Quantile-parameterized distribution
Quantile-parameterized distributions (QPDs) are probability distributions that are directly parameterized by data. They were motivated by 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 where and the functions formula_3 are continuously differentiable and linearly independent basis functions. Here, essentially, formula_4 and formula_5 are the lower and upper bounds (if they exist) of a random variable with quantile function formula_6. These  distributions are called quantile-parameterized because for a given set of quantile pairs formula_7, where formula_8, and a set of formula_9 basis functions formula_3, the coefficients formula_11 can be determined by solving a set of linear equations. If one desires to use more quantile pairs than basis functions, then the coefficients formula_11 can be chosen to minimize the sum of squared errors between the stated quantiles formula_13 and formula_14. Keelin and Powley illustrate this concept for a specific choice of basis functions that is a generalization of quantile function of the normal distribution, formula_15, for which the mean formula_16 and  standard deviation formula_17 are linear functions of cumulative probability formula_18: The result is a four-parameter distribution that can be fit to a set of four quantile/probability pairs exactly, or to any number of such pairs by linear least squares. Keelin and Powley call this the Simple Q-Normal distribution. Some skewed and symmetric Simple Q-Normal PDFs are shown in the figures below. Properties. QPD’s that meet Keelin and Powley’s definition have the following properties. Probability density function. Differentiating formula_21 with respect to formula_18 yields formula_23. The reciprocal of this quantity, formula_24, is the probability density  function (PDF) where formula_26. Note that this PDF is expressed as a function of cumulative probability formula_18 rather than formula_28. To plot it, as shown in the figures, vary formula_29 parametrically. Plot formula_30 on the horizontal axis and formula_31 on the vertical axis. Feasibility. A function of the form of formula_32 is a feasible probability distribution if and only if formula_33 for all formula_34. This implies a feasibility constraint on the set of coefficients formula_35: In practical applications, feasibility must generally be checked rather than assumed. Convexity. A QPD’s set of feasible coefficients formula_38 for all formula_39 is convex . Because convex optimization requires convex feasible sets, this property simplifies optimization applications involving QPDs. Fitting to data. The coefficients formula_40 can be determined from data by linear least squares. Given formula_41 data points formula_42 that are intended to characterize the CDF of a QPD, and formula_43 matrix formula_44 whose elements consist of formula_45, then, so long as formula_46 is invertible, coefficients' column vector formula_40 can be determined as formula_48, where formula_49 and column vector formula_50. If formula_51, this equation reduces to formula_52, where the resulting CDF runs through all data points exactly. An alternate method, implemented as a linear program , determines the coefficients by minimizing the sum of absolute distances between the CDF and the data subject to feasibility constraints. Shape flexibility. A QPD with formula_9 terms, where formula_54, has formula_55 shape parameters. Thus, QPDs can be far more flexible than the Pearson distributions, which have at most two shape parameters. For example, ten-term metalog distributions 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 K–S distance of 0.001 or less. Transformations. QPD transformations are governed  by a general property of quantile functions: for any quantile function formula_56 and increasing function formula_57 is a quantile function. For example, the quantile function of the normal distribution, formula_15, is a QPD by the Keelin and Powley definition. The natural logarithm, formula_59, is an increasing function, so formula_60 is the quantile function of the lognormal distribution with lower bound formula_61. Importantly, this transformation converts an unbounded QPD into a semi-bounded QPD. Similarly, applying this log transformation to the unbounded metalog distribution yields the semi-bounded (log) metalog distribution; likewise, applying the logit transformation, formula_62, yields the bounded (logit ) metalog distribution with lower and upper bounds formula_61 and formula_64, respectively. Moreover, by considering formula_65 to be formula_32 distributed, where formula_32 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 formula_32, which has formula_55 shape parameters; the log transformation has formula_70 shape parameters, and the logit transformation has formula_9 shape parameters. Moreover, such transformed QPDs share the same set of feasible coefficients as the underlying untransformed QPD. Moments. The formula_72 moment of a QPD is : Whether such moments exist in closed form depends on the choice of QPD basis functions formula_74. The unbounded metalog distribution and polynomial QPDs are examples of QPDs for which moments exist in closed form as functions of the coefficients formula_11. Simulation. Since the quantile function formula_76 is expressed in closed form, Keelin and Powley QPDs facilitate Monte Carlo simulation. Substituting in uniformly distributed random samples of formula_18 produces random samples of formula_28 in closed form, thereby eliminating the need to invert a CDF expressed as formula_79. Related distributions. The following probability distributions are QPDs according to Keelin and Powley’s  definition: Like the SPT metalog distributions, the Johnson Quantile-Parameterized Distributions (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 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. 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). Quantile parameterization enables a closed-form QPD representation of known distributions whose CDFs otherwise have no closed-form expression. Keelin et al. (2019) 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, cybersecurity, biases in projections of oil-field production when compared to observed production after the fact, and future Canadian population projections based on combining the probabilistic views of multiple experts. See metalog distributions and Keelin (2016) for additional applications of the metalog distribution.