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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

Comparing moments of Lambert W × Gaussian and Student's t.
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fig5: Comparing moments of Lambert W × Gaussian and Student's t.

Mentions: For Gaussian input Lambert W ×  FX equals Tukey's h, which has been studied extensively. Dutta and Babbel [40] show that (16)EZn=0,if n is odd,n<1δ,n!1−nδ−n+1/22n/2n/2!,if n is even,n<1δ,∄,if n is odd,n>1δ,∞,if n is even,n>1δ,which, in particular, implies that [28] (17)EZ=EZ3=0, if  δ<1  and  13,  respectively,(18)EZ2=11−2δ3/2, if  δ<12,Thus the kurtosis of Y equals (see Figure 5) (19)γ2δ=31−2δ31−4δ5/2 for  δ<1/4.For δ = 0, (18) and (19) reduce to the familiar Gaussian values.


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

Goerg GM - ScientificWorldJournal (2015)

Comparing moments of Lambert W × Gaussian and Student's t.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig5: Comparing moments of Lambert W × Gaussian and Student's t.
Mentions: For Gaussian input Lambert W ×  FX equals Tukey's h, which has been studied extensively. Dutta and Babbel [40] show that (16)EZn=0,if n is odd,n<1δ,n!1−nδ−n+1/22n/2n/2!,if n is even,n<1δ,∄,if n is odd,n>1δ,∞,if n is even,n>1δ,which, in particular, implies that [28] (17)EZ=EZ3=0, if  δ<1  and  13,  respectively,(18)EZ2=11−2δ3/2, if  δ<12,Thus the kurtosis of Y equals (see Figure 5) (19)γ2δ=31−2δ31−4δ5/2 for  δ<1/4.For δ = 0, (18) and (19) reduce to the familiar Gaussian values.

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