Probability density function

Differentiating ${\displaystyle x=F^{-1}(y)=\sum _{i=1}^{n}a_{i}g_{i}(y)}$ with respect to ${\displaystyle y}$ yields ${\displaystyle dx/dy}$. The reciprocal of this quantity, ${\displaystyle dy/dx}$, is the probability density function (PDF)

${\displaystyle f(y)=\left(\sum _{i=1}^{n}a_{i}{{dg_{i}(y)} \over {dy}}\right)^{-1}}$

where ${\displaystyle 0<y<1}$. Note that this PDF is expressed as a function of cumulative probability ${\displaystyle y}$ rather than ${\displaystyle x}$. To plot it, as shown in the figures, vary ${\displaystyle y\in (0,1)}$ parametrically. Plot ${\displaystyle x=F^{-1}(y)}$ on the horizontal axis and ${\displaystyle f(y)}$ on the vertical axis.

Feasibility

A function of the form of ${\displaystyle F^{-1}(y)}$ is a feasible probability distribution if and only if ${\displaystyle f(y)>0}$ for all ${\displaystyle y\in (0,1)}$.^[4] This implies a feasibility constraint on the set of coefficients ${\displaystyle {\boldsymbol {a}}=(a_{1},\ldots ,a_{n})\in \mathbb {R} ^{n}}$:

${\displaystyle \sum _{i=1}^{n}a_{i}{{dg_{i}(y)} \over {dy}}>0}$ for all ${\displaystyle y\in (0,1)}$

In practical applications, feasibility must generally be checked rather than assumed.

Convexity

A QPD’s set of feasible coefficients ${\displaystyle S_{\boldsymbol {a}}=\{{\boldsymbol {a}}\in \mathbb {R} ^{n}\mid \sum _{i=1}^{n}a_{i}dg_{i}(y)/dy>0}$ for all ${\displaystyle y\in (0,1)\}}$ is convex. Because convex optimization requires convex feasible sets, this property simplifies optimization applications involving QPDs.

Fitting to data

The coefficients ${\displaystyle {\boldsymbol {a}}}$ can be determined from data by linear least squares. Given ${\displaystyle m}$ data points ${\displaystyle (x_{i},y_{i})}$ that are intended to characterize the CDF of a QPD, and ${\displaystyle m\times n}$ matrix ${\displaystyle {\boldsymbol {Y}}}$ whose elements consist of ${\displaystyle g_{j}(y_{i})}$, then, so long as ${\displaystyle {\boldsymbol {Y}}^{T}{\boldsymbol {Y}}}$ is invertible, coefficients' column vector ${\displaystyle {\boldsymbol {a}}}$ can be determined as ${\displaystyle {\boldsymbol {a}}=({\boldsymbol {Y}}^{T}{\boldsymbol {Y}})^{-1}{\boldsymbol {Y}}^{T}{\boldsymbol {x}}}$, where ${\displaystyle m\geq n}$ and column vector ${\displaystyle {\boldsymbol {x}}=(x_{1},\ldots ,x_{m})}$. If ${\displaystyle m=n}$, this equation reduces to ${\displaystyle {\boldsymbol {a}}={\boldsymbol {Y}}^{-1}{\boldsymbol {x}}}$, 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.^[6]

Shape flexibility

A QPD with ${\displaystyle n}$ terms, where ${\displaystyle n\geq 2}$, has ${\displaystyle n-2}$ 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.^[7]

Transformations

QPD transformations are governed by a general property of quantile functions: for any quantile function ${\displaystyle x=Q(y)}$ and increasing function ${\displaystyle t(x),x=t^{-1}(Q(y))}$ is a quantile function.^[8] For example, the quantile function of the normal distribution, ${\displaystyle x=\mu +\sigma \Phi ^{-1}(y)}$, is a QPD by the Keelin and Powley definition. The natural logarithm, ${\displaystyle t(x)=\ln(x-b_{l})}$, is an increasing function, so ${\displaystyle x=b_{l}+e^{\mu +\sigma \Phi ^{-1}(y)}}$ is the quantile function of the lognormal distribution with lower bound ${\displaystyle b_{l}}$. Importantly, this transformation converts an unbounded QPD into a semi-bounded QPD. Similarly, applying this log transformation to the unbounded metalog distribution^[9] yields the semi-bounded (log) metalog distribution;^[10] likewise, applying the logit transformation, ${\displaystyle t(x)=\ln((x-b_{l})/(b_{u}-x))}$, yields the bounded (logit) metalog distribution^[10] with lower and upper bounds ${\displaystyle b_{l}}$ and ${\displaystyle b_{u}}$, respectively. Moreover, by considering ${\displaystyle t(x)}$ to be ${\displaystyle F^{-1}(y)}$ distributed, where ${\displaystyle F^{-1}(y)}$ 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 ${\displaystyle F^{-1}(y)}$, which has ${\displaystyle n-2}$ shape parameters; the log transformation has ${\displaystyle n-1}$ shape parameters, and the logit transformation has ${\displaystyle n}$ shape parameters. Moreover, such transformed QPDs share the same set of feasible coefficients as the underlying untransformed QPD.^[11]

Moments

The ${\displaystyle k^{th}}$ moment of a QPD is:^[4]

${\displaystyle E[x^{k}]=\int _{0}^{1}\left(\sum _{i=1}^{n}a_{i}g_{i}(y)\right)^{k}dy}$

Whether such moments exist in closed form depends on the choice of QPD basis functions ${\displaystyle g_{i}(y)}$. The unbounded metalog distribution and polynomial QPDs are examples of QPDs for which moments exist in closed form as functions of the coefficients ${\displaystyle a_{i}}$.

Simulation

Since the quantile function ${\displaystyle x=F^{-1}(y)}$ is expressed in closed form, Keelin and Powley QPDs facilitate Monte Carlo simulation. Substituting in uniformly distributed random samples of ${\displaystyle y}$ produces random samples of ${\displaystyle x}$ in closed form, thereby eliminating the need to invert a CDF expressed as ${\displaystyle y=F(x)}$.