SPlus Method (Analytic Quantile Solution)

The SPlus method refers to an analytical quantile solution for the S-distribution, enabling flexible and efficient statistical data modeling, random number generation, and the fit of custom distributions to empirical datasets. The S-distribution is characterized by a unique differential equation that relates the probability density function (pdf) directly to the cumulative distribution function (cdf), providing substantial generality for representing a wide variety of unimodal distributional shapes, including but not limited to many classical parametric families.

1. Mathematical Definition of the S-Distribution

The S-distribution is implicitly defined by the ordinary differential equation

$\frac{dF}{dx} = a(F^g - F^h),$

where:

• $F(x)$ denotes the cdf,
• $a > 0$ is a scale or spread parameter,
• $g, h$ are real shape parameters, typically with $h > g$,
• $F(X_0) = F_0$ specifies a location via an initial condition.

Unlike conventional distributions, the S-distribution's pdf is formulated not directly in terms of $x$, but as a function of $F$ itself, resulting in high adaptability for complex unimodal data.

2. Analytic Quantile Solution (“SPlus Method”)

A principal advancement is the derivation of an explicit solution to the quantile equation, crucial for sample generation and probabilistic modeling. For parameters $S[F_0, X_0, a, g, h]$ and with $y = h - g > 0$, the quantile function inverts $F \mapsto X$. The general case is expressed using Lerch’s transcendent: $$X = X_0 + \frac{1}{a} \left[ F^{1-g}\Phi(F^y, 1, 1 + \frac{1-g}{y}) - F_0^{1-g}\Phi(F_0^y, 1, 1 + \frac{1-g}{y}) \right],$$ where

$$\Phi(z, s, v) = \sum_{n=0}^\infty \frac{z^n}{(v+n)^s}$$

is Lerch’s transcendent.

Nongeneric cases such as $g = 1$ yield logarithmic terms in the solution. The analytical quantile solution allows direct computation of quantiles for arbitrary $F \in (0,1)$, removing reliance on numerical integration or cdf inversion.

3. Applications in Random Number Generation and Monte-Carlo Simulation

The explicit quantile function provides a direct method for generating S-distributed random samples:

1. Draw $U \sim \text{Uniform}(0,1)$.
2. Set $X = X(U)$ from the analytic quantile formula.

This approach efficiently creates large samples—on the order of $10^4$ or more—for use in Monte-Carlo experiments, enabling rapid stochastic modeling with highly flexible distributional shapes. Implementation utilizes mathematical libraries supporting Lerch's transcendent, such as Mathematica's LerchPhi.

4. Fitting S-Distributions to Observed Data

Data fitting proceeds in two stages:

• Initial estimation: Histogram the data, estimate cdf values ($F_i$), and fit the ODE model $f_i = a(F_i^g - F_i^h)$ via nonlinear regression to obtain provisional parameters $(a, g, h)$.
• Quantile refinement: Compare sample quantiles $X_i$ against theoretical quantiles from the analytic solution at $F_i$, adjusting parameters to minimize the squared differences, typically via least-squares.

A maximum likelihood estimator for $g, h$ is available (Voit, 2000). The analytically tractable quantile solution accelerates these procedures by eliminating the step of numerically integrating to obtain the cdf, markedly reducing computational cost when fitting S-distributions iteratively.

5. Design of Custom and Constrained Distributions

The SPlus method enables the construction of tailored distributions to satisfy specific constraints, notably:

• Ensuring $P(X < X_c) = 0$ (strictly truncating the distribution at $X_c$),
• Fixing the median or arbitrary quantiles at predetermined values by parameter selection.

This capability addresses scenarios such as modeling positive-definite variables in environmental or biomedical science, where negative or out-of-bounds values are nonsensical. Practically, one solves the quantile equation for the remaining parameter after imposing the desired constraint.

6. Comparison with Classical Distribution Families

The S-distribution ("SPlus method") provides broader flexibility than families such as the normal, lognormal, Weibull, and others, which are limited to specific parameterizations and often lack analytic quantiles for nonstandard forms. The ability to directly access analytic quantiles streamlines statistical reporting, probabilistic thresholding, and interval estimation.

7. Examples of Scientific and Applied Usage

Demonstrated applications include:

• Modeling mercury contamination data in fisheries, incorporating truncation at zero.
• Representing human weight distributions as functions of age in biomedical growth studies.
• Hierarchical Monte-Carlo simulations in which underlying distributions evolve dynamically.
• Fitting real clinical measurement data (e.g., intensive care patient metrics) where the observed distribution exhibits skewness or boundedness not captured by conventional parametric families.

In each case, the S-distribution's flexibility and the SPlus method's efficient computation yield models attuned to empirical complexities, with practical advantages in simulation and inferential tasks.

The SPlus method is defined by its analytical quantile solution for the S-distribution, facilitating a broad range of modeling, inference, and simulation objectives demanding flexibility beyond classical parametric approaches. Its implementation—centered on Lerch’s transcendent—permits analytic, efficient, and customizable modeling for unimodal, potentially skewed data, with significant advantages for both theoretical and applied statisticians.