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].

It 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 Distribution probability density function (PDF). Image credit: NIST [1].

Mielke’s Beta-Kappa Distribution properties

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

mielke's beta kappa distribution pdf
PDF with shape parameters k,θ, location parameter and scale parameter β.

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


[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.

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