Systems of frequency distributions for water and environmental engineering. (English) Zbl 1514.60026
Summary: A wide spectrum of frequency distributions are used in hydrologic, hydraulic, environmental and water resources engineering. These distributions may have different origins, are based on different hypotheses, and belong to different generating systems. Review of literature suggests that different systems of frequency distributions employed in science and engineering in general and environmental and water engineering in particular have been derived using different approaches which include (1) differential equations, (2) distribution elasticity, (3) genetic theory, (4) generating functions, (5) transformations, (6) Bessel function, (7) expansions, and (8) entropy maximization. This paper revisits these systems of distributions and discusses the hypotheses that are used for deriving these systems. It also proposes, based on empirical evidence, another general system of distributions and derives a number of distributions from this general system that are used in environmental and water engineering.
MSC:
| 60E05 | Probability distributions: general theory |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 62E15 | Exact distribution theory in statistics |
Keywords:
frequency distributions; generating functions; differential equations; Bessel function; transformations; distribution elasticityReferences:
| [1] | Stoppa, G., A new generating system of income distribution models, Quarderni di Statistica e Mathematica Applicata alle Science Economico-Sociali, 12, 47-55 (1990) |
| [2] | Champernowne, D., The graduation of income distribution, Econometrica, 20, 591-625 (1952) · Zbl 0048.12802 |
| [3] | Fisk, P., The graduation of income distributions, Econometrica, 29, 171-185 (1961) · Zbl 0104.38702 |
| [4] | Mandelbrot, B., The Pareto-Levy law and the distribution of income, Internat. Econom. Rev., 1, 79-106 (1960) · Zbl 0201.51101 |
| [5] | Salem, A.; Mount, T., A convenient descriptive model of income distribution: The gamma density, Econometrica, 42, 1115-1127 (1974) |
| [6] | Thurow, L. C., Analyzing the American income distribution, Amer. Econ. Rev., 48, 261-269 (1970) |
| [7] | McDonald, J. B., Some generalized functions for the size distribution of income, Econometrics, 52, 3, 647-663 (1984) · Zbl 0557.62098 |
| [8] | Chieppa, M.; Amato, P., A new estimation procedure for the three-parameter lognormal distribution, (Tsallis, C.; etal., Statistical Distributions Scientific Work, vol. 5 (1981)), 133-140 · Zbl 0472.62036 |
| [9] | Pearson, K., Contributions to the mathematical theory of evolution. II. Skew variation in homogeneous material, Trans. R. Soc. Lond., A186, 343-415 (1895) · JFM 26.0243.03 |
| [10] | Burr, I. W., Cumulative frequency functions, Ann. Math. Stat., 13, 215-232 (1942) · Zbl 0060.29602 |
| [11] | Dagum, C., Generation and properties of income distribution functions, (Dagum, C.; Zenga, M., Income and Wealth Distribution, Inequality and Poverty (1990), Springer: Springer Berlin, Germany), 1-17 |
| [12] | D’Addario, R., Richerche sulla curva dei redditi, Giornale Degli Economisti e Annali di Economia, 8, 91-114 (1949) |
| [13] | Pareto, V., Cours d’Economie Politique. New Edition by G.H. Bousquet and G. Busino (1964) (1897), Librairie Droz: Librairie Droz Generva, Switzerland |
| [14] | Pearson, K., Contributions to the mathematical theory of evolution, Phil. Trans. R. Soc., 184 (1894) · JFM 25.0347.02 |
| [15] | Dagum, C., The generation and distribution of income, the Lorenz curve and Ginni ratio, Economic Applique, 33, 327-367 (1980) |
| [16] | Dagum, C., Generating systems and properties of income distribution models, Metron, 38, 3-26 (1980) · Zbl 0533.90023 |
| [17] | Dagum, C., Income distribution models, (Kotz, S.; Johnson, N. L.; Read, C., Encyclopedia of Statistical Sciences, vol. 4 (1983), Wiley: Wiley New York), 27-34 |
| [18] | Johnson, N. L.; Kotz, S.; Balakrishnan, N., Continuous Univariate Distributions, Vol.1 (1994), John Wiley: John Wiley New York · Zbl 0811.62001 |
| [19] | Dagum, C., A new model of personal income distribution: Specification and estimation, Economic Appliquee, XXX, 3, 413-436 (1977) |
| [20] | Mielke, P. W., Another family of distributions for describing and analyzing precipitation data, J. Appl. Meteorol., 11, 275-280 (1973) |
| [21] | Papalexiou, S. M.; Koutsoyiannis, D., Entropy-based derivation of probability distributions: A case study to daily rainfall, Adv. Water Resour., 45, 51-57 (2012) |
| [22] | Hao, Z.; Singh, V. P., Entropy-based parameter estimation for extended Burr XII distribution, Stoch. Environ. Res. Risk Assess, 23, 1113-1122 (2009) · Zbl 1416.62124 |
| [23] | Hao, Z.; Singh, V. P., Entropy-based parameter estimation for extended three-parameter Burr XII distribution for low-flow frequency analysis, Trans. Amer. Soc. Agric. Biol. Engineers, 52, 4, 1-10 (2009) |
| [24] | Ducey, M. J.; Gove, J. H., Size-biased distribution in the generalized beta distribution family, with application to forestry, Forestry, 88, 143-151 (2015) |
| [25] | Brouers, F., Statistical foundation of empirical isotherms, Open J. Stat., 4, 687-701 (2014) |
| [26] | Evans, M. A., Burr critical value approximation for tests of autocorrelation and heteroscedasticity, Aust. J. Stat., 34, 3, 433-442 (1992) |
| [27] | Paranaiba, P. F.; Ortega, E. M.M.; Cordeiro, G. M.; Pescim, R. R., The beta Burr XII distribution with application to lifetime data, Comput. Statist. Data Anal., 55, 1118-1136 (2011) · Zbl 1284.62108 |
| [28] | Singh, S.; Maddala, G., A function for the size distribution of income, Econometrica, 44, 963-970 (1976) |
| [29] | Papadopoulos, A. S., The Burr distribution as a failure model from a Bayesian approach, IEEE Trans. Reliab., R-27, 5, 369-371 (1978) · Zbl 0392.62080 |
| [30] | Edgeworth, F. Y., On the representation of statistics by mathematical formulae, J. Roy. Soc., 1, 670-700 (1898) |
| [31] | Bartels, C. P.A., Economic Aspects of Regional Welfare (1977), Martinus Nijhoff: Martinus Nijhoff Leidon.G.P |
| [32] | Kloek, T.; van Dijk, H. K., Further results on efficient estimation of income distribution parameters, J. Econometrics, 8, 61-74 (1977) · Zbl 0388.62030 |
| [33] | Craig, C. C., A new exposition and chart for the Pearson system of frequency curves, Ann. Math. Stat., 7, 1, 16-28 (1936) · JFM 62.1341.01 |
| [34] | Toranzos, F. I., An asymmetric bell-shaped frequency curve, Ann. Math. Stat., 23, 3, 467-469 (1952) · Zbl 0047.12105 |
| [35] | Morlat, G., Les lois de probabilite de Halphen, Revue de Statistique Appliquee, 3, 21-43 (1956) |
| [36] | Halphen, E., Sur un nonveau type de courbe de frequence, C. R. Acad. Sci., 213, 633-635 (1941) · Zbl 0026.13703 |
| [37] | E. Halphen, Les functions factorials. Publications de l’Institut de Statistique de l’Universite de Paris, vol. IV, Fascicule I, 1955, pp. 21-39. · Zbl 0067.29605 |
| [38] | Seshadri, V., The Inverse Gaussian Distribution (1993), Clarendon Press: Clarendon Press Oxford, UK · Zbl 0468.60079 |
| [39] | Seshadri, V., Haphen’s laws, (Kotz, S.; Johnson, N. L., Encyclopedia of Statistical Sciences, Update Vil. 1 (1997)), 302-306 · Zbl 0897.62001 |
| [40] | V. Dvorok, B. Bobee, S. Boucher, F. Ashkar, Halphen Distributions and Related Systems of Frequency Functions, Research Report No. R-336, INRS-Eau, Ste-Foy, Quebec, Canada, 1988. |
| [41] | Perreault, L.; Bobee, B.; Rasmussen, P. F., Halphen distribution system. I: Mathematical and statistical properties, J. Hydrol. Eng., 4, 3, 189-199 (1999) |
| [42] | Perreault, L.; Bobee, B.; Rasmussen, P. F., Halphen distribution system. I: Parameter and quantile estimation, J. Hydrol. Eng., 4, 3, 208-209 (1999) |
| [43] | Fateh, C.; Salaheddine, E. A.; Bernard, B., Mixed estimation methods for Halphen distributions with applications in extreme hydrologic events, Stoch. Environ. Res. Risk Assess., 24, 3, 359-376 (2010) · Zbl 1420.62225 |
| [44] | T.B.M.J. Ouarde, F. Askhar, E.M. Ben Said, L. Hourani, Distribution statistiques utilisees on hydrologie: Transformation et proprieties asymptotiques. Rapport de recherché STAT-13, University of Moncton, N.B. Canada, 1994. |
| [45] | P.F. Verhulst, Nouveaux memoires de l’Academie des Sciences, des Artes et des Beaux Arts de Belgique 18, 1845. |
| [46] | (Ausloos, M.; Diricks, M., The Logistic Map and Route to Chaos: From the Beginnings to Modern Applications (2006), Springer Science and Business Media: Springer Science and Business Media Berlin) · Zbl 1085.37001 |
| [47] | Brouers, F., The Burr XII distribution family and the maximum entropy principle: Power law phenomena are not necessarily “nonextensive”, Open J. Stat., 5, 730-741 (2015) |
| [48] | Burr, I. W.; Cislak, P. J., On a general system of distributions: I. Its curve-shape characteristics: II. The sample median, J. Amer. Statist. Assoc., 63, 322, 627-635 (1968) |
| [49] | Burr, I. W., Parameters for a general system of distribution to match a grid of \(\alpha_3\) and \(\alpha_4\), Comm. Statist., 2, 1, 1-21 (1973) · Zbl 0262.62010 |
| [50] | Kleiber, C.; Kotz, S., Statistical Size Distributions in Economics and Actuarial Sciences (2003), Wiley: Wiley Hoboken, New Jersey, 332 p · Zbl 1044.62014 |
| [51] | Klugman, S. A.; Panjer, H. H.; Willmot, G. E., Loss Models (1998), John Wiley: John Wiley New York · Zbl 0905.62104 |
| [52] | Stoppa, G., Una tavola per modeli di probabilita, Metron, 51, 99-117 (1993) · Zbl 0828.62047 |
| [53] | Esteban, J., Income-share elasticity, density function and the size distribution of income, (Mimeographed Manuscript (1981), University of Barcelona: University of Barcelona Spain) · Zbl 0592.90019 |
| [54] | Stacy, E. W., A generalization of the gamma distribution, Ann. Math. Stat., 33, 1187-1192 (1962) · Zbl 0121.36802 |
| [55] | Arnold, B. C., Pareto Distributions: Pareto and Related Heavy-Tailed Distributions. Mimeographed Manuscript (1980), University of California at Riverside: University of California at Riverside California |
| [56] | Cronin, D. C., A function for the size distribution of income: A further comment, Econemetrica, 47, 773-774 (1979) · Zbl 0412.62085 |
| [57] | Johnson, N. L.; Kotz, S., Continuous Univariate Distributions (1) (1970), John Wiley & Sons: John Wiley & Sons New York · Zbl 0213.21101 |
| [58] | Tadikamalla, P. R., A look at the Burr and related distributions, Internat. Statist. Rev., 48, 337-344 (1980) · Zbl 0468.62013 |
| [59] | Kalbfleisch, J. D.; Prentice, R. L., The Statistical Analysis of Failure Time Data (1980), John Wiley & Sons: John Wiley & Sons New York · Zbl 0504.62096 |
| [60] | Venter, G., Transformed beta and gamma distributions and aggregate losses, (Proceedings of the Casualty Actuarial Society (1984)), 156-193 |
| [61] | McDonald, J. B.; Butler, R. J., Some generalized mixture distributions with an application to unemployment duration, Rev. Econ. Stat., 69, 2, 232-240 (1987) |
| [62] | Frechet, M., Sur les formules de repartition des renenus, Revue de l’Institut International de Statistique, 7, 32-38 (1939) · JFM 65.0630.01 |
| [63] | Davis, H. T., The Theory of Econometrics (1941), Principia Press: Principia Press Bloomington, Indiana · Zbl 0060.32104 |
| [64] | L. Amoroso, Richerche intorno alla curva dei reddini, in: Annali di Matematica Pura ed Applicata, Series 4-21, 1925, pp. 123-157. · JFM 51.0405.08 |
| [65] | Charlier, C. V.L., Uber die darstellung willkurliecher funktionen, Ark. Mat., Astronomi och Fysik, 2, 20, 1-35 (1905) · JFM 36.0460.01 |
| [66] | Wicksell, S. D., On the generic theory of frequency, Ark. Mat., Astronomi och Fysik, 12, 20, 1-56 (1917) · JFM 46.0777.01 |
| [67] | Kapteyn, J. C., Skew Frequency Curves in Biology and Statistics (1903), Noordhoff: Noordhoff Groningen · JFM 34.0268.03 |
| [68] | McKay, A. T., A Bessel function distribution, Biometrika, 24, 39-44 (1932) · JFM 58.1164.02 |
| [69] | Bhattacharya, G. P., The use of Mckay’s Bessel function curves for graduating frequency curves, Sankhya, 6, 175-182 (1942) · Zbl 0063.00369 |
| [70] | Khamis, S. H., Incomplete gamma functions, Bull. Int. Inst. Stat., 37, 385-396 (1958) · Zbl 0108.16101 |
| [71] | Chander, S.; Spolia, S. K.; Kumar, A., Flood frequency analysis by power transformation, J. Hydraul. Div., ASCE, 104, HY1, 1495-1504 (1978) |
| [72] | Rasheed, H. R.; Ramamoorthy, M. V.; Aldabaugh, A. S., Modified SMEMAX transformation for frequency analysis, Water Resour. Bull., 18, 3, 509-512 (1983) |
| [73] | Venugopal, K., Flood frequency analysis by SMEMAX transformation-A review, J. Hydraul. Div., ASCE, 106, HY2, 338-340 (1980) |
| [74] | Jain, D.; Singh, V. P., A comparison of transformation methods for flood frequency analysis, Water Resour. Bull., 22, 6, 903-912 (1986) |
| [75] | van Uven, M. J., Logarithmic frequency distributions, Koninklijke Akademie van Wetenschappen te Amsterdam Proceedings, 19, 4, 670-694 (1917) |
| [76] | Rietz, H. L., Frequency distributions obtained by certain transformations of normally distributed variates, Ann. of Math., 23, 4, 292-300 (1922) · JFM 49.0370.03 |
| [77] | Johnson, N. L., Systems of frequency curves generated by methods of translation, Biometrika, 36, 147-176 (1949) · Zbl 0033.07204 |
| [78] | D’Addario, R., Le transformate Euleriane, (Annali dell’Instituto di Statistica, vol. 8 (1936), Universita di Bari: Universita di Bari Italy), 67 |
| [79] | Majumder, A.; Chakravarty, S. R., Distribution of personal income: Development of a new model and its application to U.S. data, J. Appl. Econom., 5, 189-196 (1990) |
| [80] | Kloek; van Dijk, N. K., Further results on efficient estimation of income distribution parameters, J. Econometrics, 8, 61-74 (1977) · Zbl 0388.62030 |
| [81] | Kloek; van Dijk, N. K., Efficient estimation of income distribution parameters, Economic Applique, XXX, 3, 439-459 (1978) · Zbl 0388.62030 |
| [82] | Snyder, W. M.; Mills, W. C.; Knisel, W. G., A unifying set of probability transforms for stochastic hydrology, Water Resour. Bull., 14, 1, 83-98 (1978) |
| [83] | Johnson, N. L., Systems of frequency curves derived from the first law of Laplace, Trabajos de Estadistica, 5, 283-291 (1954) · Zbl 0058.35102 |
| [84] | Tadikamalla, P. R.; Johnson, N. L., Systems of frequency curves generated by transformations of logistic variables, Biometrika, 69, 461-465 (1982) |
| [85] | Johnson, N. L.; Tadikamalla, P. R., Translated families of distributions, (Balakrishnan, N., HandBook of the Logistic Distribution (1992), Mercel Dekker: Mercel Dekker New York), 189-208 |
| [86] | Tahmasebi, S.; Behboodian, J., Shannon entropy for the Feller-Pareto (FP) family and order statistics of FP subfamilies, Appl. Math. Sci., 4, 10, 495-504 (2010) · Zbl 1191.94072 |
| [87] | Mahmood, M. R.; Abd El Ghafour, A. S., Shannon entropy for the generalized feller-Pareto (GFP) family and order statistics of GPF subfamilies, Appl. Math. Sci., 7, 65, 3247-3253 (2013) |
| [88] | Arnold, B. C., Pareto Distributions (1983), International Cooperative Publishing House: International Cooperative Publishing House Fairland, Maryland · Zbl 1169.62307 |
| [89] | Klugman, S. A., Loss Models (1998), John Wiley: John Wiley New York · Zbl 0905.62104 |
| [90] | Hogg, R. V.; Klugman, S. A., On the estimation of long-tailed skewed distributions with actuarial applications, J. Econometrics, 23, 91-102 (1983) |
| [91] | Singh, V. P., Introduction to Tsallis Entropy Theory in Water Engineering (2016), CRC Press: CRC Press Boca Raton, Florida, 434 p |
| [92] | Singh, V. P., Entropy-Based Parameter Estimation in Hydrology (1997), Springer: Springer Dordrecht, the Netherlands |
| [93] | Singh, V. P., Entropy Theory and Its Application in Environmental and Water Engineering (2013), Wiley-Blackwell: Wiley-Blackwell Hoboken, New Jersey, 640 p |
| [94] | Singh, V. P.; Rajagopal, A. K.; Singh, K., Derivation of some frequency distributions using the principle of maximum entropy, Adv. Water Resour., 9, 2, 91-106 (1986) |
| [95] | Singh, V. P., Entropy Theory in Hydrologic Science and Engineering (2015), McGraw-Hill Education: McGraw-Hill Education New York, 824 p |
| [96] | Box, G. E.P.; Cox, G. C., Bayesian Inference in Statistical Analysis, 156-160 (1973), Addison Wesley Publishing Company: Addison Wesley Publishing Company New York · Zbl 0271.62044 |
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