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* Quantile functions expressed as [[polynomial]] functions of cumulative probability <math>y</math>, including [[Chebyshev polynomial]] functions.
Like the SPT metalog distributions, the Johnson Quantile-Parameterized Distributions<ref>[{{cite journal | url=https://pubsonline.informs.org/doi/abs/10.1287/deca.2016.0343 Hadlock,| Cdoi=10.C1287/deca.2016.0343 and| Bickel,title=Johnson J.E.,Quantile-Parameterized Distributions | year=2017 | last1=Hadlock | first1=Christopher C. Johnson| quantile-parameterizedlast2=Bickel distributions| first2=J. Eric | journal=Decision Analysis, | volume=14(1), pp.| pages=35–64.] }}</ref><ref>[{{cite journal | url=https://pubsonline.informs.org/doi/abs/10.1287/deca.2018.0376 Hadlock,| Cdoi=10.C1287/deca. and Bickel, J2018.E.,0376 2019.| title=The generalizedGeneralized Johnson quantileQuantile-parameterizedParameterized distributionDistribution systemSystem | year=2019 | last1=Hadlock | first1=Christopher C. | last2=Bickel | first2=J. Eric | journal=Decision Analysis, 14(1),| pp.volume=16 333.]| pages=67–85 | s2cid=159339224 }}</ref> (JQPDs) are parameterized by three quantiles. JQPDs do not meet Keelin and Powley’s QPD definition, but rather have their own properties. JQPDs are feasible for all SPT parameter sets that are consistent with the [[Probability theory|rules of probability]].
== Applications ==
The original applications of QPDs were by decision analysts wishing to conveniently convert expert-assessed quantiles (e.g., 10th, 50th, and 90th quantiles) into smooth continuous probability distributions. QPDs have also been used to fit output data from simulations in order to represent those outputs (both CDFs and PDFs) as closed-form continuous distributions.<ref>[[doi:10.1287/deca.2016.0338|Keelin, T.W. (2016), Section 6.2.2, pp. 271–274.]]</ref> Used in this way, they are typically more stable and smoother than histograms. Similarly, since QPDs can impose fewer shape constraints than traditional distributions, they have been used to fit a wide range of empirical data in order to represent those data sets as continuous distributions (e.g., reflecting bimodality that may exist in the data in a straightforward manner<ref>[[doi:10.1287/deca.2016.0338|Keelin, T.W. (2016), Section 6.1.1, Figure 10, pp 266–267.]]</ref>). Quantile parameterization enables a closed-form QPD representation of known distributions whose CDFs otherwise have no closed-form expression. Keelin et al. (2019)<ref>[{{cite book | url=https://dl.acm.org/doi/abs/10.5555/3400397.3400643 Keelin,| T.W.,isbn=9781728132839 Chrisman,| L. and Savage, S.L. (2019). “Thetitle=The metalog distributions and extremely accurate sums of lognormals in closed form.” WSC| '19:date=18 ProceedingsMay of2020 the| Winterpages=3074–3085 Simulation| Conference.last1=Mustafee 3074–3085| first1=N.] }}</ref> apply this to the sum of independent identically distributed lognormal distributions, where quantiles of the sum can be determined by a large number of simulations. Nine such quantiles are used to parameterize a semi-bounded metalog distribution that runs through each of these nine quantiles exactly. QPDs have also been applied to assess the risks of asteroid impact,<ref>[[doi:10.1111/risa.12453|Reinhardt, J.D., Chen, X., Liu, W., Manchev, P. and Pate-Cornell, M.E. (2016). “Asteroid risk assessment: A probabilistic approach.” Risk Analysis. 36 (2): 244–261]]</ref> cybersecurity,<ref name="Faber" /><ref>[{{cite journal | url=https://www.sciencedirect.com/science/article/pii/S0167404819300604 Wang,| Jdoi=10., Neil, M1016/j. and Fenton, Ncose. (2020)2019.101659 “A| title=A Bayesian network approach for cybersecurity risk assessment implementing and extending the FAIR model.” | year=2020 | last1=Wang | first1=Jiali | last2=Neil | first2=Martin | last3=Fenton | first3=Norman | journal=Computers & Security. | volume=89: | page=101659.] | s2cid=209099797 }}</ref> biases in projections of oil-field production when compared to observed production after the fact,<ref>[https://www.onepetro.org/journal-paper/SPE-195914-PA Bratvold, R.B., Mohus, E., Petutschnig, D. and Bickel, E. (2020). “Production forecasting: Optimistic and overconfident—Over and over again.” Society of Petroleum Engineers. doi:10.2118/195914-PA.]</ref> and future Canadian population projections based on combining the probabilistic views of multiple experts.<ref>[https://library.oapen.org/bitstream/handle/20.500.12657/42565/2020_Book_DevelopmentsInDemographicForec.pdf?sequence=1#page=51 Dion, P., Galbraith, N., Sirag, E. (2020). “Using expert elicitation to build long-term projection assumptions.” In Developments in Demographic Forecasting, Chapter 3, pp. 43–62. Springer]</ref> See [[Metalog distribution#Applications|metalog distributions]] and Keelin (2016)<ref name="Keelin2016" /> for additional applications of the metalog distribution.
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