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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

Schematic view of the heavy tail Lambert W ×  FX framework. Latent input X ~ FX: Hτ(X) from (6) transforms (solid arrows) X to Y~ Lambert W ×  FX and generates heavy tails (right) Observed heavy-tailed Y and y: (1) use Wτ(·) to back-transform y to latent “Normal” xτ, (2) use model ℳ𝒩 of your choice (regression, time series models, hypothesis testing, etc.) for inference on xτ, and (3) convert results back to the original “heavy-tailed world” of y (right).
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fig1: Schematic view of the heavy tail Lambert W ×  FX framework. Latent input X ~ FX: Hτ(X) from (6) transforms (solid arrows) X to Y~ Lambert W ×  FX and generates heavy tails (right) Observed heavy-tailed Y and y: (1) use Wτ(·) to back-transform y to latent “Normal” xτ, (2) use model ℳ𝒩 of your choice (regression, time series models, hypothesis testing, etc.) for inference on xτ, and (3) convert results back to the original “heavy-tailed world” of y (right).

Mentions: Figure 1 illustrates this pragmatic approach: researchers can make their observations y as Gaussian as possible (xτ) before making inference based on their favorite Gaussian model ℳ𝒩. This avoids the development of, or the data analysts waiting for, a whole new theory of ℳG and new implementations based on a particular heavy-tailed distribution G, while still improving statistical inference from heavy-tailed data y. For example, consider y = (y1,…, y500) from a standard Cauchy distribution 𝒞(0,1) in Figure 2(a): modeling heavy tails by a transformation makes it even possible to Gaussianize this Cauchy sample (Figure 2(c)). This “nice” data xτ can then be subsequently analyzed with common techniques. For example, the location can now be estimated using the sample average (Figure 2(d)). For details see Section 6.1.


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

Goerg GM - ScientificWorldJournal (2015)

Schematic view of the heavy tail Lambert W ×  FX framework. Latent input X ~ FX: Hτ(X) from (6) transforms (solid arrows) X to Y~ Lambert W ×  FX and generates heavy tails (right) Observed heavy-tailed Y and y: (1) use Wτ(·) to back-transform y to latent “Normal” xτ, (2) use model ℳ𝒩 of your choice (regression, time series models, hypothesis testing, etc.) for inference on xτ, and (3) convert results back to the original “heavy-tailed world” of y (right).
© Copyright Policy - open-access
Related In: Results  -  Collection

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fig1: Schematic view of the heavy tail Lambert W ×  FX framework. Latent input X ~ FX: Hτ(X) from (6) transforms (solid arrows) X to Y~ Lambert W ×  FX and generates heavy tails (right) Observed heavy-tailed Y and y: (1) use Wτ(·) to back-transform y to latent “Normal” xτ, (2) use model ℳ𝒩 of your choice (regression, time series models, hypothesis testing, etc.) for inference on xτ, and (3) convert results back to the original “heavy-tailed world” of y (right).
Mentions: Figure 1 illustrates this pragmatic approach: researchers can make their observations y as Gaussian as possible (xτ) before making inference based on their favorite Gaussian model ℳ𝒩. This avoids the development of, or the data analysts waiting for, a whole new theory of ℳG and new implementations based on a particular heavy-tailed distribution G, while still improving statistical inference from heavy-tailed data y. For example, consider y = (y1,…, y500) from a standard Cauchy distribution 𝒞(0,1) in Figure 2(a): modeling heavy tails by a transformation makes it even possible to Gaussianize this Cauchy sample (Figure 2(c)). This “nice” data xτ can then be subsequently analyzed with common techniques. For example, the location can now be estimated using the sample average (Figure 2(d)). For details see Section 6.1.

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