Slash Distribution

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The slash distribution (also called the slash normal or ordinary slash) is a type of ratio distribution. It is the distribution of the ratio of a normal random variable to an independent uniform random variable (i.e., N(0, 1)/U(0, 1)) with symmetric probability density [1].

The distribution is known for its extreme outliers and heavier than normal tails (i.e., greater kurtosis) [2]. It is often used as a challenging distribution for statistical procedures such as investigating how variables perform under extreme conditions [3], including robustness studies (e.g., [4, 5]) and simulation studies; it is not as pathological as the Cauchy distribution, which is used for the same purposes.

The slash distribution was given its name by William H. Rogers and John Tukey in a paper published in their 1972 paper titled Understanding some long-tailed symmetrical distributions [6]. The authors introduced the distribution with the following stochastic representation:

Y = μ + σ (Z/U^1/q)

Where

• q > 0 = shape parameter (controls tail thickness and kurtosis)
• μ ∈ ℝ = location parameter
• σ > 0 = scale parameter.

Many variants exist, including the slash (student) t distribution and the skew-slash (student) t distribution which includes degrees of freedom as an extra parameter [7].

Slash distribution properties

 

The slash distribution probability density function (PDF) can be expressed in terms of the standard normal density φ(x) [9]:

The quotient has a discontinuity at zero and is therefore undefined at that point. A workaround is to remove the discontinuity with

The slash cumulative distribution function (CDF) is computed by numerically integrating the slash probability density function [10]

Several properties do not exist: mean, variance, skewness, kurtosis and moment generating function (MGF).

 

References

1. Vidakovic, B. et al. [Eds.]. (2005). Encyclopedia of Statistical Science. Wiley.
2. Wilcox, R. (2011). Introduction to Robust Estimation and Hypothesis Testing. Elsevier Science.
3. Wang, J. & Genton, M. The multivariate skew-slash distribution. Journal of Statistical Planning and Inference. Volume 136, Issue 1, 1 January 2006, Pages 209-220
4. Jamshidian M., A note on parameter and standard error estimation in adaptive robust regression, J. Stat. Comput. Simul. 71 (2001), pp. 11–27. doi: 10.1080/00949650108812131 [Google Scholar]
5. Kashid D.N. and Kulkarni S.R., Subset selection in multiple linear regression with heavy tailed error distribution, J. Stat. Comput. Simul. 73 (2003), pp. 791–805. doi: 10.1080/0094965031000078873 [Google Scholar]
6. Rogers, W. H.; Tukey, J. W. (1972). “Understanding some long-tailed symmetrical distributions”. Statistica Neerlandica. 26 (3): 211–226. doi:10.1111/j.1467-9574.1972.tb00191.x.
7. Tan, F. et al. The multivariate slash and skew-slash student t distributions. Journal of Statistical Distribution and Applications (2015) 2:3 DOI 10.1186/s40488-015-0025-9
8. Image: Qwfp, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons
9. Olmos, N. & Bolfarine, H. (2011). An extension of the generalized half-normal distribution. Stat PapersDOI 10.1007/s00362-013-0546-6
10. Korkmaz, M. A new heavy-tailed distribution defined on the bounded interval: the logit slash distribution and its application. J Appl Stat. 2020; 47(12): 2097–2119. Published online 2019 Dec 18. doi: 10.1080/02664763.2019.1704701