Feature Request: Closed-form rule-of-thumb bandwidth for beta kernels

[egonmedhatten]

Hi Jonas,

First, thank you for maintaining kdensity! It's a fantastic package and the native support for Chen's beta kernels is incredibly useful for bounded data.

I am reaching out because I recently had a paper accepted in the Journal of Computational and Graphical Statistics (JCGS) that solves a major practical bottleneck for beta kernels: the lack of a fast, closed-form bandwidth selector. Currently, practitioners usually rely on numerical LSCV, which can be computationally expensive and occasionally unstable for extreme bounded data.

I derived the "Beta Reference Rule" (analogous to Silverman's rule), which reduces the bandwidth selection to an $O(1)$ operation with an automatic fallback heuristic for J/U-shaped distributions.

• Paper (Accepted in JCGS): A Fast, Closed-Form Bandwidth Selector for the Beta Kernel Density Estimator
• Standalone R implementation: I have translated the selector into a self-contained base R function in my repository here: beta_bw.R (There is also a Python package).

Since kdensity already has the excellent beta kernel infrastructure, I thought this might be a great fit as a new bandwidth option (e.g., bw = "beta-rot") for your users.

Here is a quick reproducible example showing how it plugs directly into your package:

# Source the standalone bandwidth selector
source("https://raw.githubusercontent.com/egonmedhatten/beta-kde/main/beta_bw.R")

# Generate bounded data
set.seed(42)
my_data <- rbeta(500, 2, 5)

# Calculate the optimal rule-of-thumb bandwidth
h_opt <- bw.beta.rot(my_data)
cat("Optimal bandwidth:", h_opt, "\n")

# Plug it directly into kdensity
library(kdensity)
fit <- kdensity(my_data, kernel = "beta", bw = h_opt)
plot(fit, main = "Beta KDE with Rule-of-Thumb Bandwidth")

I know package maintenance takes a lot of free time, so there is absolutely no pressure to implement this immediately. I just wanted to provide the math and the code in case you'd like to include it in a future CRAN release. If there is anything I can do to help (like submitting a Pull Request), please let me know!

Best regards,
Johan Hallberg Szabadváry

[JonasMoss]

That's awesome man, congrats! This sounds like a non-brainer to implement and sounds very quick 'n easy to do. I'll have a look at this later this week.

I didn't check your paper yet, but is this a closed for for just ordinary (asymptotic) MISE? Or do you use tricks like Jones & Henderson did for the Gaussian copula kernel (I can't recall specifics, i think the used a convenient weight in their MISE). Very cool regardless.

[egonmedhatten]

Hi Jonas,

Thanks for the incredibly quick and positive reply! I'm thrilled to hear you're open to including it.

To answer your question: No convenient weighted MISE tricks here! For the main selector, it is based entirely on the ordinary (standard) unweighted AMISE framework. Just as Silverman's rule uses a Gaussian reference to analytically evaluate the roughness penalty $\int (f''(x))^2 dx$, this uses a Beta distribution as the reference. Because the integrals of Beta densities evaluate nicely to Gamma functions, the standard AMISE integral completely collapses into the exact closed-form solution you see in the script.

However, in the interest of full transparency, I should mention that the script also includes a necessary "fallback" heuristic. If the data is highly U-shaped or extreme J-shaped (specifically, if the estimated Beta shape parameters are too small), the standard AMISE roughness penalty theoretically diverges because the density shoots to infinity at the boundaries.

For those specific edge cases, the standard reference rule breaks down, so the function automatically catches it and falls back to a robust finite-sample heuristic based on the variance, skewness, and kurtosis (which is what the bottom half of the bw.beta.rot function is doing).

Take your time looking at it later this week, and just let me know if you run into any questions or need anything tweaked for the package infrastructure. Thanks again!

[JonasMoss]

Used your file and made #67 Have submitted to CRAN and will merge after acceptance.

[egonmedhatten]

Fantastic news. Thank you very much for this incredibly fast integration. I am truly honored by the bw = "HS" naming convention and the README inclusion. Congratulations on the new release!