kappa-four.md

Kappa Four Distribution

Distribution index · Parameterization crosswalk

Purpose and Parameterization

The four-parameter Kappa family is a flexible location-scale-shape family that contains several distributions used in frequency analysis. RMC.Numerics constructor order is $(\xi,\alpha,\kappa,h)$, where $\alpha>0$; $\xi$ and $\alpha$ have observation units, while both shapes are dimensionless. Numerics calls the second shape Hondo.

Let $z=(x-\xi)/\alpha$ and $u=1-\kappa z$. For $\kappa h\ne0$, the implemented CDF is

$$ F(x)=\left[1-hu^{1/\kappa}\right]^{1/h}. \tag{K4.1} $$

Both inner bases must be real and nonnegative. This imposes distributional support beyond the constructor's finiteness checks.

Limiting Families and Quantiles

The CDF branches implemented by Numerics are

$$ F(x)=\begin{cases} [1-hu^{1/\kappa}]^{1/h},&\kappa\ne0,\ h\ne0,\\ \exp[-u^{1/\kappa}],&\kappa\ne0,\ h=0,\\ [1-he^{-z}]^{1/h},&\kappa=0,\ h\ne0,\\ \exp[-e^{-z}],&\kappa=0,\ h=0. \end{cases} \tag{K4.2} $$

Thus $h=0$ is GEV, $(\kappa,h)=(0,0)$ is Gumbel, and $h=1$ is generalized Pareto under the matching location/scale/shape convention. For nonzero shapes, inversion gives

$$ Q(p)=\xi+\frac{\alpha}{\kappa} \left[1-\left(\frac{1-p^h}{h}\right)^\kappa\right]. \tag{K4.3} $$

For $\kappa=0,h\ne0$, mathematical inversion of (K4.2) gives

$$ Q(p)=\xi-\alpha\log\left(\frac{1-p^h}{h}\right). \tag{K4.4} $$

The zero-primary-shape branch evaluates (K4.4) directly and is the algebraic inverse of the corresponding CDF branch.

The source reports the lower endpoint as $\xi+\alpha/\kappa$ for $h\le0,\kappa<0$; $\xi+(\alpha/\kappa)(1-h^{-\kappa})$ for $h>0,\kappa\ne0$; $\xi+\alpha\log h$ for $h>0,\kappa=0$; and $-\infty$ for $h\le0,\kappa\ge0$. The upper endpoint is infinite for $\kappa\le0$ and $\xi+\alpha/\kappa$ otherwise.

Density and Moments

For $\kappa\ne0$ the implemented density is

$$ f(x)=\frac{1}{\alpha}u^{1/\kappa-1}F(x)^{1-h}. \tag{K4.5} $$

For $\kappa=0$, the continuous density limit is $f(x)=\exp(-z)F(x)^{1-h}/\alpha$. The implementation evaluates this branch directly. Support depends on $(\kappa,h)$. Every finite shape pair defines a Kappa distribution for $\alpha>0$; moment existence is separate.

Numerics checks moment existence before integration. It integrates the standardized quantile over probability space, centers before higher powers, and applies location and scale afterward. Nonexistent or numerically unresolved moments return NaN. Mode evaluation includes support boundaries and limiting branches; it is not restricted to the central 99.8% of the distribution. Tail-dominated moments still require interpretation in light of their existence conditions and finite numerical precision.

Likelihood and Inference

Where $f$ and $F$ are finite and valid, the Kappa family enters the same exact, censored, uncertain, interval, threshold, stationary, and nonstationary likelihood as every UnivariateDistribution. The posterior multiplies that likelihood by configured priors. Initial values are obtained by an L-moment routine with a GEV fallback; optimizer bounds are then used for default uniform priors. Four flexible parameters create substantial confounding and multiple-optimum risk, especially in short records.

Compile-Checked API Example

private static (double Density, double Cdf, double Quantile) EvaluateKappaFour()
{
    var distribution = new global::Numerics.Distributions.KappaFour(
        1500.0, 400.0, 0.10, 0.20);
    return Evaluate(distribution, value: 2200.0, nonExceedanceProbability: 0.99);
}

The example uses nonzero shapes to illustrate the general branch. It demonstrates API shape, not suitability for a particular dataset.

Validation

Analytical tests verify the zero-$\kappa$ density as the derivative of the CDF, CDF/quantile inversion, support, normalization, and two-sided continuity at the limiting branch. A finite-shape regression spans all four sign combinations of $(\kappa,h)$ and verifies admissibility, monotone quantiles, CDF/quantile round trips, positive interior density, and support endpoints. Independent full-family comparisons cover nonzero-shape behavior, zero-$h$ branches, special-family identities, and tail support against Hosking's formulation.

The distribution verification matrix identifies the current independent formula and fitted-objective comparisons, retained Bayesian/MLE recovery designs, and their distinct acceptance rules. The compiled example guards API compatibility; it does not run an estimator or establish scientific accuracy by itself.

References

[1] J. R. M. Hosking, “The four-parameter kappa distribution,” IBM Journal of Research and Development, vol. 38, no. 3, pp. 251–258, 1994.

[2] J. R. M. Hosking and J. R. Wallis, Regional Frequency Analysis: An Approach Based on L-Moments. Cambridge University Press, 1997.