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== History ==
The development of quantile-parameterized distributions was inspired by the practical need for flexible continuous probability distributions that are easy to fit to data. Historically, the [[Pearson distribution|Pearson]]<ref>Johnson NL, Kotz S, Balakrishnan N. Continuous univariate distributions, Vol 1, Second Edition, John Wiley & Sons, Ltd, 1994, pp. 15–25.</ref> and [[Norman Lloyd Johnson|Johnson]]<ref>[https://www.jstor.org/stable/2332539?seq=1 Johnson, N. L. (1949). “Systems of frequency curves generated by methods of translation.” Biometrika. 36 (1/2): 149–176. doi:10.2307/2332539.]</ref><ref>[https://www.jstor.org/stable/2335422 Tadikamalla, P. R. and Johnson, N. L. (1982). “Systems of frequency curves generated by transformations of logistic variables.” Biometrika. 69 (2): 461–465.]</ref> families of distributions have been used when shape flexibility is needed. That is because both families can match the first four moments (mean, variance, skewness, and kurtosis) of any data set. In many cases, however, these distributions are either difficult to fit to data
For example, the [[beta distribution]] is a flexible Pearson distribution that is frequently used to model percentages of a population. However, if the characteristics of this population are such that the desired [[cumulative distribution function]] (CDF) should run through certain specific CDF points, there may be no beta distribution that meets this need. Because the beta distribution has only two shape parameters, it cannot, in general, match even three specified CDF points. Moreover, the beta parameters that best fit such data can be found only by nonlinear iterative methods.
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