Quantile-parameterized distribution: Difference between revisions

Revision as of 19:42, 12 February 2021

Revision as of 19:51, 12 February 2021

Line 74:

: <math>E[x^k] = \int_0^1 \left( \sum_{i=1}^n a_i g_i(y) \right)^k dy</math>

Whether such moments exist in closed form depends on the choice of QPD basis functions <math>g_i (y)</math>. The unbounded [[metalog distribution<ref name="UnboundedMetalog" />]] and polynomial QPDs are examples of QPDs for which moments exist in closed form as functions of the coefficients <math>a_i</math>.

=== Simulation ===

Line 86:

* The quantile function of the [[Cauchy distribution]], <math>x=x_0+\gamma \tan[\pi(y-0.5)]</math>.

* The quantile function of the [[logistic distribution]], <math>x=\mu+s \ln(y/(1-y) )</math>.

* The unbounded [[metalog (meta-logistic) distribution]],<ref name="UnboundedMetalog" /> which is a power series expansion of the <math>\mu</math> and <math>s</math> parameters of the logistic quantile function.

* The [https://en.wikipedia.org/wiki/Metalog_distribution#Unbounded,_semibounded,_and_bounded_metalog_distributions semi-bounded and bounded metalog distributions],<ref name="KeelinSec4" /> which are the log and logit transforms, respectively, of the unbounded metalog distribution.

* The [https://en.wikipedia.org/wiki/Metalog_distribution#SPT_metalog_distributions SPT (symmetric-percentile triplet) unbounded, semi-bounded, and bounded metalog distributions],<ref name="SPT">[[doi:10.1287/deca.2016.0338|Keelin, T.W. (2016), pp. 269–271.]]</ref> which are parameterized by three CDF points and optional upper and lower bounds.

* The Simple Q-Normal distribution<ref>[[doi:10.1287/deca.1110.0213|Keelin, T.W., and Powley, B.W. (2011), pp. 208–210]]</ref>

* The metadistributions, including the meta-normal<ref>[[doi:10.1287/deca.2016.0338|Keelin, T.W. (2016), p. 253.]]</ref>

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

Like the SPT metalog distributions,<ref name="SPT" /> the Johnson Quantile-Parameterized Distributions<ref>[https://pubsonline.informs.org/doi/abs/10.1287/deca.2016.0343 Hadlock, C.C. and Bickel, J.E., 2017. Johnson quantile-parameterized distributions. Decision Analysis, 14(1), pp. 35–64.]</ref><ref>[https://pubsonline.informs.org/doi/abs/10.1287/deca.2018.0376 Hadlock, C.C. and Bickel, J.E., 2019. The generalized Johnson quantile-parameterized distribution system. Decision Analysis, 14(1), pp. 333.]</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 ==