# BulkAndTails.jl

A package implementing the "Bulk-And-Tails" (BATs) distribution along with maximum likelihood estimation of its parameters [1,2]. Due to the fact that this estimation requires solving a nonlinear optimization problem with nonlinear constraints, this functionality is wrapped into its own package to compartmentalize the dependencies on the several required dependencies.

# Discussion

BATs is a seven-parameter univariate distribution with flexible behavior in both tails. Unlike classical methods for extremes (e.g., generalized Pareto distribution, generalized extreme value distribution), which only fit a single tail of a distribution, BATs models the entire distribution (i.e., the bulk and both tails). BATs has seven parameters (κ₀,τ₀,ϕ₀,κ₁,τ₁,ϕ₁,ν) in total: shape parameters κ which control tail behavior, location parameters ϕ, scale parameters τ, and ν degrees of freedom of a student-t distribution. The subscript 0 refers to lower tail, and the subscript 1 refers to upper tail. If κ is negative, that tail is bounded; if κ is zero (defined by continuity), that tail is thin like a Gaussian tail; and if κ is positive, that tail is a heavy tail. See [1] for more details.

# Demonstration

See the example file for a more heavily commented discussion of this same demonstration. The R_usage folder also shows a similar example script using the JuliaCall package in R. It may be necessary to try several initializations to the optimization (these are declared with keyword argument init).

using Distributions, BulkAndTails
data = rand(TDist(1.0), 5_000) # 5k samples from Cauchy distribution
(mle, obs_information_matrix) = fit_mle(BulkAndTailsDist, data)
pdf(mle, 10.0) # compare with Cauchy pdf at 10.0.

# Fitting with Covariates

[2] explores the idea of allowing BATs parameters to depend on covariates to produce a nonstationary BATs distribution. In particular, we allow the location and (log) scale parameters to depend upon covariates to model seasonality and climate change. We provide example Julia/R scripts for fitting daily average temperatures in Boston using this methodology. There are a few differences from the standard seven-parameter method described above:

• An initial guess is required here (there is no default guess).
• In Julia, a namedtuple of parameters is returned instead of a BulkAndTailsDist struct.
• Since the optimization is more difficult and time-consuming, the parameters are returned regardless of convergence. status gives the result of the optimization, with convergence indicated by zero. See the example file for a description of status values.

Note that this parameterizes the logarithm of τ as a function of covariates to ensure positivity.

# Future Enhancements

1. For plenty of distributions with a known lower bound, it is possible to simplify the model parameterization and fit the model with that known value. Not doing that can cause slight issues in some cases due to the currently enforced constraint on the tail index that guarantees second derivatives of the density at the support endpoints. Incorporating this information and re-organizing the estimation code is very doable, it just will take some time and we haven't done it yet. This would be a fine PR for users interested in studying the distribution more.

2. Similar for upper bounds, and for both lower and upper bounds.

3. In principle, analytical derivatives for the PDF could be computed and hard-coded in to avoid using automatic differentiation. But considering how fast the code already is, this is not a priority.

# References

[1] Stein, M. L. (2021). A parametric model for distributions with flexible behavior in both tails. Environmetrics, 32(2):Paper No. e2658, 24. (https://onlinelibrary.wiley.com/doi/abs/10.1002/env.2658)

[2] Krock, M., Bessac, J., Stein, M. L., and Monahan, A. H. (2022). Nonstationary seasonal model for daily mean temperature distribution bridging bulk and tails. (https://arxiv.org/pdf/2110.10046.pdf)

# Authors

Mitchell Krock mk1867@stat.rutgers.edu (active development)

Julie Bessac jbessac@anl.gov (active development)

Chris Geoga christopher.geoga@rutgers.edu (base implementation)