exponential.md

Exponential Distribution

Distribution index · Mixed-data likelihood · Priors

Purpose and Parameterization

The two-parameter exponential model is appropriate for a nonnegative excess above a known or estimated lower endpoint when the hazard is constant. In RMC.Numerics the constructor order is $(\xi,\alpha)$: location $\xi$ has the units of the observation and scale $\alpha>0$ has the same units. If an excess $Y=X-\xi$ is modeled, $Y$ is memoryless; this is a strong tail assumption, not merely a convenient curve shape.

Distribution

For $x\ge\xi$,

$$ f(x\mid\xi,\alpha)=\frac{1}{\alpha}\exp!\left[-\frac{x-\xi}{\alpha}\right],\qquad F(x)=1-\exp!\left[-\frac{x-\xi}{\alpha}\right]. \tag{EXP.1} $$

Below $\xi$, $f=F=0$. The survival and quantile functions are

$$ S(x)=\exp[-(x-\xi)/\alpha],\qquad Q(p)=\xi-\alpha\log(1-p),\quad 0<p<1. \tag{EXP.2} $$

The mean, standard deviation, median, mode, skewness, and ordinary kurtosis are $\xi+\alpha$, $\alpha$, $\xi+\alpha\log2$, $\xi$, $2$, and $9$. The upper tail is exponential and all positive moments exist.

Likelihood and Bayesian Form

For exact values $x_i$, the log likelihood is

$$ \ell(\xi,\alpha)=-n\log\alpha-\alpha^{-1}\sum_i(x_i-\xi), \quad \xi\le\min_i x_i. \tag{EXP.3} $$

RMC.BestFit substitutes (EXP.1) into the common mixed-data likelihood: exact values contribute $\log f$; low outliers and left-censored values contribute $\log F$; right-censored values contribute $\log S$; intervals contribute $\log[F(u)-F(l)]$; uncertain values contribute the measurement-error convolution; and perception-period counts contribute powers of $F$ and $S$. For nonstationary models, $\xi_t$ and/or $\alpha_t$ are trend outputs at the observation index. The posterior is proportional to this full likelihood times all parameter and quantile-prior factors.

Estimation and Numerical Behavior

UnivariateDistribution.SetDefaultParameters delegates initialization and natural-space bounds to the Numerics maximum-likelihood constraint routine using the exact sample, then creates uniform parameter priors on those returned bounds. These are data-dependent working bounds, not universal scientific limits. Scale proposals must remain positive; observations below the current location have zero probability and yield a rejected log-likelihood proposal. A free endpoint can be weakly identified by short samples and may sit close to the minimum observation.

The exponential model is the $\kappa=0$ limit of the RMC.Numerics generalized Pareto family. It should not be used for annual maxima merely because data are positive: block maxima generally require an extreme-value family, whereas exponential excesses arise naturally in peaks-over-threshold models.

Compile-Checked API Example

The example uses discharge units consistently for location, scale, evaluation magnitude, and returned quantile. A 0.99 nonexceedance probability is an annual-exceedance probability of 0.01 only when the fitted random variable is an annual maximum.

private static (double Density, double Cdf, double Quantile) EvaluateExponential()
{
    var distribution = new global::Numerics.Distributions.Exponential(1000.0, 500.0);
    return Evaluate(distribution, value: 1800.0, nonExceedanceProbability: 0.99);
}

Validation and Limitations

The chapter formula is traced to Numerics.Distributions.Exponential; API compatibility is guarded by the compiled documentation fixture. Numerical validation should additionally test normalization, $F(Q(p))=p$, the memoryless identity, and the $\kappa\to0$ generalized-Pareto limit. No claim of hydrologic adequacy follows from those programmatic checks. Threshold choice, independence of excesses, exposure, and temporal stationarity must be assessed in the analysis using this family.

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] S. Coles, An Introduction to Statistical Modeling of Extreme Values. Springer, 2001.