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Journal articles on the topic 'Beta Distribution'

Author: Grafiati

Published: 4 June 2021

Last updated: 8 September 2023

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1

Wood, A., and K. J. Beven. "On runoff generation and the distribution of storage deficits." Hydrology Research 44, no. 4 (March 14, 2013): 673–89. http://dx.doi.org/10.2166/nh.2013.119.

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A number of hydrological models use a distribution function to develop the non-linear rainfall–runoff catchment response. In this study the beta function is applied to represent a distribution of soil moisture storages in conjunction with a fast and slow pathway routing. The BETA3 and BETA4 modules, presented in this paper, have a distribution of discrete storage elements that have variable and redistributed water levels at each timestep. The PDM-BETA5 is an analytical solution with a similar structure to the commonly used probability distribution model (PDM). Model testing was performed on three catchments in the Northern Pennine region in England. The performances of the BETA models were compared with a commonly used formulation of the PDM. The BETA models performed marginally better than the PDM in calibration and parameter estimation was better with the BETA models than for the PDM. The BETA models had a small advantage in validation on the hydrologically fast responding test catchments.

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2

Gupta, Arjun K., and Daya K. Nagar. "Matrix-variate beta distribution." International Journal of Mathematics and Mathematical Sciences 24, no. 7 (2000): 449–59. http://dx.doi.org/10.1155/s0161171200002398.

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We propose matrix-variate beta type III distribution. Several properties of this distribution including Laplace transform, marginal distribution and its relationship with matrix-variate beta type I and type II distributions are also studied.

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3

Gupta, Arjun K., and Saralees Nadarajah. "Beta Bessel distributions." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–14. http://dx.doi.org/10.1155/ijmms/2006/16156.

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Three new distributions on the unit interval[0,1]are introduced which generalize the standard beta distribution. These distributions involve the Bessel function. Expression is derived for their shapes, particular cases, and thenth moments. Estimation by the method of maximum likelihood and Bayes estimation are discussed. Finally, an application to consumer price indices is illustrated to show that the proposed distributions are better models to economic data than one based on the standard beta distribution.

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4

Nagar, Daya K., and Arjun K. Gupta. "Matrix-variate Kummer-Beta distribution." Journal of the Australian Mathematical Society 73, no. 1 (August 2002): 11–26. http://dx.doi.org/10.1017/s1446788700008442.

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AbstractThis paper proposes matrix variate generalization of Kummer-Beta family of distributions which has been studied recently by Ng and Kotz. This distribution is an extension of Beta distribution. Its characteristic function has been derived and it is shown that the distribution is orthogonally invariant. Some results on distribution of random quadratic forms have also been derived.

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5

Awodutire, Phillip Oluwatobi, Oluwafemi Samson Balogun, Akintayo Kehinde Olapade, and Ethelbert Chinaka Nduka. "The modified beta transmuted family of distributions with applications using the exponential distribution." PLOS ONE 16, no. 11 (November 18, 2021): e0258512. http://dx.doi.org/10.1371/journal.pone.0258512.

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In this work, a new family of distributions, which extends the Beta transmuted family, was obtained, called the Modified Beta Transmuted Family of distribution. This derived family has the Beta Family of Distribution and the Transmuted family of distribution as subfamilies. The Modified beta transmuted frechet, modified beta transmuted exponential, modified beta transmuted gompertz and modified beta transmuted lindley were obtained as special cases. The analytical expressions were studied for some statistical properties of the derived family of distribution which includes the moments, moments generating function and order statistics. The estimates of the parameters of the family were obtained using the maximum likelihood estimation method. Using the exponential distribution as a baseline for the family distribution, the resulting distribution (modified beta transmuted exponential distribution) was studied and its properties. The modified beta transmuted exponential distribution was applied to a real life time data to assess its flexibility in which the results shows a better fit when compared to some competitive models.

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6

Díaz-García, José A. "Riesz and beta-Riesz distributions." Austrian Journal of Statistics 45, no. 2 (February 29, 2016): 35–51. http://dx.doi.org/10.17713/ajs.v45i2.55.

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This article derives several properties of the Riesz distributions, such as their corresponding Bartlett decompositions, the inverse Riesz distributions and the distribution of the generalised variance for real normed division algebras. In addition, introduce a kind of generalised beta distribution termed beta-Riesz distribution for real normed division algebras. Two versions of this distributions are proposed and some properties are studied.

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7

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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8

Okubo, Tomoya, and Shin-ichi Mayekawa. "Approximating score distributions using mixed-multivariate beta distribution." Behaviormetrika 44, no. 2 (March 20, 2017): 369–84. http://dx.doi.org/10.1007/s41237-017-0019-7.

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9

Polosin, V. G. "Shape measures for the generalized beta exponential distribution." Journal of Physics: Conference Series 2094, no. 2 (November 1, 2021): 022022. http://dx.doi.org/10.1088/1742-6596/2094/2/022022.

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Abstract This paper contains parametric and informational shape measures for various families of the generalized beta exponential distribution since it is important to determination of the distribution shape for analysing an experimental data set. A logistic parameter is used to select independent types of beta exponential distributions, that it allows to combine the distributions of different subfamilies. In this paper the use of parametric shape measures to pre-define distribution shape is discusses. In particular, the initial and standard central moments for the main types of generalized beta exponential distribution are given. In the paper it is proposes to use the entropy coefficient of unshifted distribution as an independent information measure of the shape of unshifted generalized beta exponential distributions. In order to increase the reliability of the preliminary determination of the shape of the model, expressions for the entropy coefficient of shifted families both the generalized beta exponential distributions of the first and second types, and the generalized gamma exponential distribution were obtained. For practical applied the entropy coefficients of unshifted distributions for various subfamilies of generalized beta exponential distributions can be useful.

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10

Adeleke, Maradesa. "Beta-Hyperhalfnormal Distribution and Its Application." BASRA JOURNAL OF SCIENCE 38, no. 2 (April 1, 2020): 131–56. http://dx.doi.org/10.29072/basjs.202021.

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This research developed Hyper halfnormal distribution (HHND) and Beta-hyper halfnormal distribution (BHHND). The statistical properties of those distributions were studied and BHHND is found to have bathtube hazard function. The distributions are fitted to lifetime data that seemed to have bathtube distribution. From the analysis, the definition of HHND depends on the value of p and q (mixing proportion) and coefficient of variation. For q > p, we have Hyper halfnormal and it is Hypo-halfnormal if otherwise. The BHHND is useful in modeling heavily skewed, non-normal data with bathtube hazard function.

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11

MirMostafaee, S. M. T. K., M. Mahdizadeh, and Saralees Nadarajah. "The beta Lindley distribution." Journal of Data Science 13, no. 2 (March 7, 2021): 603–26. http://dx.doi.org/10.6339/jds.201504_13(2).0011.

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12

MirMostafaee, S. M. T. K., M. Mahdizadeh, and Saralees Nadarajah. "The Beta Lindley Distribution." Journal of Data Science 13, no. 3 (April 8, 2021): 603–26. http://dx.doi.org/10.6339/jds.201507_13(3).0010.

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13

Kilany, Neveen, and H. M. Atallah. "Inverted Beta Lindley Distribution." JOURNAL OF ADVANCES IN MATHEMATICS 13, no. 1 (March 30, 2017): 7074–86. http://dx.doi.org/10.24297/jam.v13i1.5857.

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In this paper, a three-parameter continuous distribution, namely, Inverted Beta-Lindley (IBL) distribution is proposed and studied. The new model turns out to be quite flexible for analyzing positive data and has various shapes of density and hazard rate functions. Several statistical properties associated with this distribution are derived. Moreover, point estimation via method of moments and maximum likelihood method are studied and the observed information matrix is derived. An application of the new model to real data shows that it can give consistently a better fit than other important lifetime models.

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14

Akinsete, Alfred, Felix Famoye, and Carl Lee. "The beta-Pareto distribution." Statistics 42, no. 6 (December 2008): 547–63. http://dx.doi.org/10.1080/02331880801983876.

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15

Nekoukhou, V., M. H. Alamatsaz, H. Bidram, and A. H. Aghajani. "Discrete Beta-Exponential Distribution." Communications in Statistics - Theory and Methods 44, no. 10 (October 29, 2013): 2079–91. http://dx.doi.org/10.1080/03610926.2013.773348.

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16

Jones, M. C. "The complementary beta distribution." Journal of Statistical Planning and Inference 104, no. 2 (June 2002): 329–37. http://dx.doi.org/10.1016/s0378-3758(01)00260-9.

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17

Cordeiro, Gauss M., and Artur J. Lemonte. "The beta Laplace distribution." Statistics & Probability Letters 81, no. 8 (August 2011): 973–82. http://dx.doi.org/10.1016/j.spl.2011.01.017.

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18

Nadarajah, Saralees, and Samuel Kotz. "The beta exponential distribution." Reliability Engineering & System Safety 91, no. 6 (June 2006): 689–97. http://dx.doi.org/10.1016/j.ress.2005.05.008.

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19

Olkin, Ingram, and Ruixue Liu. "A bivariate beta distribution." Statistics & Probability Letters 62, no. 4 (May 2003): 407–12. http://dx.doi.org/10.1016/s0167-7152(03)00048-8.

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20

Jafari, Ali Akbar, Saeid Tahmasebi, and Morad Alizadeh. "The Beta-Gompertz Distribution." Revista Colombiana de Estadística 37, no. 1 (July 9, 2014): 141. http://dx.doi.org/10.15446/rce.v37n1.44363.

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21

Progri, Ilir F. "Exponential Generalized Beta Distribution." Journal of Geolocation, Geo-information and Geo-intelligence 2016, no. 1 (2016): 35. http://dx.doi.org/10.18610/jg3.2016.071603.

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22

Nadarajah, Saralees, and Samuel Kotz. "The beta Gumbel distribution." Mathematical Problems in Engineering 2004, no. 4 (2004): 323–32. http://dx.doi.org/10.1155/s1024123x04403068.

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The Gumbel distribution is perhaps the most widely applied statistical distribution for problems in engineering. In this paper, we introduce a generalization—referred to as the beta Gumbel distribution—generated from the logit of a beta random variable. We provide a comprehensive treatment of the mathematical properties of this new distribution. We derive the analytical shapes of the corresponding probability density function and the hazard rate function and provide graphical illustrations. We calculate expressions for thenth moment and the asymptotic distribution of the extreme order statistics. We investigate the variation of the skewness and kurtosis measures. We also discuss estimation by the method of maximum likelihood. We hope that this generalization will attract wider applicability in engineering.

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23

Moutinho Cordeiro, Gauss, and Rejane dos Santos Brito. "The beta power distribution." Brazilian Journal of Probability and Statistics 26, no. 1 (February 2012): 88–112. http://dx.doi.org/10.1214/10-bjps124.

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24

Nadarajah, Saralees. "The BivariateF2–Beta Distribution." American Journal of Mathematical and Management Sciences 27, no. 3-4 (February 2007): 351–68. http://dx.doi.org/10.1080/01966324.2007.10737705.

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25

GARCÍA, CATALINA BEATRIZ GARCÍA, JOSÉ GARCÍA PÉREZ, and SALVADOR CRUZ RAMBAUD. "THE GENERALIZED BIPARABOLIC DISTRIBUTION." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 17, no. 03 (June 2009): 377–96. http://dx.doi.org/10.1142/s0218488509005930.

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Beta distributions have been applied in a variety of fields in part due to its similarity to the normal distribution while allowing for a larger flexibility of skewness and kurtosis coverage when compared to the normal distribution. In spite of these advantages, the two-sided power (TSP) distribution was presented as an alternative to the beta distribution to address some of its short-comings, such as not possessing a cumulative density function (cdf) in a closed form and a difficulty with the interpretation of its parameters. The introduction of the biparabolic distribution and its generalization in this paper may be thought of in the same vein. Similar to the TSP distribution, the generalized biparabolic (GBP) distribution also possesses a closed form cdf, but contrary to the TSP distribution its density function is smooth at the mode. We shall demonstrate, using a moment ratio diagram comparison, that the GBP distribution provides for a larger flexibility in skewness and kurtosis coverage than the beta distribution when restricted to the unimodal domain. A detailed mean-variance comparison of GBP, beta and TSP distributions is presented in a Project Evaluation and Review Technique (PERT) context. Finally, we shall fit a GBP distribution to an example of financial European stock data and demonstrate a favorable fit of the GBP distribution compared to other distributions that have traditionally been used in that field, including the beta distribution.

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26

Polosin, V. G. "Measures of shape for the generalized beta exponential distribution." Journal of Physics: Conference Series 2094, no. 2 (November 1, 2021): 022064. http://dx.doi.org/10.1088/1742-6596/2094/2/022064.

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Abstract This paper contains parametric and informational measures of shape for various families of the generalized beta exponential distribution since it is important to determination of the distribution shape for analysing an experimental data set. A logistic parameter is used to select independent types of beta exponential distributions, that it allows to combine the distributions of different subfamilies. In this paper the use of parametric shape measures to predefine distribution shape is discusses. In particular, the initial and standard central moments for the main types of generalized beta exponential distribution are given. In the paper it is proposes to use the entropy coefficient of unshifted distribution as an independent information measure of the shape of unshifted generalized beta exponential distributions. In order to increase the reliability of the preliminary determination of the shape of the model, expressions for the entropy coefficient of shifted families both the generalized beta exponential distributions of the first and second types, and the generalized gamma exponential distribution were obtained. For practical applied the entropy coefficients of unshifted distributions for various subfamilies of generalized beta exponential distributions can be useful.

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27

Polosin, Vitaly. "Mapping of Beta Distribution for the Study of Dispersed Materials." Materials Science Forum 1049 (January 11, 2022): 295–304. http://dx.doi.org/10.4028/www.scientific.net/msf.1049.295.

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In the study of polydisperse materials, most of the experimental particle size distributions were obtained on bounded intervals. In these cases, it is also desirable to use bounded models with different shapes to simulate the results of studying polydisperse and powder materials. The beta distribution is often used to approximate results due to the fact that this distribution contains many forms for displaying realizations on a limited interval. With the development of computer technology, there has been an increased interest in the use of beta distribution in the modern practice of analyzing results. Meanwhile, there remains a limitation in the use of the beta distribution that is associated with the choice of distribution shape. The possibilities of using known shape measures for mapping beta distribution in this paper is discusses. On the example of the space of shape measure of kurtosis and skewness, the limited use of only probabilistic measures of shapes is illustrated. It is proposed to use the entropy coefficients as an additional informational parameter of the beta distribution shape. On the base of a features comparison of the entropy coefficients for biased and unbiased beta distributions, recommendations for their application are given. By using the example of beta distributions mapping in the space of asymmetry and the entropy coefficient, it is shown that the synergistic combination of probabilistic and informational measures of the shape allows expanding the possibilities of estimating the shape parameters beta distributions. Two methods to display the positions of realizations of beta distributions is proposed. There are trajectories on a constant ratio of shape and realizations position curve on equal values of one parameter. In particular, the features of the choice of beta distributions with negative skewness are discussed.

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28

Badmus, N. I., Olanrewaju Faweya, and K. A. Adeleke. "Generalized Beta-Exponential Weibull Distribution and its Applications." Journal of Statistics: Advances in Theory and Applications 24, no. 1 (December 10, 2020): 1–33. http://dx.doi.org/10.18642/jsata_7100122158.

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In this article, we investigate a distribution called the generalized beta-exponential Weibull distribution. Beta exponential x family of link function which is generated from family of generalized distributions is used in generating the new distribution. Its density and hazard functions have different shapes and contains special case of distributions that have been proposed in literature such as beta-Weibull, beta exponential, exponentiated-Weibull and exponentiated-exponential distribution. Various properties of the distribution were obtained namely; moments, generating function, Renyi entropy and quantile function. Estimation of model parameters through maximum likelihood estimation method and observed information matrix are derived. Thereafter, the proposed distribution is illustrated with applications to two different real data sets. Lastly, the distribution clearly shown that is better fitted to the two data sets than other distributions.

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29

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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30

Hurairah, Ahmed Ali, and Saeed A. Hassen. "SOME PROPERTIES OF BETA TRANSMUTED DAGUM DISTRIBUTION WITH APPLICATIONS." Indonesian Journal of Statistics and Its Applications 4, no. 2 (July 31, 2020): 327–40. http://dx.doi.org/10.29244/ijsa.v4i2.646.

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In this paper, we introduce a new family of continuous distributions called the beta transmuted Dagum distribution which extends the beta and transmuted familys. The genesis of the beta distribution and transmuted map is used to develop the so-called beta transmuted Dagum (BTD) distribution. The hazard function, moments, moment generating function, quantiles and stress-strength of the beta transmuted Dagum distribution (BTD) are provided and discussed in detail. The method of maximum likelihood estimation is used for estimating the model parameters. A simulation study is carried out to show the performance of the maximum likelihood estimate of parameters of the new distribution. The usefulness of the new model is illustrated through an application to a real data set.

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31

Polosin, V. G. "Shape measures of generalized beta distributions." Journal of Physics: Conference Series 2094, no. 2 (November 1, 2021): 022009. http://dx.doi.org/10.1088/1742-6596/2094/2/022009.

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Abstract This paper presents shape measures for generalized beta distributions that unit many subfamilies of distributions. For the study of complex systems, the information entropy of the whole family of the generalized beta distribution is obtained. The paper uses the interval of entropy uncertainty as an estimate of the entropy uncertainty for probable models, which are given in units of an observable random variable. The entropy uncertainty interval was used to construct the entropy coefficient of unbiased subfamilies of the generalized beta distribution. Particular entropy coefficients are given for frequently used subfamilies of beta distribution, that greatly facilitates the use of coefficients as independent information measures in determining the shape of models. The paper contains the most general formulas for probabilistic measures of the distributions shape also.

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32

Rodrigues, J. A., A. P. C. M. Silva, and G. G. Hamedani. "The beta exponentiated Lindley distribution." Journal of Statistical Theory and Applications 14, no. 1 (2015): 60. http://dx.doi.org/10.2991/jsta.2015.14.1.6.

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33

Moitra, Soumyo D. "Skewness and the Beta Distribution." Journal of the Operational Research Society 41, no. 10 (October 1990): 953. http://dx.doi.org/10.2307/2583273.

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34

Khan, Muhammad Nauman. "The Modified Beta Weibull Distribution." Hacettepe Journal of Mathematics and Statistics 45, no. 40 (November 23, 2014): 1. http://dx.doi.org/10.15672/hjms.2014408152.

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35

Barreto-Souza, Wagner, Alessandro H. S. Santos, and Gauss M. Cordeiro. "The beta generalized exponential distribution." Journal of Statistical Computation and Simulation 80, no. 2 (February 2010): 159–72. http://dx.doi.org/10.1080/00949650802552402.

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36

Castellares, F., L. C. Montenegro, and G. M. Cordeiro. "The beta log-normal distribution." Journal of Statistical Computation and Simulation 83, no. 2 (February 2013): 203–28. http://dx.doi.org/10.1080/00949655.2011.599809.

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37

Bidram, H., J. Behboodian, and M. Towhidi. "The beta Weibull-geometric distribution." Journal of Statistical Computation and Simulation 83, no. 1 (January 2013): 52–67. http://dx.doi.org/10.1080/00949655.2011.603089.

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38

Cordeiro, Gauss M., Antonio Eduardo Gomes, Cibele Queiroz da-Silva, and Edwin M. M. Ortega. "The beta exponentiated Weibull distribution." Journal of Statistical Computation and Simulation 83, no. 1 (January 2013): 114–38. http://dx.doi.org/10.1080/00949655.2011.615838.

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39

Cordeiro, Gauss M., and Artur J. Lemonte. "The Beta-Half-Cauchy Distribution." Journal of Probability and Statistics 2011 (2011): 1–18. http://dx.doi.org/10.1155/2011/904705.

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On the basis of the half-Cauchy distribution, we propose the called beta-half-Cauchy distribution for modeling lifetime data. Various explicit expressions for its moments, generating and quantile functions, mean deviations, and density function of the order statistics and their moments are provided. The parameters of the new model are estimated by maximum likelihood, and the observed information matrix is derived. An application to lifetime real data shows that it can yield a better fit than three- and two-parameter Birnbaum-Saunders, gamma, and Weibull models.

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40

Nagar, Daya K., Edwin Zarrazola, and Jessica Serna-Morales. "Generalized Bivariate Kummer-Beta Distribution." Ingeniería y Ciencia 16, no. 32 (November 2020): 7–31. http://dx.doi.org/10.17230/ingciencia.16.32.1.

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A new bivariate beta distribution based on the Humbert’s confluent hypergeometric function of the second kind is introduced. Various representations are derived for its product moments, marginal densities, marginal moments, conditional densities and entropies.

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41

Nadarajah, Saralees. "THE BIVARIATE F3-BETA DISTRIBUTION." Communications of the Korean Mathematical Society 21, no. 2 (April 1, 2006): 363–74. http://dx.doi.org/10.4134/ckms.2006.21.2.363.

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42

Cordeiro, Gauss M., Giovana O. Silva, and Edwin M. M. Ortega. "The beta-Weibull geometric distribution." Statistics 47, no. 4 (August 2013): 817–34. http://dx.doi.org/10.1080/02331888.2011.577897.

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43

Cordeiro, Gauss M., Fredy Castellares, Lourdes C. Montenegro, and Mário de Castro. "The beta generalized gamma distribution." Statistics 47, no. 4 (August 2013): 888–900. http://dx.doi.org/10.1080/02331888.2012.658397.

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44

Nadarajah, Saralees, Shou Hsing Shih, and Daya K. Nagar. "A new bivariate beta distribution." Statistics 51, no. 2 (October 5, 2016): 455–74. http://dx.doi.org/10.1080/02331888.2016.1240681.

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45

Pereira, Gustavo H. A., Denise A. Botter, and Mônica C. Sandoval. "The Truncated Inflated Beta Distribution." Communications in Statistics - Theory and Methods 41, no. 5 (March 2012): 907–19. http://dx.doi.org/10.1080/03610926.2010.530370.

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46

Moitra, Soumyo D. "Skewness and the Beta Distribution." Journal of the Operational Research Society 41, no. 10 (October 1990): 953–61. http://dx.doi.org/10.1057/jors.1990.147.

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47

L'Ecuyer, Pierre, and Richard Simard. "Inverting the symmetrical beta distribution." ACM Transactions on Mathematical Software 32, no. 4 (December 2006): 509–20. http://dx.doi.org/10.1145/1186785.1186786.

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48

Díaz-García, José A., and Ramón Gutiérrez Jáimez. "Singular matrix variate beta distribution." Journal of Multivariate Analysis 99, no. 4 (April 2008): 637–48. http://dx.doi.org/10.1016/j.jmva.2007.02.006.

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49

Cordeiro, Gauss M., and Artur J. Lemonte. "The McDonald inverted beta distribution." Journal of the Franklin Institute 349, no. 3 (April 2012): 1174–97. http://dx.doi.org/10.1016/j.jfranklin.2012.01.006.

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50

Alavi, Sayed Mohammad Reza, Safura Alibabaie, and Rahim Chinipardaz. "Size Biased Inflated Beta Distribution." Journal of Statistical Sciences 11, no. 2 (March 1, 2018): 285–95. http://dx.doi.org/10.29252/jss.11.2.285.

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