Tukey lambda distribution

Abstract: The Tukey depth (Proceedings of the International Congress of Mathematicians, vol. 2, pp. 523---531, 1975) of a point p with respect to a finite set S of points is the minimum number of elements of S contained in any closed halfspace that contains p. Algorithms for computing the Tukey depth of a point in various dimensions are considered. The running times of these algorithms depend on the value of the output, making them suited to situations, such as outlier removal, where the value of the output is typically small.

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.

Abstract: When a normal distribution of forecast errors is assumed one often ends up with a decision rule for the safety factor k involving the tail area of the unit normal distribution. Some form of table or rational approximation is necessary to find the k value once the desired tail area is known. Here we develop a simple approximation based on Tukey's lambda distribution.