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=== 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 \varphiPhi^{-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 \varphiPhi^{-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 unbounded metalog distribution<ref name="UnboundedMetalog">[[doi:10.1287/deca.2016.0338|Keelin, T.W. (2016), Section 3, pp. 249–257.]]</ref> yields the semi-bounded (log) metalog distribution;<ref name="KeelinSec4">[[doi:10.1287/deca.2016.0338|Keelin, T.W. (2016), Section 4.]]</ref> likewise, applying the logit transformation, <math>t(x)=\ln((x-b_l)/(b_u-x))</math>, yields the 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>
=== Moments ===
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