Journal articles: 'Distributions'

1

Ejsmont, Wiktor. "Free Meixner Distributions." Didactics of Mathematics 13, no. 17 (2016): 13–16. http://dx.doi.org/10.15611/dm.2016.13.02.

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Omer, Abdeen. "Medicines Distribution, Regulatory Privatisation, Social Welfare Services and Financing Alternatives." International Journal of Medical Reviews and Case Reports 2, Reports in Surgery and Dermatolo (2018): 1. http://dx.doi.org/10.5455/ijmrcr.medicine-distributions-sudan.

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Sulewski, Piotr, and Marcin Szymkowiak. "Modelling income distributions based on theoretical distributions derived from normal distributions." Wiadomości Statystyczne. The Polish Statistician 2023, no. 6 (June 30, 2023): 1–23. http://dx.doi.org/10.59139/ws.2023.06.1.

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In income modelling studies, such well-known distributions as the Dagum, the lognormal or the Zenga distributions are often used as approximations of the observed distributions. The objective of the research described in the article is to verify the possibility of using other type of distributions, i.e. asymmetric distributions derived from normal distribution (ND) in the context of income modelling. Data from the 2011 EU-SILC survey on the monthly gross income per capita in Poland were used to assess the most important characteristics of the discussed distributions. The probability distributions were divided into two groups: I – distributions commonly used for income modelling (e.g. the Dagum distribution) and II – distributions derived from ND (e.g. the SU Johnson distribution). In addition to the visual evaluation of the usefulness of the analysed probability distributions, various numerical criteria were applied: information criteria for econometric models (such as the Akaike Information Criterion, Schwarz’s Bayesian Information Criterion and the Hannan-Quinn Information Criterion), measures of agreement, as well as empirical and theoretical characteristics, including a measure based on quantiles, specifically defined by the authors for the purposes of this article. The research found that the SU Johnson distribution (Group II), similarly to the Dagum distribution (Group I), can be successfully used for income modelling.

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Panton, Don B. "Distribution function values for logstable distributions." Computers & Mathematics with Applications 25, no. 9 (May 1993): 17–24. http://dx.doi.org/10.1016/0898-1221(93)90128-i.

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TRANDAFIR, Romică, Vasile PREDA, Sorin DEMETRIU, and Ion MIERLUŞ-MAZILU. "ON MIXING CONTINUOUS DISTRIBUTIONS WITH DISCRETE DISTRIBUTIONS USED IN RELIABILITY." Review of the Air Force Academy 16, no. 2 (October 31, 2018): 5–16. http://dx.doi.org/10.19062/1842-9238.2018.16.2.1.

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Marengo, James E., David L. Farnsworth, and Lucas Stefanic. "A Geometric Derivation of the Irwin-Hall Distribution." International Journal of Mathematics and Mathematical Sciences 2017 (2017): 1–6. http://dx.doi.org/10.1155/2017/3571419.

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The Irwin-Hall distribution is the distribution of the sum of a finite number of independent identically distributed uniform random variables on the unit interval. Many applications arise since round-off errors have a transformed Irwin-Hall distribution and the distribution supplies spline approximations to normal distributions. We review some of the distribution’s history. The present derivation is very transparent, since it is geometric and explicitly uses the inclusion-exclusion principle. In certain special cases, the derivation can be extended to linear combinations of independent uniform random variables on other intervals of finite length. The derivation adds to the literature about methodologies for finding distributions of sums of random variables, especially distributions that have domains with boundaries so that the inclusion-exclusion principle might be employed.

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Aguirre, M. A., and C. K. Li. "The distributional products of particular distributions." Applied Mathematics and Computation 187, no. 1 (April 2007): 20–26. http://dx.doi.org/10.1016/j.amc.2006.08.098.

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Clancy, Damian, and Philip K. Pollett. "A note on quasi-stationary distributions of birth–death processes and the SIS logistic epidemic." Journal of Applied Probability 40, no. 03 (September 2003): 821–25. http://dx.doi.org/10.1017/s002190020001977x.

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For Markov processes on the positive integers with the origin as an absorbing state, Ferrari, Kesten, Martínez and Picco studied the existence of quasi-stationary and limiting conditional distributions by characterizing quasi-stationary distributions as fixed points of a transformation Φ on the space of probability distributions on {1, 2, …}. In the case of a birth–death process, the components of Φ(ν) can be written down explicitly for any given distributionν. Using this explicit representation, we will show that Φ preserves likelihood ratio ordering between distributions. A conjecture of Kryscio and Lefèvre concerning the quasi-stationary distribution of the SIS logistic epidemic follows as a corollary.

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Cousineau, Denis, Jean-Philippe Thivierge, Bradley Harding, and Yves Lacouture. "Constructing a group distribution from individual distributions." Canadian Journal of Experimental Psychology/Revue canadienne de psychologie expérimentale 70, no. 3 (2016): 253–77. http://dx.doi.org/10.1037/cep0000069.

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Moya-Cessa, J. R., H. Moya-Cessa, L. R. Berriel-Valdos, O. Aguilar-Loreto, and P. Barberis-Blostein. "Unifying distribution functions: some lesser known distributions." Applied Optics 47, no. 22 (April 24, 2008): E13. http://dx.doi.org/10.1364/ao.47.000e13.

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Deshmukh, S. R., and M. S. Kasture. "BIVARIATE DISTRIBUTION WITH TRUNCATED POISSON MARGINAL DISTRIBUTIONS." Communications in Statistics - Theory and Methods 31, no. 4 (May 14, 2002): 527–34. http://dx.doi.org/10.1081/sta-120003132.

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12

Li, Kaican, and Zhi Geng. "The Noncentral Wishart Distribution and Related Distributions." Communications in Statistics - Theory and Methods 32, no. 1 (January 3, 2003): 33–45. http://dx.doi.org/10.1081/sta-120017798.

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13

Castillo, E., and J. Galambos. "Conditional distributions and the bivariate normal distribution." Metrika 36, no. 1 (December 1989): 209–14. http://dx.doi.org/10.1007/bf02614094.

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14

Voit, Eberhard O., and Shuiyang Yu. "The S-Distribution: Approximation of Discrete Distributions." Biometrical Journal 36, no. 2 (1994): 205–19. http://dx.doi.org/10.1002/bimj.4710360215.

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15

Suleiman, Ahmad Abubakar, Hanita Daud, Narinderjit Singh Sawaran Singh, Mahmod Othman, Aliyu Ismail Ishaq, and Rajalingam Sokkalingam. "A Novel Odd Beta Prime-Logistic Distribution: Desirable Mathematical Properties and Applications to Engineering and Environmental Data." Sustainability 15, no. 13 (June 28, 2023): 10239. http://dx.doi.org/10.3390/su151310239.

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In parametric statistical modeling, it is important to construct new extensions of existing probability distributions (PDs) that can make modeling data more flexible and help stakeholders make better decisions. In the present study, a new family of probability distributions (FPDs) called the odd beta prime generalized (OBP-G) FPDs is proposed to improve the traditional PDs. A new PD called the odd beta prime-logistic (OBP-logistic) distribution has been developed based on the developed OBP-G FPDs. Some desirable mathematical properties of the proposed OBP-logistic distribution, including the moments, moment-generating function, information-generating function, quantile function, stress–strength, order statistics, and entropies, are studied and derived. The proposed OBP-logistic distribution’s parameters are determined by adopting the maximum likelihood estimation (MLE) method. The applicability of the new PD was demonstrated by employing three data sets and these were compared by the known extended logistic distributions, such as the gamma generalized logistic distribution, new modified exponential logistic distribution, gamma-logistic distribution, exponential modified Weibull logistic distribution, exponentiated Weibull logistic distribution, and transmuted Weibull logistic distribution. The findings reveal that the studied distribution provides better results than the competing PDs. The empirical results showed that the new OBP-logistic distribution performs better than the other PDs based on several statistical metrics. We hoped that the newly constructed OBP-logistic distribution would be an alternative to other well-known extended logistic distributions for the statistical modeling of symmetric and skewed data sets.

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Alhaji Modu Isa, Sule Omeiza Bashiru, Alhaji Buhari Ali, Akeem Ajibola Adepoju, and Ibrahim Ismaila Itopa. "Sine-Exponential Distribution: Its Mathematical Properties and Application to Real Dataset." UMYU Scientifica 1, no. 1 (September 30, 2022): 127–31. http://dx.doi.org/10.56919/usci.1122.017.

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To increase flexibility or to develop covariate models in various ways, new parameters can be added to existing families of distributions or a new family of distributions can be compounded with well-known standard normal distribution. In this paper, a trigonometric-type distribution was developed in order to come up with flexible distribution without adding parameters, considering Exponential distribution as the baseline distribution and Sine-G as the generator. The proposed distribution is referred to as Sine Exponential Distribution. Statistical features, including the moment, moment generating function, entropy, and order statistics were obtained. The proposed distribution's parameters were estimated using the Maximum Likelihood method. Using real datasets, the model's importance was demonstrated. The newly developed model was proven to be better than its competitors.

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17

Lindenberg, Björn, Jonas Nordqvist, and Karl-Olof Lindahl. "Conjugated Discrete Distributions for Distributional Reinforcement Learning." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 7 (June 28, 2022): 7516–24. http://dx.doi.org/10.1609/aaai.v36i7.20716.

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In this work we continue to build upon recent advances in reinforcement learning for finite Markov processes. A common approach among previous existing algorithms, both single-actor and distributed, is to either clip rewards or to apply a transformation method on Q-functions to handle a large variety of magnitudes in real discounted returns. We theoretically show that one of the most successful methods may not yield an optimal policy if we have a non-deterministic process. As a solution, we argue that distributional reinforcement learning lends itself to remedy this situation completely. By the introduction of a conjugated distributional operator we may handle a large class of transformations for real returns with guaranteed theoretical convergence. We propose an approximating single-actor algorithm based on this operator that trains agents directly on unaltered rewards using a proper distributional metric given by the Cramér distance. To evaluate its performance in a stochastic setting we train agents on a suite of 55 Atari 2600 games using sticky-actions and obtain state-of-the-art performance compared to other well-known algorithms in the Dopamine framework.

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18

Tamraz, Maissa, and Raluca Vernic. "ON THE EVALUATION OF MULTIVARIATE COMPOUND DISTRIBUTIONS WITH CONTINUOUS SEVERITY DISTRIBUTIONS AND SARMANOV'S COUNTING DISTRIBUTION." ASTIN Bulletin 48, no. 02 (January 17, 2018): 841–70. http://dx.doi.org/10.1017/asb.2017.46.

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AbstractIn this paper, we present closed-type formulas for some multivariate compound distributions with multivariate Sarmanov counting distribution and independent Erlang distributed claim sizes. Further on, we also consider a type-II Pareto dependency between the claim sizes of a certain type. The resulting densities rely on the special hypergeometric function, which has the advantage of being implemented in the usual software. We numerically illustrate the applicability and efficiency of such formulas by evaluating a bivariate cumulative distribution function, which is also compared with the similar function obtained by the classical recursion-discretization approach.

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19

Withers, Christopher S., and Saralees Nadarajah. "The distribution and quantiles of functionals of weighted empirical distributions when observations have different distributions." Statistics & Probability Letters 80, no. 13-14 (July 2010): 1093–102. http://dx.doi.org/10.1016/j.spl.2010.03.003.

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20

Alsolami, Eatemad, and Dawlah Alsulami. "Combining Two Exponentiated Families to Generate a New Family of Distributions." Symmetry 14, no. 8 (August 20, 2022): 1739. http://dx.doi.org/10.3390/sym14081739.

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This article presents a new technique to generate distributions that have the ability to fit any complex data called the exponentiated exponentiated Weibull-X (EEW-X) family, and the exponentiated exponentiated Weibull exponential (EEWE) distribution is presented as a member of this family. The new distribution’s unknown parameters were calculated by applying the maximum likelihood method. Some statistical properties, such as quantile, Rényi entropy, order statistics, and median are obtained for the proposed distribution. A simulation study was performed for different cases to investigate the estimation method’s performance. Three real datasets have been applied in which the new distribution has shown more flexibility compared to some other distributions.

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21

Lingappaiah, Giri S. "Generalized length biased distributions." Applications of Mathematics 33, no. 5 (1988): 354–61. http://dx.doi.org/10.21136/am.1988.104316.

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22

Kucera, Jan, and Carlos Bosch. "Multipliers of temperate distributions." Mathematica Bohemica 130, no. 3 (2005): 225–29. http://dx.doi.org/10.21136/mb.2005.134095.

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23

Kabir, Umar, and Terna Godfrey IEREN. "On the inferences and applications of transmuted exponential Lomax distribution." International Journal of Advanced Statistics and Probability 6, no. 1 (January 13, 2018): 30. http://dx.doi.org/10.14419/ijasp.v6i1.8129.

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This article proposed a new distribution referred to as the transmuted Exponential Lomax distribution as an extension of the popular Lomax distribution in the form of Exponential Lomax by using the Quadratic rank transmutation map proposed and studied in earlier research. Using the transmutation map, we defined the probability density function (PDF) and cumulative distribution function (CDF) of the transmuted Exponential Lomax distribution. Some properties of the new distribution were extensively studied after derivation. The estimation of the distribution’s parameters was also done using the method of maximum likelihood estimation. The performance of the proposed probability distribution was checked in comparison with some other generalizations of Lomax distribution using three real-life data sets. The results obtained indicated that TELD performs better than the other distributions comprising power Lomax, Exponential-Lomax, and the Lomax distributions.

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Mukherjee, Asmita, Sreeraj Nair, and Vikash Kumar Ojha. "Wigner Distributions of Quark." International Journal of Modern Physics: Conference Series 40 (January 2016): 1660055. http://dx.doi.org/10.1142/s2010194516600557.

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Wigner distribution functions are the quantum analogue of the classical phase space distribution and being quantum implies that they are not genuine phase space distribution and thus lack any probabilistic interpretation. Nevertheless, Wigner distributions are still interesting since they can be related to both generalized parton distributions (GPDs) and transverse momentum dependent parton distributions (TMDs) under some limit. We study the Wigner distribution of quarks and also the orbital angular momentum (OAM) of quarks in the dressed quark model.

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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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26

Papalexiou, S. M., D. Koutsoyiannis, and C. Makropoulos. "How extreme is extreme? An assessment of daily rainfall distribution tails." Hydrology and Earth System Sciences Discussions 9, no. 5 (May 2, 2012): 5757–78. http://dx.doi.org/10.5194/hessd-9-5757-2012.

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Abstract. The upper part of a probability distribution, usually known as the tail, governs both the magnitude and the frequency of extreme events. The tail behaviour of all probability distributions may be, loosely speaking, categorized in two families: heavy-tailed and light-tailed distributions, with the latter generating more "mild" and infrequent extremes compared to the former. This emphasizes how important for hydrological design is to assess correctly the tail behaviour. Traditionally, the wet-day daily rainfall has been described by light-tailed distributions like the Gamma, although heavier-tailed distributions have also been proposed and used, e.g. the Lognormal, the Pareto, the Kappa, and others. Here, we investigate the issue of tails for daily rainfall by comparing the upper part of empirical distributions of thousands of records with four common theoretical tails: those of the Pareto, Lognormal, Weibull and Gamma distributions. Specifically, we use 15 029 daily rainfall records from around the world with record lengths from 50 to 163 yr. The analysis shows that heavier-tailed distributions are in better agreement with the observed rainfall extremes than the more often used lighter tailed distributios, with clear implications on extreme event modelling and engineering design.

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Abu El Azm, Wael S., Ehab M. Almetwally, Sundus Naji AL-Aziz, Abd Al-Aziz H. El-Bagoury, Randa Alharbi, and O. E. Abo-Kasem. "A New Transmuted Generalized Lomax Distribution: Properties and Applications to COVID-19 Data." Computational Intelligence and Neuroscience 2021 (October 7, 2021): 1–14. http://dx.doi.org/10.1155/2021/5918511.

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A new five-parameter transmuted generalization of the Lomax distribution (TGL) is introduced in this study which is more flexible than current distributions and has become the latest distribution theory trend. Transmuted generalization of Lomax distribution is the name given to the new model. This model includes some previously unknown distributions. The proposed distribution's structural features, closed forms for an rth moment and incomplete moments, quantile, and Rényi entropy, among other things, are deduced. Maximum likelihood estimate based on complete and Type-II censored data is used to derive the new distribution's parameter estimators. The percentile bootstrap and bootstrap-t confidence intervals for unknown parameters are introduced. Monte Carlo simulation research is discussed in order to estimate the characteristics of the proposed distribution using point and interval estimation. Other competitive models are compared to a novel TGL. The utility of the new model is demonstrated using two COVID-19 real-world data sets from France and the United Kingdom.

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C M, Latha, and Sandhya E. "CG and CEG Distributions with Uniform Secondary Distribution." International Journal of Scientific and Research Publications (IJSRP) 9, no. 12 (December 6, 2019): p9635. http://dx.doi.org/10.29322/ijsrp.9.12.2019.p9635.

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29

Dahel, S., and N. Giri. "Some distributions related to a noncentral wishart distribution." Communications in Statistics - Theory and Methods 23, no. 1 (January 1994): 229–37. http://dx.doi.org/10.1080/03610929408831249.

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30

Pudovkin, Alexander I., and Lutz Bornmann. "Approximation of citation distributions to the Poisson distribution." COLLNET Journal of Scientometrics and Information Management 12, no. 1 (January 2, 2018): 49–53. http://dx.doi.org/10.1080/09737766.2017.1332605.

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31

Nadarajah, Saralees, and Samuel Kotz. "Sampling distributions associated with the multivariate t distribution." Statistica Neerlandica 59, no. 2 (May 2005): 214–34. http://dx.doi.org/10.1111/j.1467-9574.2005.00288.x.

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32

Steliga, Katarzyna, and Dominik Szynal. "On counting distributions related to the Delaporte distribution." Applicationes Mathematicae 46, no. 1 (2019): 1–38. http://dx.doi.org/10.4064/am2337-8-2018.

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33

Schatzki, Thomas F. "Distribution of Aflatoxin in Pistachios. 1. Lot Distributions." Journal of Agricultural and Food Chemistry 43, no. 6 (June 1995): 1561–65. http://dx.doi.org/10.1021/jf00054a027.

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34

Boyd, Albert V. "Fitting the Truncated Pareto Distribution to Loss Distributions." Journal of the Staple Inn Actuarial Society 31 (March 1988): 151–58. http://dx.doi.org/10.1017/s2049929900010291.

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Hogg and Klugman use the truncated Pareto distribution with probability density functionwhere δ≥0 is specified and α > 0 and λ > 0 are unknown parameters, to describe insurance claims. This is fitted first of all by the method of moments, using the estimatorsand where is the mean of a simple random sample, and the (biased) varianceThe authors then suggest, on pp. 113–16, that these estimates be used as starting values in a Newton iteration to get the maximum likelihood estimates of the parameters, but this technique can fail as a result of convergence problems. The object of this note is to show that this has led Hogg and Klugman to underestimate seriously the area in the tail of a fitted loss distribution, and to discuss a method of circumventing this difficulty.

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35

Khmaladze, Estate. "Distribution free testing for conditional distributions given covariates." Statistics & Probability Letters 129 (October 2017): 348–54. http://dx.doi.org/10.1016/j.spl.2017.06.026.

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36

Liu, Tong, Ping Zhang, Wu-Sheng Dai, and Mi Xie. "An intermediate distribution between Gaussian and Cauchy distributions." Physica A: Statistical Mechanics and its Applications 391, no. 22 (November 2012): 5411–21. http://dx.doi.org/10.1016/j.physa.2012.06.035.

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37

Khmaladze, Estate. "Note on distribution free testing for discrete distributions." Annals of Statistics 41, no. 6 (December 2013): 2979–93. http://dx.doi.org/10.1214/13-aos1176.

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38

Newhall, Bruce, and Juan Arvelo. "Relating the distribution of bathymetry to clutter distributions." Journal of the Acoustical Society of America 118, no. 3 (September 2005): 2041. http://dx.doi.org/10.1121/1.4785846.

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39

Hady, Dina H. Abdel. "USE OF EXPONENTIAL DISTRIBUTION FOR HYBRIDIZATION OF DISTRIBUTIONS." Advances and Applications in Statistics 58, no. 1 (September 20, 2019): 57–75. http://dx.doi.org/10.17654/as058010057.

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40

Cirillo, Pasquale, Frank Redig, and Wioletta Ruszel. "Duality and stationary distributions of wealth distribution models." Journal of Physics A: Mathematical and Theoretical 47, no. 8 (February 10, 2014): 085203. http://dx.doi.org/10.1088/1751-8113/47/8/085203.

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41

Ahmadabadi, M. Nili, Y. Farjami, and M. B. Moghadam. "Approximating Distributions by Extended Generalized Lambda Distribution (XGLD)." Communications in Statistics - Simulation and Computation 41, no. 1 (January 2012): 1–23. http://dx.doi.org/10.1080/03610911003681503.

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Gharib, M. "Characterizations of the exponential distribution via mixing distributions." Microelectronics Reliability 36, no. 3 (March 1996): 293–305. http://dx.doi.org/10.1016/0026-2714(95)00131-x.

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43

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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44

Nguyen, T. T. "Conditional Distributions and Characterizations of Multivariate Stable Distribution." Journal of Multivariate Analysis 53, no. 2 (May 1995): 181–93. http://dx.doi.org/10.1006/jmva.1995.1031.

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45

Tomaselli, Domenico, Paul Stursberg, Michael Metzger, and Florian Steinke. "Learning probability distributions over georeferenced distribution grid models." Electric Power Systems Research 235 (October 2024): 110636. http://dx.doi.org/10.1016/j.epsr.2024.110636.

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46

Mačutek, Ján, Gejza Wimmer, and Michaela Koščová. "On a Parametrization of Partial-Sums Discrete Probability Distributions." Mathematics 10, no. 14 (July 16, 2022): 2476. http://dx.doi.org/10.3390/math10142476.

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For every discrete probability distribution, there is one and only one partial summation which leaves the distribution unchanged. This invariance property is reconsidered for distributions with one parameter. We show that if we change the parameter value in the function which defines the summation, two families of distributions can be observed. The first of them consists of distributions which are resistant to the change. For these distributions, the change of the parameter is reversed by the normalization constant, and the distributions remain unchanged. The other contains distributions sensitive to the change. Partial summations with the changed parameter value applied to sensitive distributions result in new distributions with two parameters. A necessary and sufficient condition for a distribution to be resistant to the parameter change is presented.

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Schneider, Uwe, Stephan Radonic, and Jürgen Besserer. "Tumor Volume Distributions Based on Weibull Distributions of Maximum Tumor Diameters." Applied Sciences 13, no. 19 (October 2, 2023): 10925. http://dx.doi.org/10.3390/app131910925.

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(1) Background: The distribution of tumor volumes is important for various aspects of cancer research. Unfortunately, tumor volume is rarely documented in tumor registries; usually only maximum tumor diameter is. This paper presents a method to derive tumor volume distributions from tumor diameter distributions. (2) Methods: The hypothesis is made that tumor maximum diameters d are Weibull distributed, and tumor volume is proportional to dk, where k is a parameter from the Weibull distribution of d. The assumption is tested by using a test dataset of 176 segmented tumor volumes and comparing the k obtained by fitting the Weibull distribution of d and from a direct fit of the volumes. Finally, tumor volume distributions are calculated from the maximum diameters of the SEER database for breast, NSCLC and liver. (3) Results: For the test dataset, the k values obtained from the two separate methods were found to be k = 2.14 ± 0.36 (from Weibull distribution of d) and 2.21 ± 0.25 (from tumor volume). The tumor diameter data from the SEER database were fitted to a Weibull distribution, and the resulting parameters were used to calculate the corresponding exponential tumor volume distributions with an average volume obtained from the diameter fit. (4) Conclusions: The agreement of the fitted k using independent data supports the presented methodology to obtain tumor volume distributions. The method can be used to obtain tumor volume distributions when only maximum tumor diameters are available.

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48

Müller, K., and W. D. Richter. "Exact distributions of order statistics from ln,p-symmetric sample distributions." Dependence Modeling 5, no. 1 (August 28, 2017): 221–45. http://dx.doi.org/10.1515/demo-2017-0013.

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Abstract We derive the exact distributions of order statistics from a finite number of, in general, dependent random variables following a joint ln,p-symmetric distribution. To this end,we first review the special cases of order statistics fromspherical aswell as from p-generalized Gaussian sample distributions from the literature. To study the case of general ln,p-dependence, we use both single-out and cone decompositions of the events in the sample space that correspond to the cumulative distribution function of the kth order statistic if they are measured by the ln,p-symmetric probability measure.We show that in each case distributions of the order statistics from ln,p-symmetric sample distribution can be represented as mixtures of skewed ln−ν,p-symmetric distributions, ν ∈ {1, . . . , n − 1}.

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49

Korolev, Victor. "Analytic and Asymptotic Properties of the Generalized Student and Generalized Lomax Distributions." Mathematics 11, no. 13 (June 27, 2023): 2890. http://dx.doi.org/10.3390/math11132890.

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Analytic and asymptotic properties of the generalized Student and generalized Lomax distributions are discussed, with the main focus on the representation of these distributions as scale mixtures of the laws that appear as limit distributions in classical limit theorems of probability theory, such as the normal, folded normal, exponential, Weibull, and Fréchet distributions. These representations result in the possibility of proving some limit theorems for statistics constructed from samples with random sizes in which the generalized Student and generalized Lomax distributions are limit laws. An overview of known properties of the generalized Student distribution is given, and some simple bounds for its tail probabilities are presented. An analog of the ‘multiplication theorem’ is proved, and the identifiability of scale mixtures of generalized Student distributions is considered. The normal scale mixture representation for the generalized Student distribution is discussed, and the properties of the mixing distribution in this representation are studied. Some simple general inequalities are proved that relate the tails of the scale mixture with that of the mixing distribution. It is proved that for some values of the parameters, the generalized Student distribution is infinitely divisible and admits a representation as a scale mixture of Laplace distributions. Necessary and sufficient conditions are presented that provide the convergence of the distributions of sums of a random number of independent random variables with finite variances and other statistics constructed from samples with random sizes to the generalized Student distribution. As an example, the convergence of the distributions of sample quantiles in samples with random sizes is considered. The generalized Lomax distribution is defined as the distribution of the absolute value of the random variable with the generalized Student distribution. It is shown that the generalized Lomax distribution can be represented as a scale mixture of folded normal distributions. The convergence of the distributions of maximum and minimum random sums to the generalized Lomax distribution is considered. It is demonstrated that the generalized Lomax distribution can be represented as a scale mixture of Weibull distributions or that of Fréchet distributions. As a consequence, it is demonstrated that the generalized Lomax distribution can be limiting for extreme statistics in samples with random size. The convergence of the distributions of mixed geometric random sums to the generalized Lomax distribution is considered, and the corresponding extension of the famous Rényi theorem is proved. The law of large numbers for mixed Poisson random sums is presented, in which the limit random variable has a generalized Lomax distribution.

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Darwish, Jumanah Ahmed, Saman Hanif Shahbaz, Lutfiah Ismail Al-Turk, and Muhammad Qaiser Shahbaz. "Some bivariate and multivariate families of distributions: Theory, inference and application." AIMS Mathematics 7, no. 8 (2022): 15584–611. http://dx.doi.org/10.3934/math.2022854.

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<abstract> <p>The bivariate and multivariate probability distributions are useful in joint modeling of several random variables. The development of bivariate and multivariate distributions is relatively tedious as compared with the development of univariate distributions. In this paper we have proposed a new method of developing bivariate and multivariate families of distributions from the univariate marginals. The properties of the proposed families of distributions have been studies. These properties include marginal and conditional distributions; product, ratio and conditional moments; joint reliability function and dependence measures. Statistical inference about the proposed families of distributions has also been done. The proposed bivariate family of distributions has been studied for Weibull baseline distribution giving rise to a new bivariate Weibull distribution. The properties of the proposed bivariate Weibull distribution have been studied alongside maximum likelihood estimation of the unknown parameters. The proposed bivariate Weibull distribution has been used for modeling of real bivariate data sets and we have found that the proposed bivariate Weibull distribution has been a suitable choice for the modeling of data used.</p> </abstract>

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Journal articles on the topic 'Distributions'

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