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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

Transformation and inverse transformation for δℓ = 0 and δr = 1/10: identity on the left (same tail behavior) and a heavy-tailed transformation in the right tail of input U.
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fig3: Transformation and inverse transformation for δℓ = 0 and δr = 1/10: identity on the left (same tail behavior) and a heavy-tailed transformation in the right tail of input U.

Mentions: A parametric transformation is the basis of Tukey's h random variables [17](2)Z=Uexph2U2, h≥0,where U is standard Normal random variable and h is the heavy tail parameter. The random variable Z has tail parameter a = 1/h [17] and reduces to the Gaussian for h = 0. Morgenthaler and Tukey [26] extend the h distribution to the skewed, heavy-tailed family of hh random variables (3)Z=Uexpδl2U2,if  U≤0,Uexpδr2U2,if  U>0,where again U ~ 𝒩(0,1). Here δℓ ≥ 0 and δr ≥ 0 shape the left and right tail of Z, respectively; thus transformation (3) can model skewed and heavy-tailed data; see Figure 3(a). For the sake of brevity let Hδ(u)∶ = uexp((δ/2)u2).


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

Goerg GM - ScientificWorldJournal (2015)

Transformation and inverse transformation for δℓ = 0 and δr = 1/10: identity on the left (same tail behavior) and a heavy-tailed transformation in the right tail of input U.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig3: Transformation and inverse transformation for δℓ = 0 and δr = 1/10: identity on the left (same tail behavior) and a heavy-tailed transformation in the right tail of input U.
Mentions: A parametric transformation is the basis of Tukey's h random variables [17](2)Z=Uexph2U2, h≥0,where U is standard Normal random variable and h is the heavy tail parameter. The random variable Z has tail parameter a = 1/h [17] and reduces to the Gaussian for h = 0. Morgenthaler and Tukey [26] extend the h distribution to the skewed, heavy-tailed family of hh random variables (3)Z=Uexpδl2U2,if  U≤0,Uexpδr2U2,if  U>0,where again U ~ 𝒩(0,1). Here δℓ ≥ 0 and δr ≥ 0 shape the left and right tail of Z, respectively; thus transformation (3) can model skewed and heavy-tailed data; see Figure 3(a). For the sake of brevity let Hδ(u)∶ = uexp((δ/2)u2).

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