The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map

By William T. Shaw, Ian Buckley. 2009

512 citations recorded by the source corpus.

Motivated by the need for parametric families of rich and yet tractable distributions in financial mathematics, both in pricing and risk management settings, but also considering wider statistical applications, we investigate a novel technique for introducing skewness or kurtosis into a symmetric or other distribution. We use a "transmutation" map, which is the functional composition of the cumulative distribution function of one distribution with the inverse cumulative distribution (quantile) function of another. In contrast to the Gram-Charlier approach, this is done without resorting to an asymptotic expansion, and so avoids the pathologies that are often associated with it. Examples of parametric distributions that we can generate in this way include the skew-uniform, skew-exponential, skew-normal, and skew-kurtotic-normal.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/235d9508e5e53f9d2ccd2034a9a5054ee4fa0a4a

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Citations in the corpus

Transmuted Generalized Rayleigh Distribution (2014, unknown): “, (9) for p,ν,μ > 0 (see Gradshtein and Ryzhnik(2000), Sec.”
The Transmuted Topp-Leone G Family of Distributions: Theory, Characterizations and Applications (2021, methodology): “For an arbitrary baseline cdf G(x ), Shaw and Buckley (2007) de ned the TG family with cdf and probability density function (pdf) given by F (x) =F (x; ; ) =H (x; ) [1 + H(x; )] (1) and f (x) =f (x; ; ) =h(x; ) [1 + 2 H(x; )] ; (2) respectively, where j j 1 is a shape parameters,x > 0 and = ( k) =…”
Transmuted Lomax Distribution (2013, methodology): “λ − ≤ ≤ An extensive information about the quadratic rank transmutation map is given in [8].”
A New Kumaraswamy Generalized Family of Distributions with Properties, Applications, and Bivariate Extension (2020, background): “[4] (beta-G), Gleaton and Lynch [5] (odd log-logistic-G (OLL-G)), Shaw and Buckley [6] (transmuted-G), Zografos and Balakrishnan [7] (ZBgamma-G), Cordeiro and de-Castro [8](Kumaraswamy-G (Kw-G)), Alexander et al.”
Transmuted Log-Logistic Distribution (2013, background): “[15] to define a new model which generalizes the log-logistic (LLog) model.”
A General Transmuted Family of Distributions (2018, methodology): “Another method of generating families of distributions has been proposed by Shaw and Buckley (2007), which used quadratic transmutation map to generate new probability distribution using any baseline distribution.”
The Kumaraswamy Transmuted-G Family of Distributions: Properties and Applications (2021, background): “maraswamy transmuted-G family which extends the transmuted class defined by Shaw and Buckley (2007).”
The Transmuted Odd Fréchet-G Family of Distributions: Theory and Applications (2020, background): “First of all, the transmuted-G (T-G) family by [9] is defined by the cumulative distribution function (cdf) and probability density function (pdf) given by”
The Transmuted Marshall-Olkin Fr\'{e}chet Distribution: Properties and Applications (2015, methodology): “In fact, based on the transmuted-G (T-G) family pioneered by Shaw and Buckley (2007), we construct a new distribution called the transmuted Marshall-Olkin Fréchet (henceforth in short TMOFr) distribution and provide a comprehensive description of some of its mathematical properties.”