Mielke’s Beta-Kappa Distribution

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Mielke’s beta-kappa distribution (sometimes called Mielke’s distribution) is a continuous probability distribution defined over positive real numbers. It is a generalized F distribution [1] that can be applied to a diverse range of practical data such as precipitation and stream flow data, defining scale thresholds for geomagnetic storms [2] and incompleteness of electronic health records [3].

Mielke’s distribution is also equivalent to the Dagum distribution with a different parameterization: while Dagum [4] used parameters (β, δ, λ) in his 1977 paper, Mielke used (α, β, θ) four years earlier — in 1973 — when he proposed the distribution for a meteorological application to model daily rainfall [5]. Mielke initially called his distribution the kappa distribution but later referred to (with co-author Johnson) as the Beta-K distribution [6].

Mielke’s Beta-Kappa Distribution properties

The general form of the probability density function (pdf) is:

The standard form of Mielke’s Beta-Kappa distribution, with location parameter (μ) = 0 and scale parameter (β) = 1 is:

References

1. Statistical Engineering Division, Dataplot. MIEPDF. Retrieved May 14, 2021 from: https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/miepdf.htm
2. Palacios et. al. Defining scale thresholds for geomagnetic storms through statistics. Nat. Hazards Earth Syst. Sci. Discuss., https://doi.org/10.5194/nhess-2017-367
3. Gurupur et. al. Analyzing the Data Completeness of Patients’ Records Using a Random Variable Approach to Predict the Incompleteness of Electronic Health Records. Appl. Sci. 2022, 12, 10746.
4. Dagum, C. (1977). A new model of personal income distribution: Specification and estimation. Economie Appliqu´ee, 30, 413–437.
5. Mielke, P.W. (1973). Another family of distributions for describing and analyzing precipitation data. Journal of Applied Meteorology, 12, 275–280.
6. Mielke, P.W., and Johnson, E.S. (1974). Some generalized beta distributions of the second kind having desirable application features in hydrology and meteorology. Water Resources Research, 10, 223–226.