index.md

Probability Distributions

Back to Technical Reference · Common univariate model · Parameterization crosswalk

RMC.BestFit wraps RMC.Numerics probability distributions with mixed-observation likelihoods, parameter trends, priors, posterior inference, diagnostics, and hydrologic analysis outputs. The family chapters define the exact Numerics 2.2.0 constructor order and sign/log conventions; those definitions control the likelihood used by BestFit.

Supported Univariate Families

Family	API order	Essential convention	Chapter
Exponential	$(\xi,\alpha)$	Location, scale	Exponential
Gamma	$(\theta,k)$	Scale before shape	Gamma
Generalized extreme value	$(\xi,\alpha,\kappa)$	Numerics shape is negative of common Coles shape	GEV
Generalized logistic	$(\xi,\alpha,\kappa)$	Same support/sign transform as Numerics GEV	GLO
Generalized Normal	$(\xi,\alpha,\kappa)$	Hosking GNO, not exponential-power family	GNO
Generalized Pareto	$(\xi,\alpha,\kappa)$	Negative Numerics shape is heavy upper tail	GPD
Gumbel	$(\xi,\alpha)$	Zero-shape GEV	Gumbel
Kappa Four	$(\xi,\alpha,\kappa,h)$	Two shape parameters with explicit zero-shape limits	Kappa Four
Ln-Normal	$(m,s)$	Natural-space arithmetic mean and SD	Ln-Normal
Logistic	$(\xi,\alpha)$	Symmetric exponential tails	Logistic
Log-Normal	$(\mu_Y,\sigma_Y)$	Moments of $Y=\log_{10}X$	Log-Normal
Log-Pearson III	$(\mu_Y,\sigma_Y,\gamma_Y)$	Moments of $Y=\log_{10}X$	LP3
Normal	$(\mu,\sigma)$	Arithmetic mean and SD	Normal
Pearson III	$(\mu,\sigma,\gamma)$	Arithmetic moments; reflected for negative skew	PIII
Weibull	$(\lambda,k)$	Scale before shape; zero lower endpoint	Weibull

Cross-Family Modeling Layers

The common univariate model explains how a selected family participates in exact, uncertain, censored, interval, threshold-count, stationary, and nonstationary likelihoods. The data-frame chapter defines each observation contribution, and parameters and priors defines bounds, fixed coefficients, Jeffreys scale priors, and quantile priors.

Advanced constructions have separate formulations:

• Mixture models represent latent populations.
• Competing risks combine maxima or minima of processes.
• Point-process models combine threshold exceedance occurrence and magnitude.
• Composite analysis combines model outputs rather than asserting one parent density.

Selection Discipline

Distribution selection must consider the sampling scheme, support, endpoint, tail class, generating process, parameter uncertainty, and extrapolation objective. Information criteria summarize relative predictive fit under their assumptions; they do not certify physical plausibility. Rare return levels should be reported with posterior uncertainty and the fitted tail/endpoint convention.

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