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The Lambert Way to Gaussianize Heavy-Tailed Data with the Inverse of Tukey's h Transformation as a Special Case.

Goerg GM - ScientificWorldJournal (2015)

Bottom Line: For X being Gaussian it reduces to Tukey's h distribution.Parameters can be estimated by maximum likelihood and applications to S&P 500 log-returns demonstrate the usefulness of the presented methodology.The R package Lambert W implements most of the introduced methodology and is publicly available on CRAN.

View Article: PubMed Central - PubMed

Affiliation: Department of Statistics, Carnegie Mellon University, Pittsburgh, PA 15213, USA.

ABSTRACT
I present a parametric, bijective transformation to generate heavy tail versions of arbitrary random variables. The tail behavior of this heavy tail Lambert W × F X random variable depends on a tail parameter δ ≥ 0: for δ = 0, Y ≡ X, for δ > 0 Y has heavier tails than X. For X being Gaussian it reduces to Tukey's h distribution. The Lambert W function provides an explicit inverse transformation, which can thus remove heavy tails from observed data. It also provides closed-form expressions for the cumulative distribution (cdf) and probability density function (pdf). As a special case, these yield analytic expression for Tukey's h pdf and cdf. Parameters can be estimated by maximum likelihood and applications to S&P 500 log-returns demonstrate the usefulness of the presented methodology. The R package Lambert W implements most of the introduced methodology and is publicly available on CRAN.

No MeSH data available.


Related in: MedlinePlus

Pdf (left) and cdf (right) of a heavy tail (a) “noncentral, nonscaled,” (b) “scale,” and (c and d) “location-scale” Lambert W ×  FX random variable Y for various degrees of heavy tails (color, dashed lines).
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fig4: Pdf (left) and cdf (right) of a heavy tail (a) “noncentral, nonscaled,” (b) “scale,” and (c and d) “location-scale” Lambert W ×  FX random variable Y for various degrees of heavy tails (color, dashed lines).

Mentions: The input is not necessarily Gaussian (Tukey's h) but can be any other location-scale continuous random variable, for example, from a uniform distribution, X ~ U(a, b) (see Figure 4).


The Lambert Way to Gaussianize Heavy-Tailed Data with the Inverse of Tukey's h Transformation as a Special Case.

Goerg GM - ScientificWorldJournal (2015)

Pdf (left) and cdf (right) of a heavy tail (a) “noncentral, nonscaled,” (b) “scale,” and (c and d) “location-scale” Lambert W ×  FX random variable Y for various degrees of heavy tails (color, dashed lines).
© Copyright Policy - open-access
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4562338&req=5

fig4: Pdf (left) and cdf (right) of a heavy tail (a) “noncentral, nonscaled,” (b) “scale,” and (c and d) “location-scale” Lambert W ×  FX random variable Y for various degrees of heavy tails (color, dashed lines).
Mentions: The input is not necessarily Gaussian (Tukey's h) but can be any other location-scale continuous random variable, for example, from a uniform distribution, X ~ U(a, b) (see Figure 4).

Bottom Line: For X being Gaussian it reduces to Tukey's h distribution.Parameters can be estimated by maximum likelihood and applications to S&P 500 log-returns demonstrate the usefulness of the presented methodology.The R package Lambert W implements most of the introduced methodology and is publicly available on CRAN.

View Article: PubMed Central - PubMed

Affiliation: Department of Statistics, Carnegie Mellon University, Pittsburgh, PA 15213, USA.

ABSTRACT
I present a parametric, bijective transformation to generate heavy tail versions of arbitrary random variables. The tail behavior of this heavy tail Lambert W × F X random variable depends on a tail parameter δ ≥ 0: for δ = 0, Y ≡ X, for δ > 0 Y has heavier tails than X. For X being Gaussian it reduces to Tukey's h distribution. The Lambert W function provides an explicit inverse transformation, which can thus remove heavy tails from observed data. It also provides closed-form expressions for the cumulative distribution (cdf) and probability density function (pdf). As a special case, these yield analytic expression for Tukey's h pdf and cdf. Parameters can be estimated by maximum likelihood and applications to S&P 500 log-returns demonstrate the usefulness of the presented methodology. The R package Lambert W implements most of the introduced methodology and is publicly available on CRAN.

No MeSH data available.


Related in: MedlinePlus