEIBULL DISTRIBUTIONS." International Journal of Reliability, Quality and Safety Engineering 11, no. 02 (June 2004): 133–40. http://dx.doi.org/10.1142/s0218539304001403.

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The Weibull distribution is widely adopted as a lifetime distribution. One of the characteristics the Weibull distribution possesses is that its cumulative distribution function can be expressed by closed form. Parameter estimation for the Weibull distribution has been discussed by many authors. Various methods have been proposed for constructing confidence intervals and joint confidence regions for the parameters of the Weibull distribution based on censored data. This paper discusses those methods that deal with exact confidence intervals or exact joint confidence regions for the parameters. One of the applications of the joint confidence regions of the parameters is to find confidence bounds for the functions of the parameters. In this paper, confidence bounds for the mean lifetime and reliability function for the Weibull distributions are discussed. Some unresolved problems for the exact confidences and joint confidence regions are mentioned in the discussion section.

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12

Mi, Zichuan, Saddam Hussain, and Christophe Chesneau. "On a Special Weighted Version of the Odd Weibull-Generated Class of Distributions." Mathematical and Computational Applications 26, no. 3 (August 29, 2021): 62. http://dx.doi.org/10.3390/mca26030062.

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In recent advances in distribution theory, the Weibull distribution has often been used to generate new classes of univariate continuous distributions. They find many applications in important disciplines such as medicine, biology, engineering, economics, informatics, and finance; their usefulness is synonymous with success. In this study, a new Weibull-generated-type class is presented, called the weighted odd Weibull generated class. Its definition is based on a cumulative distribution function, which combines a specific weighted odd function with the cumulative distribution function of the Weibull distribution. This weighted function was chosen to make the new class a real alternative in the first-order stochastic sense to two of the most famous existing Weibull generated classes: the Weibull-G and Weibull-H classes. Its mathematical properties are provided, leading to the study of various probabilistic functions and measures of interest. In a consequent part of the study, the focus is on a special three-parameter survival distribution of the new class defined with the standard exponential distribution as a reference. The exploratory analysis reveals a high level of adaptability of the corresponding probability density and hazard rate functions; the curves of the probability density function can be decreasing, reversed N shaped, and unimodal with heterogeneous skewness and tail weight properties, and the curves of the hazard rate function demonstrate increasing, decreasing, almost constant, and bathtub shapes. These qualities are often required for diverse data fitting purposes. In light of the above, the corresponding data fitting methodology has been developed; we estimate the model parameters via the likelihood function maximization method, the efficiency of which is proven by a detailed simulation study. Then, the new model is applied to engineering and environmental data, surpassing several generalizations or extensions of the exponential model, including some derived from established Weibull-generated classes; the Weibull-G and Weibull-H classes are considered. Standard criteria give credit to the proposed model; for the considered data, it is considered the best.

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Smadi, Mahmoud M., and Mahmoud H. Alrefaei. "New extensions of Rayleigh distribution based on inverted-Weibull and Weibull distributions." International Journal of Electrical and Computer Engineering (IJECE) 11, no. 6 (December 1, 2021): 5107. http://dx.doi.org/10.11591/ijece.v11i6.pp5107-5118.

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The Rayleigh distribution was proposed in the fields of acoustics and optics by lord Rayleigh. It has wide applications in communication theory, such as description of instantaneous peak power of received radio signals, i.e. study of vibrations and waves. It has also been used for modeling of wave propagation, radiation, synthetic aperture radar images, and lifetime data in engineering and clinical studies. This work proposes two new extensions of the Rayleigh distribution, namely the Rayleigh inverted-Weibull (RIW) and the Rayleigh Weibull (RW) distributions. Several fundamental properties are derived in this study, these include reliability and hazard functions, moments, quantile function, random number generation, skewness, and kurtosis. The maximum likelihood estimators for the model parameters of the two proposed models are also derived along with the asymptotic confidence intervals. Two real data sets in communication systems and clinical trials are analyzed to illustrate the concept of the proposed extensions. The results demonstrated that the proposed extensions showed better fitting than other extensions and competing models.

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14

Fang, Zhigang, Burton R. Patterson, and Malcolm E. Turner. "Modeling particle size distributions by the Weibull distribution function." Materials Characterization 31, no. 3 (October 1993): 177–82. http://dx.doi.org/10.1016/1044-5803(93)90058-4.

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15

Makubate, Boikanyo, Broderick O. Oluyede, Gofaone Motobetso, Shujiao Huang, and Adeniyi F. Fagbamigbe. "The Beta Weibull-G Family of Distributions: Model, Properties and Application." International Journal of Statistics and Probability 7, no. 2 (January 18, 2018): 12. http://dx.doi.org/10.5539/ijsp.v7n2p12.

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A new family of generalized distributions called the beta Weibull-G (BWG) distribution is proposed and developed. This new class of distributions has several new and well known distributions including exponentiated-G, Weibull-G, Rayleigh-G, exponential-G, beta exponential-G, beta Rayleigh-G, beta Rayleigh exponential, beta-exponential-exponential, Weibull-log-logistic distributions, as well as several other distributions such as beta Weibull-Uniform, beta Rayleigh-Uniform, beta exponential-Uniform, beta Weibull-log logistic and beta Weibull-exponential distributions as special cases. Series expansion of the density function, hazard function, moments, mean deviations, Lorenz and Bonferroni curves, R\'enyi entropy, distribution of order statistics and maximum likelihood estimates of the model parameters are given. Application of the model to real data set is presented to illustrate the importance and usefulness of the special case beta Weibull-log-logistic distribution.

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M. Kaliraja and K. Perarasan. "A Mathematical WGED – Model Approach on Short-Term High in Density Exercise Training, Attenuated Acute Exercise – Induced Growth Hormone Response." Mathematical Journal of Interdisciplinary Sciences 8, no. 1 (September 11, 2019): 1–5. http://dx.doi.org/10.15415/mjis.2019.81001.

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In the current manuscript, we have demonstrated the recent generalization of Weibull-G exponential distribution (three-parameter) and it is a very familiar distribution as compared to other distribution.It has been found that Weibull-G exponential distribution (WGED) can be utilized pretty efficiently to evaluate the biological data in the position of gamma and log-normal Weibull distributions. It has two shape parameters and the three scale parameters namely, a, b, λ. Some of its statistical properties are acquired, which includes reserved hazard function, probability-density function, hazard-rate function and survival function. Our aim is to shore-up the results of life-time using three-parameter Weibull generalized exponential distribution. Hence, the corresponding probability functions, hazard-rate function, survival function as well as reserved hazard-rate function has been analyzed in the 3 weeks of high-intensity exercise training in short-term. The outcomes of the present study supporting the results of life-time data that the interim elevated intensity exercise activity attenuated an acute exercise induced growth hormone release.

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Pogány, Tibor, and Abdus Saboor. "The gamma exponentiated exponential-Weibull distribution." Filomat 30, no. 12 (2016): 3159–70. http://dx.doi.org/10.2298/fil1612159p.

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Anewfour-parameter model called the gamma-exponentiated exponential-Weibull distribution is being introduced in this paper. The new model turns out to be quite flexible for analyzing positive data. Representations of certain statistical functions associated with this distribution are obtained. Some special cases are pointed out as well. The parameters of the proposed distribution are estimated by making use of the maximum likelihood approach. This density function is utilized to model two actual data sets. The new distribution is shown to provide a better fit than related distributions as measured by the Anderson-Darling and Cram?r-von Mises goodness-of-fit statistics. The proposed distribution may serve as a viable alternative to other distributions available in the literature for modeling positive data arising in various fields of scientific investigation such as the physical and biological sciences, hydrology, medicine, meteorology and engineering.

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Tahir, M. H., Morad Alizadeh, M. Mansoor, Gauss M. Cordeiro, and M. Zubair. "THE WEIBULL-POWER FUNCTION DISTRIBUTION WITH APPLICATIONS." Hacettepe Journal of Mathematics and Statistics 45, no. 42 (December 8, 2014): 1. http://dx.doi.org/10.15672/hjms.2014428212.

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Makubate, Boikanyo, Broderick O. Oluyede, Neo Dingalo, and Adeniyi Francis Fagbamigbe. "The Beta Log-Logistic Weibull Distribution: Model, Properties and Application." International Journal of Statistics and Probability 7, no. 6 (September 12, 2018): 49. http://dx.doi.org/10.5539/ijsp.v7n6p49.

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We propose and develop the properties of a new generalized distribution called the beta log-logistic Weibull (BLLoGW) distribution. This model contain several new distributions such as beta log-logistic Rayleigh, beta log-logistic exponential, exponentiated log-logistic Weibull, exponentiated log-logistic Rayleigh, exponentiated log-logistic exponential,  log-logistic Weibull, log-logistic Rayleigh and log-logistic distributions as special cases. Structural properties of this generalized distribution including series expansion of the probability density function and cumulative distribution function, hazard function, reverse hazard function, quantile function, moments, conditional moments, mean deviations, Bonferroni and Lorenz curves, R\'enyi entropy and distribution of order statistics are presented. The parameters of the distribution are estimated using maximum likelihood estimation technique. A Monte Carlo simulation study is conducted to examine the bias and mean square error of the maximum likelihood estimates. A real dataset is used to illustrate the applicability and usefulness of the new generalized distribution.

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Dokur, Emrah, Salim Ceyhan, and Mehmet Kurban. "Finsler Geometry for Two-Parameter Weibull Distribution Function." Mathematical Problems in Engineering 2017 (2017): 1–6. http://dx.doi.org/10.1155/2017/9720946.

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To construct the geometry in nonflat spaces in order to understand nature has great importance in terms of applied science. Finsler geometry allows accurate modeling and describing ability for asymmetric structures in this application area. In this paper, two-dimensional Finsler space metric function is obtained for Weibull distribution which is used in many applications in this area such as wind speed modeling. The metric definition for two-parameter Weibull probability density function which has shape (k) and scale (c) parameters in two-dimensional Finsler space is realized using a different approach by Finsler geometry. In addition, new probability and cumulative probability density functions based on Finsler geometry are proposed which can be used in many real world applications. For future studies, it is aimed at proposing more accurate models by using this novel approach than the models which have two-parameter Weibull probability density function, especially used for determination of wind energy potential of a region.

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Arshad, Rana Muhammad Imran, Christophe Chesneau, and Farrukh Jamal. "The Odd Gamma Weibull-Geometric Model: Theory and Applications." Mathematics 7, no. 5 (May 2, 2019): 399. http://dx.doi.org/10.3390/math7050399.

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In this paper, we study a new four-parameter distribution called the odd gamma Weibull-geometric distribution. Having the qualities suggested by its name, the new distribution is a special member of the odd-gamma-G family of distributions, defined with the Weibull-geometric distribution as baseline, benefiting of their respective merits. Firstly, we present a comprehensive account of its mathematical properties, including shapes, asymptotes, quantile function, quantile density function, skewness, kurtosis, moments, moment generating function and stochastic ordering. Then, we focus our attention on the statistical inference of the corresponding model. The maximum likelihood estimation method is used to estimate the model parameters. The performance of this method is assessed by a Monte Carlo simulation study. An empirical illustration of the new distribution is presented by the analyses two real-life data sets. The results of the proposed model reveal to be better as compared to those of the useful beta-Weibull, gamma-Weibull and Weibull-geometric models.

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Jiang, Lichun, and John R. Brooks. "Predicting Diameter Distributions for Young Longleaf Pine Plantations in Southwest Georgia." Southern Journal of Applied Forestry 33, no. 1 (February 1, 2009): 25–28. http://dx.doi.org/10.1093/sjaf/33.1.25.

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Abstract Parameter prediction equations for the Weibull distribution function were developed based on four percentile functions and a parameter recovery method for longleaf pine (Pinus palustris Mill.) in Southwest Georgia. Four percentiles were expressed as functions of stand-level characteristics based on stepwise regression and seemingly unrelated regression. Using a percentile-based parameter recovery method (PCT), estimated diameter distributions were obtained from available stand-level variables. The PCT method was also compared with a cumulative distribution function (CDF) regression method. The PCT method produced consistently better goodness-of-fit statistics than the CDF method. The results indicate that diameter distribution in longleaf pine stands can be successfully characterized with the Weibull function.

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Aminu Adamu, Abubakar Yahaya, and Hussaini Garba Dikko. "APPLICATIONS OF INVERSE WEIBULL RAYLEIGH DISTRIBUTION TO FAILURE RATES AND VINYL CHLORIDE DATA SETS." FUDMA JOURNAL OF SCIENCES 5, no. 2 (June 22, 2021): 89–99. http://dx.doi.org/10.33003/fjs-2021-0502-479.

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In this work, a new three parameter distribution called the Inverse Weibull Rayleigh distribution is proposed. Some of its statistical properties were presented. The PDF plot of Inverse Weibull Rayleigh distribution showed that it is good for modeling positively skewed and symmetrical datasets. The plot of the hazard function showed that the proposed distribution can fit datasets with bathtub shape. Method of maximum likelihood estimation was employed to estimate the parameters of the distribution, the estimators of the parameters of Inverse Weibull Rayleigh distribution is asymptotically unbiased and asymptotically efficient from the result of the simulation carried out. Applying the new distribution to a positively skewed Vinyl Chloride data set shows that the distribution performs better than Rayleigh, Generalized Rayleigh, Weibull Rayleigh, Inverse Weibull, Inverse Weibull Weibull, Inverse Weibull Inverse Exponential and Inverse Weibull Pareto distribution in fitting the data as it has the smallest AIC value. Also, applying the new distribution to a negatively skewed bathtub shape failure rates data shows that the distribution is a competitive model after Weibull Rayleigh and Inverse Weibull Weibull distributions in fitting the data because it has the third least AIC value.

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Maghsoodloo, Saeed, and Dilcu Helvaci. "Renewal and Renewal-Intensity Functions with Minimal Repair." Journal of Quality and Reliability Engineering 2014 (March 19, 2014): 1–10. http://dx.doi.org/10.1155/2014/857437.

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The renewal and renewal-intensity functions with minimal repair are explored for the Normal, Gamma, Uniform, and Weibull underlying lifetime distributions. The Normal, Gamma, and Uniform renewal, and renewal-intensity functions are derived by the convolution method. Unlike these last three failure distributions, the Weibull except at shape β=1 does not have a closed-form function for the n-fold convolution. Since the Weibull is the most important failure distribution in reliability analyses, the approximate renewal and renewal-intensity functions of Weibull were obtained by the time-discretizing method using the Mean-Value Theorem for Integrals. A Matlab program outputs all reliability and renewal measures.

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Lai, C. D., Michael B. C. Khoo, K. Muralidharan, and M. Xie. "Weibull Model Allowing Nearly Instantaneous Failures." Journal of Applied Mathematics and Decision Sciences 2007 (September 24, 2007): 1–11. http://dx.doi.org/10.1155/2007/90842.

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A generalized Weibull model that allows instantaneous or early failures is modified so that the model can be expressed as a mixture of the uniform distribution and the Weibull distribution. Properties of the resulting distribution are derived; in particular, the probability density function, survival function, and the hazard rate function are obtained. Some selected plots of these functions are also presented. An R script was written to fit the model parameters. An application of the modified model is illustrated.

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O. Oluyede, Broderick, Huybrechts F. Bindele, Boikanyo Makubate, and Shujiao Huang. "A New Generalized Log-logistic and Modified Weibull Distribution with Applications." International Journal of Statistics and Probability 7, no. 3 (April 17, 2018): 72. http://dx.doi.org/10.5539/ijsp.v7n3p72.

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A new generalized distribution called the {\em log-logistic modified Weibull} (LLoGMW) distribution is presented. This distribution includes many submodels such as the log-logistic modified Rayleigh, log-logistic modified exponential, log-logistic Weibull, log-logistic Rayleigh, log-logistic exponential, log-logistic, Weibull, Rayleigh and exponential distributions as special cases. Structural properties of the distribution including the hazard function, reverse hazard function, quantile function, probability weighted moments, moments, conditional moments, mean deviations, Bonferroni and Lorenz curves, distribution of order statistics, L-moments and R\'enyi entropy are derived. Model parameters are estimated based on the method of maximum likelihood. Finally, real data examples are presented to illustrate the usefulness and applicability of the model.

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Nofal, Zohdy M., Ahmed Z. Afify, Haitham M. Yousof, Daniele C. T. Granzotto, and Francisco Louzada. "Kumaraswamy Transmuted Exponentiated Additive Weibull Distribution." International Journal of Statistics and Probability 5, no. 2 (February 22, 2016): 78. http://dx.doi.org/10.5539/ijsp.v5n2p78.

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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.

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Dessoky, Shimaa A., and Ahmed M. T. Abd El-Bar. "A New Five Parameter Lifetime Distribution: Properties and Application." JOURNAL OF ADVANCES IN MATHEMATICS 13, no. 3 (April 30, 2017): 7205–18. http://dx.doi.org/10.24297/jam.v13i3.6045.

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This paper deals with a new generalization of the Weibull distribution. This distribution is called exponentiated exponentiated exponential-Weibull (EEE-W) distribution. Various structural properties of the new probabilistic model are considered, such as hazard rate function, moments, moment generating function, quantile function, skewness, kurtosis, Shannon entropy and RÃ©nyi entropy. The maximum likelihood estimates of its unknown parameters are obtained. Finally, areal data set is analyzed and it observed that the present distribution can provide a better fit than some other known distributions.

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Marthin, Pius, and Gadde Srinivasa Rao. "Generalized Weibull–Lindley (GWL) Distribution in Modeling Lifetime Data." Journal of Mathematics 2020 (August 31, 2020): 1–15. http://dx.doi.org/10.1155/2020/2049501.

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In this manuscript, we have derived a new lifetime distribution named generalized Weibull–Lindley (GWL) distribution based on the T-X family of distribution specifically the generalized Weibull-X family of distribution. We derived and investigated the shapes of its probability density function (pdf), hazard rate function, and survival function. Some statistical properties such as quantile function, mode, median, order statistics, Shannon entropy, Galton skewness, and Moors kurtosis have been derived. Parameter estimation was done through maximum likelihood estimation (MLE) method. Monte Carlo simulation was conducted to check the performance of the parameter estimates. For the inference purpose, two real-life datasets were applied and generalized Weibull–Lindley (GWL) distribution appeared to be superior over its competitors including Lindley distribution, Akash distribution, new Weibull-F distribution, Weibull–Lindley (WL) distribution, and two-parameter Lindley (TPL) distribution.

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Anwar, Masood, and Amna Bibi. "The Half-Logistic Generalized Weibull Distribution." Journal of Probability and Statistics 2018 (2018): 1–12. http://dx.doi.org/10.1155/2018/8767826.

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A new three-parameter generalized distribution, namely, half-logistic generalized Weibull (HLGW) distribution, is proposed. The proposed distribution exhibits increasing, decreasing, bathtub-shaped, unimodal, and decreasing-increasing-decreasing hazard rates. The distribution is a compound distribution of type I half-logistic-G and Dimitrakopoulou distribution. The new model includes half-logistic Weibull distribution, half-logistic exponential distribution, and half-logistic Nadarajah-Haghighi distribution as submodels. Some distributional properties of the new model are investigated which include the density function shapes and the failure rate function, raw moments, moment generating function, order statistics, L-moments, and quantile function. The parameters involved in the model are estimated using the method of maximum likelihood estimation. The asymptotic distribution of the estimators is also investigated via Fisher’s information matrix. The likelihood ratio (LR) test is used to compare the HLGW distribution with its submodels. Some applications of the proposed distribution using real data sets are included to examine the usefulness of the distribution.

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Araki, Junichi, Hiromi Matsubara, Juichiro Shimizu, Takeshi Mikane, Satoshi Mohri, Ju Mizuno, Miyako Takaki, Tohru Ohe, Masahisa Hirakawa, and Hiroyuki Suga. "Weibull distribution function for cardiac contraction: integrative analysis." American Journal of Physiology-Heart and Circulatory Physiology 277, no. 5 (November 1, 1999): H1940—H1945. http://dx.doi.org/10.1152/ajpheart.1999.277.5.h1940.

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The Weibull distribution is widely used to analyze the cumulative loss of performance, i.e., breakdown, of a complex system in systems engineering. We found for the first time that the difference curve of two Weibull distribution functions almost identically fitted the isovolumically contracting left ventricular (LV) pressure-time curve [P( t)] in all 345 beats (3 beats at each of 5 volumes in 23 canine hearts; r = 0.999953 ± 0.000027; mean ± SD). The first derivative of the difference curve also closely fitted the first derivative of the P( t) curve. These results suggest the possibility that the LV isovolumic P( t) curve may be characterized by two counteracting cumulative breakdown systems. Of these, the first breakdown system causes a gradual pressure rise and the second breakdown system causes a gradual pressure fall. This Weibull-function model of the heart seems to give a new systems engineering or integrative physiological view of the logic underlying LV isovolumic pressure generation.

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Merovci, Faton, Ibrahim Elbatal, and Alaa Ahmed. "The Transmuted Generalized Inverse Weibull Distribution." Austrian Journal of Statistics 43, no. 2 (May 11, 2014): 119–31. http://dx.doi.org/10.17713/ajs.v43i2.28.

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A generalization of the generalized inverse Weibull distribution the so-called transmuted generalized inverse Weibull distribution is proposed and studied. We will use the quadratic rank transmutation map (QRTM) in order to generate a flexible family of probability distributions taking the generalized inverseWeibull distribution as the base value distribution by introducing a new parameter that would offer more distributional flexibility. Various structural properties including explicit expressions for the moments, quantiles, and moment generating function of the new distribution are derived. We propose the method of maximum likelihood for estimating the model parameters and obtain the observed information matrix. A real data set are used to compare the flexibility of the transmuted version versus the generalized inverse Weibull distribution.

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Kang, Byongjun, Soonyu Yoo, and Kyoohong Park. "Estimation of sewer deterioration by Weibull distribution function." Journal of the Korean Society of Water and Wastewater 34, no. 4 (August 30, 2020): 251–58. http://dx.doi.org/10.11001/jksww.2020.34.4.251.

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Gul, Ahtasham, Muhammad Mohsin, Muhammad Adil, and Mansoor Ali. "A modified truncated distribution for modeling the heavy tail, engineering and environmental sciences data." PLOS ONE 16, no. 4 (April 6, 2021): e0249001. http://dx.doi.org/10.1371/journal.pone.0249001.

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Truncated models are imperative to efficiently analyze the finite data that we observe in almost all the real life situations. In this paper, a new truncated distribution having four parameters named Weibull-Truncated Exponential Distribution (W-TEXPD) is developed. The proposed model can be used as an alternative to the Exponential, standard Weibull and shifted Gamma-Weibull and three parameter Weibull distributions. The statistical characteristics including cumulative distribution function, hazard function, cumulative hazard function, central moments, skewness, kurtosis, percentile and entropy of the proposed model are derived. The maximum likelihood estimation method is employed to evaluate the unknown parameters of the W-TEXPD. A simulation study is also carried out to assess the performance of the model parameters. The proposed probability distribution is fitted on five data sets from different fields to demonstrate its vast application. A comparison of the proposed model with some extant models is given to justify the performance of the W-TEXPD.

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Ahmad, Zubair, and Zawar Hussain. "Modified New Flexible Weibull Distribution." Circulation in Computer Science 2, no. 6 (July 20, 2017): 7–13. http://dx.doi.org/10.22632/ccs-2017-252-30.

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The present paper is devoted to introduce a four-parameter modification of new flexible Weibull distribution. The proposed model will be called modified new flexible Weibull distribution, able to model lifetime phenomena with increasing or bathtub-shaped failure rates. Some of its mathematical properties will be studied. The approach of maximum likelihood will be used for estimating the model parameters. A brief mathematical description for the reliability function will also be discussed. The usefulness of the proposed distribution will be illustrated by an application to a real data set.

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AL-Moisheer, A. S., Refah Mohammed Alotaibi, Ghadah A. Alomani, and H. Rezk. "Bivariate Mixture of Inverse Weibull Distribution: Properties and Estimation." Mathematical Problems in Engineering 2020 (March 28, 2020): 1–12. http://dx.doi.org/10.1155/2020/5234601.

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In this study, we construct a mixture of bivariate inverse Weibull distribution. We assumed that the parameters of two marginals have Bernoulli distributions. Several properties of the proposed model are obtained, such as probability marginal density function, probability marginal cumulative function, the product moment, the moment of the two variables x and y, the joint moment-generating function, and the correlation between x and y. The real dataset has been analyzed. We observed that the mixture bivariate inverse Weibull distribution provides a better fit than the other model.

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Cordeiro, Gaussm, Abdus Saboor, Muhammad Khan, and Serge Provost. "The transmuted generalized modified Weibull distribution." Filomat 31, no. 5 (2017): 1395–412. http://dx.doi.org/10.2298/fil1705395c.

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Canada EM provost@stats.uwo.ca AU Ortega Edwinm M. AF Universidade de S?o Paulo, Departamento de Ci?ncias Exatas, Piracicaba, Brazil EM edwin@usp.br KW Generalized modifiedWeibull distribution % Goodness-of-fit statistic % Lifetime data % Transmuted family % Weibull distribution KR nema A profusion of new classes of distributions has recently proven useful to applied statisticians working in various areas of scientific investigation. Generalizing existing distributions by adding shape parameters leads to more flexible models. We define a new lifetime model called the transmuted generalized modified Weibull distribution from the family proposed by Aryal and Tsokos [1], which has a bathtub shaped hazard rate function. Some structural properties of the new model are investigated. The parameters of this distribution are estimated using the maximum likelihood approach. The proposed model turns out to be quite flexible for analyzing positive data. In fact, it can provide better fits than related distributions as measured by the Anderson-Darling (A*) and Cram?r-von Mises (W*) statistics, which is illustrated by applying it to two real data sets. It may serve as a viable alternative to other distributions for modeling positive data arising in several fields of science such as hydrology, biostatistics, meteorology and engineering.

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Shahzad, Mirza Naveed, Ehsan Ullah, and Abid Hussanan. "Beta Exponentiated Modified Weibull Distribution: Properties and Application." Symmetry 11, no. 6 (June 12, 2019): 781. http://dx.doi.org/10.3390/sym11060781.

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One of the most prominent statistical distributions is the Weibull distribution. The recent modifications in this distribution have enhanced its application but only in specific fields. To introduce a more generalized Weibull distribution, in this work beta exponentiated modified Weibull distribution is established. This distribution consolidate the exponential, skewed and symmetric shapes into one density. The proposed distribution also contains nineteen lifetime distributions as a special case, which shows the flexibility of the distribution. The statistical properties of the proposed model are derived and discussed, including reliability analysis and order statistics. The hazard function of the proposed distribution can have a unimodal, decreasing, bathtub, upside-down bathtub, and increasing shape that make it effective in reliability analysis. The parameters of the proposed model are evaluated by maximum likelihood and least squares estimation methods. The significance of the beta exponentiated modified Weibull distribution for modeling is illustrated by the study of real data. The numerical study indicates that the new proposed distribution gives better results than other comparable distributions.

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Louzada, Francisco, Vitor Marchi, and James Carpenter. "The Complementary Exponentiated Exponential Geometric Lifetime Distribution." Journal of Probability and Statistics 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/502159.

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We proposed a new family of lifetime distributions, namely, complementary exponentiated exponential geometric distribution. This new family arises on a latent competing risk scenario, where the lifetime associated with a particular risk is not observable but only the maximum lifetime value among all risks. The properties of the proposed distribution are discussed, including a formal proof of its probability density function and explicit algebraic formulas for its survival and hazard functions, moments,rth moment of theith order statistic, mean residual lifetime, and modal value. Inference is implemented via a straightforwardly maximum likelihood procedure. The practical importance of the new distribution was demonstrated in three applications where our distribution outperforms several former lifetime distributions, such as the exponential, the exponential-geometric, the Weibull, the modified Weibull, and the generalized exponential-Poisson distribution.

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Salem, Hamdy M. "The Marshall–Olkin Generalized Inverse Weibull Distribution: Properties and Application." Modern Applied Science 13, no. 2 (January 3, 2019): 54. http://dx.doi.org/10.5539/mas.v13n2p54.

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In this paper, a new distribution namely, The Marshall–OlkinGeneralized Inverse Weibull Distribution is illustrated and studied. The new distribution is very flexible and contains sub-models such asinverse exponential, inverse Rayleigh, Weibull, inverse Weibull, Marshall–Olkininverse Weibull and Fréchetdistributions. Also, the hazard function of the new distribution can produce variety of forms:an increase, a decrease and an upside-down bathtub. Some properties such as hazard function, quintile function, entropy, moment generating function and order statistics are obtained. Different estimation approaches namely, maximum likelihood estimators, interval estimators, least square estimators, fisher information matrix and asymptotic confidence intervals are described. To illustrate the superior performance of the proposed distribution, a simulation study and a real data analysis are investigated against other models.

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Li, Meishen, and Xianguo Li. "MEP-type distribution function: a better alternative to Weibull function for wind speed distributions." Renewable Energy 30, no. 8 (July 2005): 1221–40. http://dx.doi.org/10.1016/j.renene.2004.10.003.

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Najarzadegan, Hossein, Mohammad Hossein Alamatsaz, and Saied Hayati. "Truncated Weibull-G More Flexible and More Reliable than Beta-G Distribution." International Journal of Statistics and Probability 6, no. 5 (July 20, 2017): 1. http://dx.doi.org/10.5539/ijsp.v6n5p1.

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Our purpose in this study includes introducing a new family of distributions as an alternative to beta-G (B-G) distribution with flexible hazard rate and greater reliability which we call Truncated Weibull-G (TW-G) distribution. We shall discuss several submodels of the family in detail. Then, its mathematical properties such as expansions, probability density function and cumulative distribution function, moments, moment generating function, order statistics, entropies, unimodality, stochastic comparison with the B-G distribution and stress-strength reliability function are studied. Moreover, we study shape of the density and hazard rate functions, and based on the maximum likelihood method, estimate parameters of the model. Finally, we apply the model to a real data set and compare B-G distribution with our proposed model.

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Zhang, Xian Shang, Guang Cai Wen, Qing Ming Long, and Hui Ming Yang. "Research on Log-Weibull Distribution and its Application in Statistical Distribution of Coal Cuttings Particle Size." Advanced Materials Research 634-638 (January 2013): 3472–77. http://dx.doi.org/10.4028/www.scientific.net/amr.634-638.3472.

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In this work, we reveals the statistical distribution law of 1-3mm coal particle size by MATLAB and the definition and probability distribution function of log-Weibull distribution. In order to get the theoretical probability model of the particle size distribution, we studied the parameter estimation of Weibull, lognormal and log-Weibull distribution though using the maximum likelihood estimation method. The comparison criterion Cc was introduced to evaluate the difference between the theoretical probability density model and the actual experimental data distribution. The results of comparison criterion shows when the particle size has a smaller intervals, log-Weibull distribution is better to predict the experimental data than Weibull distribution and has a smaller change with the different intervals of particle size than others.

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Badmus, N. I., K. A. Adeleke, and A. A. Olufolabo. "A NEW CLASS OF TRANSMUTED MODIFIED WEIGHTED WEIBULL DISTRIBUTION AND ITS PROPERTIES." Advances in Mathematics: Scientific Journal 10, no. 5 (May 14, 2021): 2527–36. http://dx.doi.org/10.37418/amsj.10.5.17.

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An additional parameter was added to Modified Weighted Weibull distribution with method of quadratic rank transmutation which led to a newly developed distribution called Transmuted Modified Weighted Weibull distribution. Two distributions that emanated from the new distribution are Transmuted Modified Rayleigh and Transmuted Modified Exponential distributions. Some properties of the distribution that were obtained include; the survival rate, hazard rate, reverse hazard rate function; and moment generating function, mean and variance. Also, parameters of the model were estimated using maximum likelihood estimation method. The model was applied to a life time data set of total milk production of the first birth of 107 cows which showed a better performance compared to some existing known distributions.

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Abid, Salah Hamza, Nadia Hashim Al-Noor, and Mohammad Abd Alhussein Boshi. "The Generalized Gamma – Exponentiated Weibull Distribution with its Properties." Al-Mustansiriyah Journal of Science 31, no. 2 (April 15, 2020): 30. http://dx.doi.org/10.23851/mjs.v31i2.775.

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In this paper, we present the Generalized Gamma-Exponentiated Weibull distribution as a special case of new generated Generalized Gamma - G family of probability distribution. The cumulative distribution, probability density, reliability and hazard rate functions are introduced. Furthermore, the most vital statistical properties, for instance, the r-th moment, characteristic function, quantile function, simulated data, Shannon and relative entropies besides the stress-strength model are obtained.

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Shoaib, Muhammad, Saif Ur Rehman, Imran Siddiqui, Shafiqur Rehman, Shamim Khan, and Zia Ibrahim. "Comparison of Weibull and Gaussian Mixture Models for Wind Speed Data Analysis." International Journal of Economic and Environmental Geology 11, no. 1 (July 6, 2020): 10–16. http://dx.doi.org/10.46660/ijeeg.vol11.iss1.2020.405.

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In order to have a reliable estimate of wind energy potential of a site, high frequency wind speed and direction data recorded for an extended period of time is required. Weibull distribution function is commonly used to approximate the recorded data distribution for estimation of wind energy. In the present study a comparison of Weibull function and Gaussian mixture model (GMM) as theoretical functions are used. The data set used for the study consists of hourly wind speeds and wind directions of 54 years duration recorded at Ijmuiden wind site located in north of Holland. The entire hourly data set of 54 years is reduced to 12 sets of hourly averaged data corresponding to 12 months. Authenticity of data is assessed by computing descriptive statistics on the entire data set without average and on monthly 12 data sets. Additionally, descriptive statistics show that wind speeds are positively skewed and most of the wind data points are observed to be blowing in south-west direction. Cumulative distribution and probability density function for all data sets are determined for both Weibull function and GMM. Wind power densities on monthly as well as for the entire set are determined from both models using probability density functions of Weibull function and GMM. In order to assess the goodness-of-fit of the fitted Weibull function and GMM, coefficient of determination (R2) and Kolmogorov-Smirnov (K-S) tests are also determined. Although R2 test values for Weibull function are much closer to ‘1’ compared to its values for GMM. Nevertheless, overall performance of GMM is superior to Weibull function in terms of estimated wind power densities using GMM which are in good agreement with the power densities estimated using wind data for the same duration. It is reported that wind power densities for the entire wind data set are 307 W/m2 and 403.96 W/m2 estimated using GMM and Weibull function, respectively.

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Shoaib, Muhammad, Saif Ur Rehman, Imran Siddiqui, Shafiqur Rehman, Shamim Khan, and Zia Ibrahim. "Comparison of Weibull and Gaussian Mixture Models for Wind Speed Data Analysis." International Journal of Economic and Environmental Geology 11, no. 1 (July 6, 2020): 10–16. http://dx.doi.org/10.46660/ojs.v11i1.405.

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In order to have a reliable estimate of wind energy potential of a site, high frequency wind speed and direction data recorded for an extended period of time is required. Weibull distribution function is commonly used to approximate the recorded data distribution for estimation of wind energy. In the present study a comparison of Weibull function and Gaussian mixture model (GMM) as theoretical functions are used. The data set used for the study consists of hourly wind speeds and wind directions of 54 years duration recorded at Ijmuiden wind site located in north of Holland. The entire hourly data set of 54 years is reduced to 12 sets of hourly averaged data corresponding to 12 months. Authenticity of data is assessed by computing descriptive statistics on the entire data set without average and on monthly 12 data sets. Additionally, descriptive statistics show that wind speeds are positively skewed and most of the wind data points are observed to be blowing in south-west direction. Cumulative distribution and probability density function for all data sets are determined for both Weibull function and GMM. Wind power densities on monthly as well as for the entire set are determined from both models using probability density functions of Weibull function and GMM. In order to assess the goodness-of-fit of the fitted Weibull function and GMM, coefficient of determination (R2) and Kolmogorov-Smirnov (K-S) tests are also determined. Although R2 test values for Weibull function are much closer to ‘1’ compared to its values for GMM. Nevertheless, overall performance of GMM is superior to Weibull function in terms of estimated wind power densities using GMM which are in good agreement with the power densities estimated using wind data for the same duration. It is reported that wind power densities for the entire wind data set are 307 W/m2 and 403.96 W/m2 estimated using GMM and Weibull function, respectively.

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Pieper, Patrick, André Düsterhus, and Johanna Baehr. "A universal Standardized Precipitation Index candidate distribution function for observations and simulations." Hydrology and Earth System Sciences 24, no. 9 (September 21, 2020): 4541–65. http://dx.doi.org/10.5194/hess-24-4541-2020.

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Abstract. The Standardized Precipitation Index (SPI) is a widely accepted drought index. Its calculation algorithm normalizes the index via a distribution function. Which distribution function to use is still disputed within the literature. This study illuminates that long-standing dispute and proposes a solution that ensures the normality of the index for all common accumulation periods in observations and simulations. We compare the normality of SPI time series derived with the gamma, Weibull, generalized gamma, and the exponentiated Weibull distribution. Our normality comparison is based on a complementary evaluation. Actual compared to theoretical occurrence probabilities of SPI categories evaluate the absolute performance of candidate distribution functions. Complementary, the Akaike information criterion evaluates candidate distribution functions relative to each other while analytically punishing complexity. SPI time series, spanning 1983–2013, are calculated from the Global Precipitation Climatology Project's monthly precipitation dataset, and seasonal precipitation hindcasts are from the Max Planck Institute Earth System Model. We evaluate these SPI time series over the global land area and for each continent individually during winter and summer. While focusing on regional performance disparities between observations and simulations that manifest in an accumulation period of 3 months, we additionally test the drawn conclusions for other common accumulation periods (1, 6, 9, and 12 months). Our results suggest that calculating SPI with the commonly used gamma distribution leads to deficiencies in the evaluation of ensemble simulations. Replacing it with the exponentiated Weibull distribution reduces the area of those regions where the index does not have any skill for precipitation obtained from ensemble simulations by more than one magnitude. The exponentiated Weibull distribution maximizes also the normality of SPI obtained from observational data and a single ensemble simulation. We demonstrate that calculating SPI with the exponentiated Weibull distribution delivers better results for each continent and every investigated accumulation period, irrespective of the heritage of the precipitation data. Therefore, we advocate the employment of the exponentiated Weibull distribution as the basis for SPI.

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Qiu, Li Juan, Cheng Ming Jin, and Ming Ma. "Research on Modeling of Sensor Nodes Reliability." Applied Mechanics and Materials 203 (October 2012): 247–51. http://dx.doi.org/10.4028/www.scientific.net/amm.203.247.

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To establish the model of wireless sensor network reliability, the sensor nodes’ reliability was assumed to be exponential distribution. By the methods of MLE and K-S , reliability distribution function of sensor nodes was verified in line with two parameters Weibull distribution . Basing on experimental dates , reliability distribution function of sensor nodes accorded with two parameters Weibull distribution and exponential distribution were in Eq.9 and Eq.10. Comparative results in Fig.5 shows that the two parameters Weibull distribution exceeds the exponential distribution to describe the distribution of sensor nodes.

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Chandrasekhar, B. K. "Estimation of Weibull parameter with a modified weight function." Journal of Materials Research 12, no. 10 (October 1997): 2638–42. http://dx.doi.org/10.1557/jmr.1997.0351.

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The Weibull modulus is widely used for estimating the reliability of ceramic components in engineering applications. An improvement in the evaluation of the Weibull modulus is achieved by using an appropriate weight function to the data points while fitting a straight line to the Weibull plot by the least square method. The conventional weight function is a function of the probability of failure. This paper describes an alternate method of obtaining the weight function based on first principles. This modified weight function is a function of the stress at failure rather than probability of failure. Evaluation of the two-parameter Weibull modulus was estimated on simulated strength distribution data with both the weight functions. A comparative analysis indicates that the modified weight function gives a different result than the conventional weight function. The paper also highlights the effect and importance of uncertainties in the measurement of strength on the calculated Weibull modulus.

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Journal articles on the topic 'Weibull distribution function'

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