Quantile-parameterized distribution: Difference between revisions

Revision as of 13:34, 15 May 2023

Revision as of 03:47, 28 July 2023

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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>[{{cite journal | url=https://www.jstor.org/stable/2332539?seq=1 Johnson,| N.jstor=2332539 L.| (1949). “Systemstitle=Systems of frequencyFrequency curvesCurves generatedGenerated by methodsMethods of translationTranslation | last1=Johnson | first1=N.” BiometrikaL. | journal=Biometrika | year=1949 | volume=36 (| issue=1/2): | pages=149–176. | doi:=10.2307/2332539.] | pmid=18132090 }}</ref><ref>[{{cite journal | url=https://www.jstor.org/stable/2335422 Tadikamalla,| P.jstor=2335422 R.| andtitle=Systems Johnson,of N.Frequency L.Curves (1982).Generated “Systemsby Transformations of frequencyLogistic curvesVariables generated| bylast1=Tadikamalla transformations| offirst1=Pandu logisticR. variables| last2=Johnson | first2=Norman L.” | journal=Biometrika. | year=1982 | volume=69 (| issue=2): | pages=461–465 | doi=10.]1093/biomet/69.2.461 }}</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 or not flexible enough to fit the data appropriately.

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.

Practitioners of [[decision analysis]], needing distributions easily parameterized by three or more CDF points (e.g., because such points were specified as the result of an [[Expert elicitation|expert-elicitation process]]), originally invented quantile-parameterized distributions for this purpose. Keelin and Powley (2011)<ref name="KeelinPowley">[[{{Cite journal |doi:=10.1287/deca.1110.0213 |Keelin,title=Quantile-Parameterized T.W.Distributions and|year=2011 Powley,|last1=Keelin |first1=Thomas B.W. (2011).|last2=Powley “Quantile-parameterized|first2=Bradford distributionsW.” |journal=Decision Analysis. |volume=8 (|issue=3): |pages=206–219.]] }}</ref> provided the original definition. Subsequently, Keelin (2016)<ref name="Keelin2016">[[{{Cite journal |doi:=10.1287/deca.2016.0338 |Keelin,title=The T.W.Metalog (Distributions |year=2016). “The|last1=Keelin Metalog|first1=Thomas DistributionsW.” |journal=Decision Analysis. |volume=13 (|issue=4): |pages=243–277.]] }}</ref> developed the [[metalog distribution]]s, a family of quantile-parameterized distributions that has virtually unlimited shape flexibility, simple equations, and closed-form moments.

== Definition ==

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=== Shape flexibility ===

A QPD with <math>n</math> terms, where <math>n\ge 2</math>, has <math>n-2</math> shape parameters. Thus, QPDs can be far more flexible than the [[Pearson distribution]]s, which have at most two shape parameters. For example, ten-term [[metalog distribution]]s parameterized by 105 CDF points from 30 traditional source distributions (including normal, student-t, lognormal, gamma, beta, and extreme value) have been shown to approximate each such source distribution within a [[Kolmogorov–Smirnov test|K–S]] distance of 0.001 or less.<ref>[[{{Cite journal |doi:=10.1287/deca.2016.0338|at=Table 8 |title=The Metalog Distributions |year=2016 |last1=Keelin, T.|first1=Thomas W. (2016),|journal=Decision TableAnalysis 8]]|volume=13 |issue=4 }}</ref>

=== Transformations ===

QPD transformations are governed by a general property of quantile functions: for any [[quantile function]] <math>x=Q(y)</math> and increasing function <math>t(x), x=t^{-1} (Q(y))</math> is a [[quantile function]].<ref>Gilchrist, W., 2000. Statistical modelling with quantile functions. CRC Press.</ref> For example, the [[quantile function]] of the [[normal distribution]], <math>x=\mu+\sigma \Phi^{-1} (y)</math>, is a QPD by the Keelin and Powley definition. The natural logarithm, <math>t(x)=\ln(x-b_l)</math>, is an increasing function, so <math>x=b_l+e^{\mu+\sigma \Phi^{-1} (y)}</math> is the [[quantile function]] of the [[Log-normal distribution|lognormal distribution]] with lower bound <math>b_l</math>. Importantly, this transformation converts an unbounded QPD into a semi-bounded QPD. Similarly, applying this log transformation to the [[Metalog distribution#Unbounded,_semibounded,_and_bounded_metalog_distributions|unbounded metalog distribution]]<ref name="UnboundedMetalog">[[{{Cite journal |doi:=10.1287/deca.2016.0338|Keelin, T.W. (2016), at=Section 3, pp. 249–257 |title=The Metalog Distributions |year=2016 |last1=Keelin |first1=Thomas W.]] |journal=Decision Analysis |volume=13 |issue=4 }}</ref> yields the [[Metalog distribution#Unbounded,_semibounded,_and_bounded_metalog_distributions|semi-bounded (log) metalog distribution]];<ref name="KeelinSec4">[[{{Cite journal |doi:=10.1287/deca.2016.0338|at=Section 4 |title=The Metalog Distributions |year=2016 |last1=Keelin, T.|first1=Thomas W. (2016),|journal=Decision SectionAnalysis |volume=13 |issue=4.]] }}</ref> likewise, applying the logit transformation, <math>t(x)=\ln((x-b_l)/(b_u-x))</math>, yields the [[Metalog distribution#Unbounded,_semibounded,_and_bounded_metalog_distributions|bounded (logit) metalog distribution]]<ref name="KeelinSec4" /> with lower and upper bounds <math>b_l</math> and <math>b_u</math>, respectively. Moreover, by considering <math>t(x)</math> to be <math>F^{-1} (y)</math> distributed, where <math>F^{-1} (y)</math> is any QPD that meets Keelin and Powley’s definition, the transformed variable maintains the above properties of feasibility, convexity, and fitting to data. Such transformed QPDs have greater shape flexibility than the underlying <math>F^{-1} (y)</math>, which has <math>n-2</math> shape parameters; the log transformation has <math>n-1</math> shape parameters, and the logit transformation has <math>n</math> shape parameters. Moreover, such transformed QPDs share the same set of feasible coefficients as the underlying untransformed QPD.<ref>[http://metalogdistributions.com/images/Powley_Dissertation_2013-augmented.pdf Powley, B.W. (2013). “Quantile Function Methods For Decision Analysis”. Corollary 12, p 30. PhD Dissertation, Stanford University]</ref>

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* The [[Metalog distribution#Unbounded,_semibounded,_and_bounded_metalog_distributions|semi-bounded and bounded metalog distributions]], which are the log and logit transforms, respectively, of the unbounded metalog distribution.

* The [[Metalog distribution#SPT_metalog_distributions|SPT (symmetric-percentile triplet) unbounded, semi-bounded, and bounded metalog distributions]], which are parameterized by three CDF points and optional upper and lower bounds.

* The Simple Q-Normal distribution<ref>[[{{Cite journal |doi:=10.1287/deca.1110.0213|at=pp. 208–210 |title=Quantile-Parameterized Distributions |year=2011 |last1=Keelin, T.|first1=Thomas W., and |last2=Powley, B.|first2=Bradford W. (2011),|journal=Decision pp.Analysis 208–210]]|volume=8 |issue=3 }}</ref>

* The metadistributions, including the meta-normal<ref>[[{{Cite journal |page=253 |doi:=10.1287/deca.2016.0338 |title=The Metalog Distributions |year=2016 |last1=Keelin, T.|first1=Thomas W. (2016),|journal=Decision p.Analysis 253.]]|volume=13 |issue=4 }}</ref>

* Quantile functions expressed as [[polynomial]] functions of cumulative probability <math>y</math>, including [[Chebyshev polynomial]] functions.