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Journal articles on the topic 'Binomial distribution'

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Published: 4 June 2021

Last updated: 29 July 2024

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1

Sampath, S. "Hybrid Binomial Distribution." International Journal of Fuzzy System Applications 2, no. 4 (October 2012): 64–75. http://dx.doi.org/10.4018/ijfsa.2012100104.

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Buckley and Eslami (2003) introduced a discrete distribution, namely Fuzzy Binomial distribution, that is intended to suit situations wherein impreciseness and randomness coexist. Using the approach of Liu (2008), a hybrid (combination of imprecision and randomness) version of Binomial distribution is developed. With the help of genetic algorithms the process of computing the chance distributions, expectation and variance of the distribution that is developed in this paper are illustrated. Illustrative examples are given to justify the usefulness of the distribution.

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2

Kalev, Krasimir. "APPLICATION OF BINOMIAL DISTRIBUTION IN LOGISTICS SYSTEMS." Journal Scientific and Applied Research 6, no. 1 (November 12, 2014): 114–20. http://dx.doi.org/10.46687/jsar.v6i1.147.

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A downtime interval during which a machine is performing no work due to lack of a spare part is an important economical issue for companies. It has to know in advance the need for spare elements to ensure reliable operation of the machines. In order to determine the required amount of spare elements used a scientific approach. In this paper is proposed a well-known statistical approach to inventory management. The binomial distribution permits to analyze without difficulties the operational reliability and calculating the spare parts demand. Some of results are given by engineering software.

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3

Čekanavičius, V., and B. Roos. "Binomial Approximation to the Markov Binomial Distribution." Acta Applicandae Mathematicae 96, no. 1-3 (March 23, 2007): 137–46. http://dx.doi.org/10.1007/s10440-007-9114-1.

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4

Ehm, Werner. "Binomial approximation to the Poisson binomial distribution." Statistics & Probability Letters 11, no. 1 (January 1991): 7–16. http://dx.doi.org/10.1016/0167-7152(91)90170-v.

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5

Handayani, Deby. "Karakterisasi Sebaran Binomial Negatif-Binomial Negatif." Jurnal Penelitian Dan Pengkajian Ilmiah Eksakta 1, no. 2 (July 26, 2022): 94–97. http://dx.doi.org/10.47233/jppie.v1i2.558.

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This study discusses the convolution or the sum of independent and identical random variables, where the random variables are two distribution of Negative Binomial distribution so that the resulting distribution is known as the Negative Binomial - Negative Binomial. The purpose of this study is to find the characteristics of the distribution including the expected value, the variance value, the moment generating function and the characteristic function. This property is obtained by using theorems and lemmas that relate to the properties of a distribution. It is found that the expected value, variance value, moment generating function and characteristic function of the Negative Binomial-Exponential distribution are

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6

Anjela, Wanjala, and George Muhua. "Negative Binomial Three Parameter Lindley Distribution and Its Properties." International Journal of Theoretical and Applied Mathematics 10, no. 1 (June 14, 2024): 1–5. http://dx.doi.org/10.11648/j.ijtam.20241001.11.

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Many researchers have proposed mixed distributions as one of the most important methods for obtaining new probability distributions. Several studies have shown that mixed Negative Binomial distributions fits count data better than Poisson and Negative Binomial distribution itself. In this paper, we introduce a mixed distribution by mixing the distributions of negative binomial and three Parameter Lindley distribution. This new distribution has a thick tail and may be considered as an alternative for fitting count data with over dispersion. The parameters of the new distribution are estimated using MLE method and properties studied. Special cases of the new distribution and also identified. A simulation study carried out shows that the ML estimators give the parameter estimates close to the parameter when the sample is large, that is, the bias and variance of the parameter estimates decrease with increase in sample size showing the consistent nature of the new compound distribution. The study also compares the performance of the new distribution over distributions of Poisson, Negative Binomial, Negative Binomial oneParameter Lindley Distribution, Negative Binomial two Parameter distribution, three parameter Lindley distribution using a real count over dispersed dataset and the results shows that Negative Binomial three parameter Lindley distribution gave the smallest Kolmogorov Smirnov test statistic, AIC and BIC as compared to other distributions, hence the new distribution provided a better fit compared to other distributions under study for fitting over dispersed count data.

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7

Iso, C., and K. Mori. "Negative binomial multiplicity distribution from binomial cluster production." Zeitschrift für Physik C: Particles and Fields 46, no. 1 (March 1990): 59–61. http://dx.doi.org/10.1007/bf02440833.

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8

MUNTEANU, Bogdan-Gheorghe. "QUALITATIVE ASPECTS OF THE MIN PARETO BINOMIAL DISTRIBUTION." Review of the Air Force Academy 15, no. 2 (October 20, 2017): 63–68. http://dx.doi.org/10.19062/1842-9238.2017.15.2.8.

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9

Ross, G. J. S., and D. A. Preece. "The Negative Binomial Distribution." Statistician 34, no. 3 (1985): 323. http://dx.doi.org/10.2307/2987659.

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10

Omey, E., J. Santos, and Gulck Van. "A Markov-binomial distribution." Applicable Analysis and Discrete Mathematics 2, no. 1 (2008): 38–50. http://dx.doi.org/10.2298/aadm0801038o.

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11

Supanekar, S. R. "A Bivariate Binomial Distribution." International Journal of Scientific Research in Mathematical and Statistical Sciences 5, no. 5 (October 31, 2018): 60–64. http://dx.doi.org/10.26438/ijsrmss/v5i5.6064.

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12

Kuş, Coşkun, Yunus Akdoğan, Akbar Asgharzadeh, İsmail Kınacı, and Kadir Karakaya. "Binomial-Discrete Lindley Distribution." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 68, no. 1 (April 11, 2018): 401–11. http://dx.doi.org/10.31801/cfsuasmas.424228.

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13

Drezner, Zvi, and Nicholas Farnum. "A generalized binomial distribution." Communications in Statistics - Theory and Methods 22, no. 11 (January 1993): 3051–63. http://dx.doi.org/10.1080/03610929308831202.

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14

Pudprommarat, Chookait, and Winai Bodhisuwan. "Stochastic Orders Comparisons of Negative Binomial Distribution with Negative Binomial—Lindley Distribution." Open Journal of Statistics 02, no. 02 (2012): 208–12. http://dx.doi.org/10.4236/ojs.2012.22025.

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15

Rupassara, Upul, and Bishnu Sedai. "On the Convergence of Hypergeometric to Binomial Distributions." Computer and Information Science 16, no. 3 (July 24, 2023): 15. http://dx.doi.org/10.5539/cis.v16n3p15.

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This study presents a measure-theoretic approach to estimate the upper bound on the total variation of the di erence between hypergeometric and binomial distributions using the Kullback-Leibler information divergence. The binomial distribution can be used to find the probabilities associated with the binomial experiments. But if the sample size is large relative to the population size, the experiment may not be binomial, and a binomial distribution is not a good choice to find the probabilities associated with the experiment. The hypergeometric probability distribution is the appropriate probability model to be used when the sample size is large compared to the population size. An upper bound for the total variation in the distance between the hypergeometric and binomial distributions is derived using only the sample and population sizes. This upper bound is used to demonstrate how the hypergeometric distribution uniformly converges to the binomial distribution when the population size increases relative to the sample size.

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16

Lopez, Gonzalo, and Juan Aparicio. "Simple models for macro-parasite distributions in hosts." Brazilian Journal of Biometrics 41, no. 2 (June 20, 2023): 191–203. http://dx.doi.org/10.28951/bjb.v41i2.616.

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The Negative binomial distribution is the most used distribution to model macro-parasite burden in hosts. However, reliable maximum likelihood parameter estimation from data is far from trivial. No closed formula is available and numerical estimation requires sophisticated methods. Using data from the literature, we show that simple alternatives to negative binomial, like zero-inflated geometric or hurdle geometric distributions, produce in some cases a better fit to data than the negative binomial distribution. We derived simple closed formulas for the maximum likelihood parameter estimators which constitutes a significant advantage of these distributions over the negative binomial distribution.

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17

Nawaz, Rubab, Saira Zainab, Fairouz Tchier, Qin Xin, Afis Saliu, and Sarfraz Nawaz Malik. "Partial sums of analytic functions defined by binomial distribution and negative binomial distribution." Applied Mathematics in Science and Engineering 30, no. 1 (January 2, 2022): 554–72. http://dx.doi.org/10.1080/27690911.2022.2109630.

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18

Kostadinova, Krasimira Y., and Meglena D. Lazarova. "COUNTING DISTRIBUTIONS IN RISK THEORY." Annals of the Academy of Romanian Scientists Series on Mathematics and Its Application 13, no. 1-2 (2021): 20–44. http://dx.doi.org/10.56082/annalsarscimath.2022.1-2.20.

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In this paper we introduce some signiﬁcant counting distributionsin risk theory. The ﬁrst one is the I-Delaporte distribution. It is ageneralization of the Non-central negative binomial distribution. Thesecond distribution is the Non-central P´olya-Aeppli distribution. It isa sum of two independent random variables, one that is a Poisson andanother one, a P´olya-Aeppli distributed. The P´olya-Aeppli-Lindley,the compound P´olya and compound binomial distributions are alsoconsidered. They are mixed P´olya-Aeppli distribution with Lindleymixing distribution, compound negative binomial and compound bi-nomial distribution with geometric compounding distribution. Themain application of these distributions is that they can be used ascorresponding counting processes’ distributions in risk models

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19

Kostadinova, Krasimira Y., and Meglena D. Lazarova. "COUNTING DISTRIBUTIONS IN RISK THEORY." Annals of the Academy of Romanian Scientists Series on Mathematics and Its Application 13, no. 1-2 (2021): 20–44. http://dx.doi.org/10.56082/annalsarscimath.2021.1-2.20.

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In this paper we introduce some signiﬁcant counting distributionsin risk theory. The ﬁrst one is the I-Delaporte distribution. It is ageneralization of the Non-central negative binomial distribution. Thesecond distribution is the Non-central P´olya-Aeppli distribution. It isa sum of two independent random variables, one that is a Poisson andanother one, a P´olya-Aeppli distributed. The P´olya-Aeppli-Lindley,the compound P´olya and compound binomial distributions are alsoconsidered. They are mixed P´olya-Aeppli distribution with Lindleymixing distribution, compound negative binomial and compound bi-nomial distribution with geometric compounding distribution. Themain application of these distributions is that they can be used ascorresponding counting processes’ distributions in risk models

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20

Sirbiladze, Gia, Janusz Kacprzyk, Teimuraz Manjafarashvili, Bidzina Midodashvili, and Bidzina Matsaberidze. "New Fuzzy Extensions on Binomial Distribution." Axioms 11, no. 5 (May 9, 2022): 220. http://dx.doi.org/10.3390/axioms11050220.

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The use of discrete probabilistic distributions is relevant to many practical tasks, especially in present-day situations where the data on distribution are insufficient and expert knowledge and evaluations are the only instruments for the restoration of probability distributions. However, in such cases, uncertainty arises, and it becomes necessary to build suitable approaches to overcome it. In this direction, this paper discusses a new approach of fuzzy binomial distributions (BDs) and their extensions. Four cases are considered: (1) When the elementary events are fuzzy. Based on this information, the probabilistic distribution of the corresponding fuzzy-random binomial variable is calculated. The conditions of restrictions on this distribution are obtained, and it is shown that these conditions depend on the ratio of success and failure of membership levels. The formulas for the generating function (GF) of the constructed distribution and the first and second order moments are also obtained. The Poisson distribution is calculated as the limit case of a fuzzy-random binomial experiment. (2) When the number of successes is of a fuzzy nature and is represented as a fuzzy subset of the set of possible success numbers. The formula for calculating the probability of convolution of binomial dependent fuzzy events is obtained, and the corresponding GF is built. As a result, the scheme for calculating the mathematical expectation of the number of fuzzy successes is defined. (3) When the spectrum of the extended distribution is fuzzy. The discussion is based on the concepts of a fuzzy-random event and its probability, as well as the notion of fuzzy random events independence. The fuzzy binomial upper distribution is specifically considered. In this case the fuzziness is represented by the membership levels of the binomial and non-binomial events of the complete failure complex. The GF of the constructed distribution and the first-order moment of the distribution are also calculated. Sufficient conditions for the existence of a limit distribution and a Poisson distribution are also obtained. (4) As is known, based on the analysis of lexical material, the linguistic spectrum of the statistical process of word-formation becomes two-component when switching to vocabulary. For this, two variants of the hybrid fuzzy-probabilistic process are constructed, which can be used in the analysis of the linguistic spectrum of the statistical process of word-formation. A fuzzy extension of standard Fuchs distribution is also presented, where the fuzziness is reflected in the growing numbers of failures. For better representation of the results, the examples of fuzzy BD are illustrated in each section.

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21

Vellaisamy, P., and Abraham P. Punnen. "On the nature of the binomial distribution." Journal of Applied Probability 38, no. 1 (March 2001): 36–44. http://dx.doi.org/10.1239/jap/996986641.

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We examine how the binomial distribution B(n,p) arises as the distribution Sn = ∑i=1nXi of an arbitrary sequence of Bernoulli variables. It is shown that B(n,p) arises in infinitely many ways as the distribution of dependent and non-identical Bernoulli variables, and arises uniquely as that of independent Bernoulli variables. A number of illustrative examples are given. The cases B(2,p) and B(3,p) are completely analyzed to bring out some of the intrinsic properties of the binomial distribution. The conditions under which Sn follows B(n,p), given that Sn-1 is not necessarily a binomial variable, are investigated. Several natural characterizations of B(n,p), including one which relates the binomial distributions and the Poisson process, are also given. These results and characterizations lead to a better understanding of the nature of the binomial distribution and enhance the utility.

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22

Wiuf, Carsten, and Michael P. H. Stumpf. "Binomial subsampling." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2068 (January 17, 2006): 1181–95. http://dx.doi.org/10.1098/rspa.2005.1622.

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In this paper, we discuss statistical families with the property that if the distribution of a random variable X is in , then so is the distribution of Z ∼Bi( X , p ) for 0≤ p ≤1. (Here we take Z ∼Bi( X , p ) to mean that given X = x , Z is a draw from the binomial distribution Bi( x , p ).) It is said that the family is closed under binomial subsampling. We characterize such families in terms of probability generating functions and for families with finite moments of all orders we give a necessary and sufficient condition for the family to be closed under binomial subsampling. The results are illustrated with power series and other examples, and related to examples from mathematical biology. Finally, some issues concerning inference are discussed.

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23

Vellaisamy, P., and Abraham P. Punnen. "On the nature of the binomial distribution." Journal of Applied Probability 38, no. 01 (March 2001): 36–44. http://dx.doi.org/10.1017/s0021900200018489.

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We examine how the binomial distribution B(n,p) arises as the distribution S n = ∑ i=1 n X i of an arbitrary sequence of Bernoulli variables. It is shown that B(n,p) arises in infinitely many ways as the distribution of dependent and non-identical Bernoulli variables, and arises uniquely as that of independent Bernoulli variables. A number of illustrative examples are given. The cases B(2,p) and B(3,p) are completely analyzed to bring out some of the intrinsic properties of the binomial distribution. The conditions under which S n follows B(n,p), given that S n-1 is not necessarily a binomial variable, are investigated. Several natural characterizations of B(n,p), including one which relates the binomial distributions and the Poisson process, are also given. These results and characterizations lead to a better understanding of the nature of the binomial distribution and enhance the utility.

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24

Sasvari, Zoltan, and Richard Stong. "Approximating the Binomial Distribution: 10876." American Mathematical Monthly 110, no. 4 (April 2003): 341. http://dx.doi.org/10.2307/3647892.

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25

Binet, Francis E. "Fitting the Negative Binomial Distribution." Biometrics 42, no. 4 (December 1986): 989. http://dx.doi.org/10.2307/2530715.

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26

Montgomery, David B. "Comment: On Negative Binomial Distribution." Journal of Business & Economic Statistics 6, no. 2 (April 1988): 163. http://dx.doi.org/10.2307/1391553.

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27

Chalikias, Miltiadis S. "The Binomial Distribution in Shooting." Teaching Statistics 31, no. 3 (September 2009): 87–89. http://dx.doi.org/10.1111/j.1467-9639.2009.00344.x.

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28

Minkova, Leda D., and Edward Omey. "A New Markov Binomial Distribution." Communications in Statistics - Theory and Methods 43, no. 13 (June 5, 2014): 2674–88. http://dx.doi.org/10.1080/03610926.2012.681538.

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29

Bevrani, H., and S. Sharifi Far. "An Approximation to Binomial Distribution." Journal of Statistical Theory and Practice 9, no. 2 (July 7, 2014): 305–12. http://dx.doi.org/10.1080/15598608.2014.892445.

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30

Barone, Lorenzo, and Georgios Z. Voulgaridis. "Observations on inverse binomial distribution." Journal of Information and Optimization Sciences 21, no. 2 (May 2000): 323–35. http://dx.doi.org/10.1080/02522667.2000.10699455.

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31

Zhang, Zhongxin, Barbara A. Burtness, and Daniel Zelterman. "The maximum negative binomial distribution." Journal of Statistical Planning and Inference 87, no. 1 (May 2000): 1–19. http://dx.doi.org/10.1016/s0378-3758(99)00177-9.

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32

Biswas, Atanu, and Jing-Shiang Hwang. "A new bivariate binomial distribution." Statistics & Probability Letters 60, no. 2 (November 2002): 231–40. http://dx.doi.org/10.1016/s0167-7152(02)00323-1.

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33

Eryilmaz, Serkan. "Compound Markov negative binomial distribution." Journal of Computational and Applied Mathematics 292 (January 2016): 1–6. http://dx.doi.org/10.1016/j.cam.2015.06.026.

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34

Yousry, M. A., and R. C. Srivastava. "The Hyper-Negative Binomial Distribution." Biometrical Journal 29, no. 7 (1987): 875–84. http://dx.doi.org/10.1002/bimj.4710290720.

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35

Tripsiannis, Gregory A., Afroditi A. Papathanasiou, and Andreas N. Philippou. "Generalized distributions of orderkassociated with success runs in Bernoulli trials." International Journal of Mathematics and Mathematical Sciences 2003, no. 13 (2003): 801–15. http://dx.doi.org/10.1155/s0161171203207250.

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In a sequence of independent Bernoulli trials, by counting multidimensional lattice paths in order to compute the probability of a first-passage event, we derive and study a generalized negative binomial distribution of orderk, typeI, which extends to distributions of orderk, the generalized negative binomial distribution of Jain and Consul (1971), and includes as a special case the negative binomial distribution of orderk, typeI, of Philippou et al. (1983). This new distribution gives rise in the limit to generalized logarithmic and Borel-Tanner distributions and, by compounding, to the generalized Pólya distribution of the same order and type. Limiting cases are considered and an application to observed data is presented.

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36

Roos, Bero, and Bero Roos. "Binomial approximation to the Poisson binomial distribution: The Krawtchouk expansion." Teoriya Veroyatnostei i ee Primeneniya 45, no. 2 (2000): 328–44. http://dx.doi.org/10.4213/tvp466.

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37

Jaioun, K., and K. Teerapabolarn. "An improved negative binomial approximation for the beta binomial distribution." Applied Mathematical Sciences 8 (2014): 5529–32. http://dx.doi.org/10.12988/ams.2014.47568.

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38

Roos, B. "Binomial Approximation to the Poisson Binomial Distribution: The Krawtchouk Expansion." Theory of Probability & Its Applications 45, no. 2 (January 2001): 258–72. http://dx.doi.org/10.1137/s0040585x9797821x.

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39

Pethes, Róbert, and Levente Kovács. "An Exact and an Approximation Method to Compute the Degree Distribution of Inhomogeneous Random Graph Using Poisson Binomial Distribution." Mathematics 11, no. 6 (March 16, 2023): 1441. http://dx.doi.org/10.3390/math11061441.

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Inhomogeneous random graphs are commonly used models for complex networks where nodes have varying degrees of connectivity. Computing the degree distribution of such networks is a fundamental problem and has important applications in various fields. We define the inhomogeneous random graph as a random graph model where the edges are drawn independently and the probability of a link between any two vertices can be different for each node pair. In this paper, we present an exact and an approximation method to compute the degree distribution of inhomogeneous random graphs using the Poisson binomial distribution. The exact algorithm utilizes the DFT-CF method to compute the distribution of a Poisson binomial random variable. The approximation method uses the Poisson, binomial, and Gaussian distributions to approximate the Poisson binomial distribution.

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40

Kucukoglu, Irem, Burcin Simsek, and Yilmaz Simsek. "Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function." Axioms 8, no. 4 (October 11, 2019): 112. http://dx.doi.org/10.3390/axioms8040112.

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The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution.

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41

Whitaker, Thomas B., Francis G. Giesbrecht, Jeremy Wu, Winston M. Hagler, and Floyd E. Dowell. "Predicting the Distribution of Aflatoxin Test Results from Farmers’ Stock Peanuts." Journal of AOAC INTERNATIONAL 77, no. 3 (May 1, 1994): 659–66. http://dx.doi.org/10.1093/jaoac/77.3.659.

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Abstract Suitability of the negative binomial function for use in estimating the distribution of sample aflatoxin test results associated with testing farmers1 stock peanuts for aflatoxin was studied. A 900 kg portion of peanut pods was removed from each of 40 contaminated farmers1 stock lots. The lots averaged about 4100 kg. Each 900 kg portion was divided into fifty 2.26 kg samples, fifty 4.21 kg samples, and fifty 6.91 kg samples. The aflatoxin in each sample was quantified by liquid chromatography. An observed distribution of sample aflatoxin test results consisted of 50 aflatoxin test results for each lot and each sample size. The mean aflatoxin concentration, m; the variance, s2xamong the 50 sample aflatoxin test results; and the shape parameter, k, for the negative binomial function were determined for each of the 120 observed distributions (40 lots times 3 sample sizes). Regression analysis indicated the functional relationship between k and m to be k = 0.000006425m0.8047. The 120 observed distributions of sample aflatoxin test results were compared to the negative binomial function by using the Kolmogorov–Smirnov (KS) test. The null hypothesis that the true unknown distribution function was negative binomial was not rejected at the 5% significance level for 114 of the 120 distributions. The negative binomial function failed the KS test at a sample concentration of 0 ng/g in all 6 of the distributions where the negative binomial function was rejected. The negative binomial function always predicted a smaller percentage of samples testing 0 ng/g than was actually observed. However, the negative binomial function did fit the observed distribution for sample test results at a concentration greater than 0 in 4 of the 6 cases. As a result, the negative binomial function provides an accurate estimate of the acceptance probabilities associated with accepting contaminated lots of farmers' stock peanuts for various sample sizes and various sample acceptance levels greater than 0 ng/g.

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42

O’Kane, Dominic. "Approximating independent loss distributions with an adjusted binomial distribution." Journal of Credit Risk 7, no. 4 (December 2011): 103–17. http://dx.doi.org/10.21314/jcr.2012.133.

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43

Wiles, Lori J., Glenn W. Oliver, Alan C. York, Harvey J. Gold, and Gail G. Wilkerson. "Spatial Distribution of Broadleaf Weeds in North Carolina Soybean (Glycine max) Fields." Weed Science 40, no. 4 (December 1992): 554–57. http://dx.doi.org/10.1017/s0043174500058124.

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Spatial distribution of broadleaf weeds within 14 North Carolina soybean fields was characterized by fitting negative binomial distributions to frequency distributions of weed counts in each field. In most cases, the data could be represented by a negative binomial distribution. Estimated values of the parameter K of this distribution were small, often less than one, indicating a high degree of patchiness. The data also indicated that the population as a whole was patchy. Counts of individual species were positively correlated with each other in some fields and total weed count could be represented by a negative binomial for 12 of the 14 fields.

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44

Sahatsathatsana, Chanokgan, and Sattra Sahatsathatsana. "On Approximation of the Panjer Distribution by the Poisson and Binomial Distributions." WSEAS TRANSACTIONS ON MATHEMATICS 20 (September 30, 2021): 520–23. http://dx.doi.org/10.37394/23206.2021.20.55.

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The aim of this paper is to approximate the panjer distribution by the poisson and binomial distributions, where each bound is obtained by using the z-function and the Stein-Chen identity. For these bounds, it is indicated that a result of each of the Poisson and Binomial approximations yields a good approximation if both α and λ are small.

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45

Habib, Elsayed Ali. "Estimation of Log-Linear-Binomial Distribution with Applications." Journal of Probability and Statistics 2010 (2010): 1–13. http://dx.doi.org/10.1155/2010/423654.

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Log-linear-binomial distribution was introduced for describing the behavior of the sum of dependent Bernoulli random variables. The distribution is a generalization of binomial distribution that allows construction of a broad class of distributions. In this paper, we consider the problem of estimating the two parameters of log-linearbinomial distribution by moment and maximum likelihood methods. The distribution is used to fit genetic data and to obtain the sampling distribution of the sign test under dependence among trials.

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46

Raul, S. R. "On the beta-correlated binomial (bcb) distribution - a three parameter generalization of the binomial distribution." Communications in Statistics - Theory and Methods 16, no. 5 (January 1987): 1473–78. http://dx.doi.org/10.1080/03610928708829449.

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47

Whitaker, Thomas, Francis Giesbrecht, and Jeremy Wu. "Suitability of Several Statistical Models to Simulate Observed Distribution of Sample Test Results in Inspections of Aflatoxin-Contaminated Peanut Lots." Journal of AOAC INTERNATIONAL 79, no. 4 (July 1, 1996): 981–88. http://dx.doi.org/10.1093/jaoac/79.4.981.

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Abstract The acceptability of 10 theoretical distributions to simulate observed distribution of sample aflatoxin test results was evaluated by using 2 parameter estimation methods and 3 goodness of fit (GOF) tests. All theoretical distributions were compared with 120 observed distributions of aflatoxin test results of farmers' stock peanuts. For a given parameter estimation method and GOF test, the negative binomial distribution had the highest percentage of statistically acceptable fits. The log normal and Poisson-gamma (gamma shape parameter = 0.5) distributions had slightly fewer but an almost equal percentage of acceptable fits. For the 3 most acceptable statistical models, the negative binomial had the greatest percentage of best or closest fits. Both the parameter estimation method and the GOF test had an influence on which theoretical distribution had the largest number of acceptable fits. All theoretical distributions, except the negative binomial distribution, had more acceptable fits when model parameters were determined by the maximum likelihood method. The negative binomial had slightly more acceptable fits when model parameters were estimated by the method of moments. The results also demonstrated the importance of using the same GOF test for comparing the acceptability of several theoretical distributions.

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48

Peköz, Erol A., Adrian Röllin, Vydas Čekanavičius, and Michael Shwartz. "A Three-Parameter Binomial Approximation." Journal of Applied Probability 46, no. 4 (December 2009): 1073–85. http://dx.doi.org/10.1239/jap/1261670689.

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We approximate the distribution of the sum of independent but not necessarily identically distributed Bernoulli random variables using a shifted binomial distribution, where the three parameters (the number of trials, the probability of success, and the shift amount) are chosen to match the first three moments of the two distributions. We give a bound on the approximation error in terms of the total variation metric using Stein's method. A numerical study is discussed that shows shifted binomial approximations are typically more accurate than Poisson or standard binomial approximations. The application of the approximation to solving a problem arising in Bayesian hierarchical modeling is also discussed.

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49

Peköz, Erol A., Adrian Röllin, Vydas Čekanavičius, and Michael Shwartz. "A Three-Parameter Binomial Approximation." Journal of Applied Probability 46, no. 04 (December 2009): 1073–85. http://dx.doi.org/10.1017/s0021900200006148.

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Abstract:

We approximate the distribution of the sum of independent but not necessarily identically distributed Bernoulli random variables using a shifted binomial distribution, where the three parameters (the number of trials, the probability of success, and the shift amount) are chosen to match the first three moments of the two distributions. We give a bound on the approximation error in terms of the total variation metric using Stein's method. A numerical study is discussed that shows shifted binomial approximations are typically more accurate than Poisson or standard binomial approximations. The application of the approximation to solving a problem arising in Bayesian hierarchical modeling is also discussed.

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50

Gil, A., J. Segura, and N. M. Temme. "Asymptotic inversion of the binomial and negative binomial cumulative distribution functions." ETNA - Electronic Transactions on Numerical Analysis 52 (2020): 270–80. http://dx.doi.org/10.1553/etna_vol52s270.

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