A quantile-parameterized distribution (QPD) is a probability distributions that is directly parameterized by data. They were created to meet the need for easy-to-use continuous probability distributions flexible enough to represent a wide range of uncertainties, such as those commonly encountered in business, economics, engineering, and science. Because QPDs are directly parameterized by data, they have the practical advantage of avoiding the intermediate step of parameter estimation, a time-consuming process that typically requires non-linear iterative methods to estimate probability-distribution parameters from data. Some QPDs have virtually unlimited shape flexibility and closed-form moments as well.
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 and Johnson 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 process), originally invented quantile-parameterized distributions for this purpose. Keelin and Powley (2011) provided the original definition. Subsequently, Keelin (2016) developed the metalog distributions, a family of quantile-parameterized distributions that has virtually unlimited shape flexibility, simple equations, and closed-form moments.
Definition Keelin and Powley define a quantile-parameterized distribution as one whose quantile function (inverse CDF) can be written in the form
F − 1 ( y ) = { L 0 for y = 0 ∑ i = 1 n a i g i ( y ) for 0 < y < 1 L 1 for y = 1 {\displaystyle F^{-1}(y)=\left\{{\begin{array}{cl}L_{0}&{\text{for }}y=0\\\sum _{i=1}^{n}a_{i}g_{i}(y)&{\text{for }}0<y<1\\L_{1}&{\mbox{for }}y=1\end{array}}\right.}
where
L 0 = lim y → 0 + F − 1 ( y ) L 1 = lim y → 1 − F − 1 ( y ) {\displaystyle {\begin{array}{rcl}L_{0}&=&\lim _{y\rightarrow 0^{+}}F^{-1}(y)\\L_{1}&=&\lim _{y\rightarrow 1^{-}}F^{-1}(y)\end{array}}}
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