BulkAndTailsDist(κ₀,τ₀,ϕ₀,κ₁,τ₁,ϕ₁,ν) The Bulk-And-Tails (BATs) distribution has cumulative distribution function

\[F(x) = T_ν(H(x))\]

where T_ν is the student-t cdf with ν degrees of freedom and H is a monotone increasing function. ϕ₀ and ϕ₁ are location parameters for the lower and upper tails. τ₀ and τ₁ are scale parameters for the lower and upper tails. κ₀ and κ₁ are shape parameters for the lower and upper tails. Negative κ indicates a bounded tail. Positive κ indicates a heavy tail. The case κ=0, defined by continuity, indicates a thin Gaussian tail.

fitbats provides maximum likelihood estimation for these parameters. In addition, we provide pdf, cdf, logpdf, logcdf, quantile, and rand functions for BulkAndTailsDist, all of which follow the Distributions.jl framework. We also provide R-friendly versions of these functions and show how to call them from R.

BulkAndTailsDist(κ₀,τ₀,ϕ₀,κ₁,τ₁,ϕ₁,ν)     # BATs distribution
params(d)        # Get the parameters
minimum(d)       # lower bound of support
maximum(d)       # upper bound of support

External links

• Stein, M. L. (2021). A parametric model for distributions with flexible behavior in both tails. Environmetrics, 32(2):Paper No. e2658, 24.