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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

Log-likelihood decomposition for Lambert W ×  FX distributions.
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Related In: Results  -  Collection


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fig6: Log-likelihood decomposition for Lambert W ×  FX distributions.

Mentions: Decomposition (28) shows the difference between the exact MLE based on y and the approximate MLE based on xτ alone: if we knew τ = (μX, σX, δ) beforehand, then we could back-transform y to xτ and estimate from xτ (maximize (29) with respect to β). In practice, however, τ must also be estimated and this enters the likelihood via the additive term ℛ(τ; y). A little calculation shows that for any yi ∈ ℝ, logR(μX, σX, δ; yi) ≤ 0 if δ ≥ 0, with equality if and only if δ = 0. Thus ℛ(τ; y) can be interpreted as a penalty for transforming the data. Maximizing (28) faces a trade-off between transforming the data to follow fX(x∣β) (and increasing ) and the penalty of a more extreme transformation (and decreasing ℛ(τ; y)); see Figure 6(b).


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

Goerg GM - ScientificWorldJournal (2015)

Log-likelihood decomposition for Lambert W ×  FX distributions.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig6: Log-likelihood decomposition for Lambert W ×  FX distributions.
Mentions: Decomposition (28) shows the difference between the exact MLE based on y and the approximate MLE based on xτ alone: if we knew τ = (μX, σX, δ) beforehand, then we could back-transform y to xτ and estimate from xτ (maximize (29) with respect to β). In practice, however, τ must also be estimated and this enters the likelihood via the additive term ℛ(τ; y). A little calculation shows that for any yi ∈ ℝ, logR(μX, σX, δ; yi) ≤ 0 if δ ≥ 0, with equality if and only if δ = 0. Thus ℛ(τ; y) can be interpreted as a penalty for transforming the data. Maximizing (28) faces a trade-off between transforming the data to follow fX(x∣β) (and increasing ) and the penalty of a more extreme transformation (and decreasing ℛ(τ; y)); see Figure 6(b).

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