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millsratio: Interactive Analysis of Mills Ratios and Tail Thickness

Overview

The millsratio package provides tools for computing and visualizing Mills ratios across different probability distributions. It includes an interactive dashboard for exploring how Mills ratios reveal fundamental differences between distribution tails, particularly highlighting the t(30) paradox where distributions appear similar centrally but diverge substantially in the tails.

This package implements and extends concepts from John D. Cook’s blog post on Mills ratios.

Key Concepts

The Mills ratio is defined as:

m(x) = [1 - F(x)] / f(x)

where F(x) is the CDF and f(x) is the PDF.

What Mills Ratios Reveal:

  • Decreasing Mills ratio → Thin tails (e.g., Normal: m(x) ~ 1/x)
  • Increasing Mills ratio → Fat tails (e.g., t-distribution: m(x) ~ x/df)
  • Constant Mills ratio → Exponential tails (e.g., Exponential: m(x) = 1/rate)

Installation

From GitHub (Standard R) 

# Install development version from GitHub
remotes::install_github("JohnGavin/millsratio")

# Load the package
library(millsratio)

From Nix (Reproducible Environment)

For a fully reproducible environment with all dependencies:

# Clone and use the project's Nix shell
git clone https://github.com/JohnGavin/millsratio.git
cd millsratio

# Enter Nix shell with GC root (creates default.nix automatically)
chmod +x default.sh
./default.sh  # First run: builds environment, subsequent runs: fast

# Now in Nix shell, R has all dependencies
R
> library(millsratio)

Using with rix

For integration with existing R projects:

library(rix)
rix(
  r_ver = "4.5.0",
  r_pkgs = c("tidyverse", "plotly", "shiny"),
  git_pkgs = list(
    millsratio = list(
      package_source = "github",
      repo_url = "JohnGavin/millsratio",
      branch = "main"
    )
  ),
  ide = "code",
  project_path = "."
)
# Then: nix-shell

Quick Start

Launch Interactive Dashboard 

# Launch the interactive dashboard
launch_dashboard()

The dashboard provides: - 12 interactive pages organized into Analysis, Theory, and Playground sections - Real-time visualization of Mills ratios - Comparison tools for multiple distributions - The t(30) paradox demonstration - Custom code playground for experiments

Basic Usage 

# Calculate Mills ratio for normal distribution
x <- seq(1, 5, by = 0.5)
m_normal <- mills_ratio_normal(x)
print(round(m_normal, 3))

# Calculate for t-distribution
m_t30 <- mills_ratio_t(x, df = 30)
print(round(m_t30, 3))

# Compare distributions
x_compare <- c(1, 2, 3, 4)
comparison <- compare_mills_ratios(x_compare, c("normal", "t30", "exponential"))
print(comparison)

Visualize the Comparison 

# Generate curves for visualization
curves <- simulate_mills_curves(
  x_range = c(0.5, 5),
  distributions = c("normal", "t30", "exponential")
)

# Create plot
plot_mills_curves(curves, log_y = TRUE)

The t(30) Paradox 

# Analyze the t(30) paradox
paradox_data <- analyze_t30_paradox(x_range = c(0, 5), n_points = 100)

# Show how t(30) diverges from normal in the tails
x_vals <- c(1, 2, 3, 4, 5)
ratio <- mills_ratio_t(x_vals, df = 30) / mills_ratio_normal(x_vals)
cat("t(30)/Normal Mills ratio divergence:\n")
for (i in seq_along(x_vals)) {
  cat(sprintf("x = %d: ratio = %.3f\n", x_vals[i], ratio[i]))
}
# Visualize the paradox
plot_t30_paradox(paradox_data, focus = "mills")

Main Functions

Core Mills Ratio Functions

Analysis Functions

Simulation Functions

Visualization Functions

Dashboard

Mathematical Background

Asymptotic Behavior

Distribution Mills Ratio Behavior Tail Type
Normal m(x) ~ 1/x as x→∞ Thin
Student t(df) m(x) ~ x/df as x→∞ Fat
Exponential m(x) = 1/rate (constant) Medium

The t(30) Paradox

The t(30) distribution is often considered “practically normal” for many applications. However: - Central region: Nearly indistinguishable from normal - Tail region: Substantially different Mills ratios - Implication: Risk models based on normality assumptions may severely underestimate tail probabilities

Examples

Example 1: Verify Mills Ratio Properties 

# Verify that normal has decreasing Mills ratio (thin tails)
x <- seq(1, 5, by = 0.5)
m <- mills_ratio_normal(x)
cat("Normal Mills ratio decreasing:", all(diff(m) < 0), "\n")

# Verify that t(3) has increasing Mills ratio (fat tails)
m_t <- mills_ratio_t(x, df = 3)
cat("t(3) Mills ratio increasing:", all(diff(m_t) > 0), "\n")

# Verify exponential has constant Mills ratio
m_exp <- mills_ratio_exp(x)
cat("Exponential Mills ratio constant:", all(abs(diff(m_exp)) < 1e-10), "\n")

Example 2: Interactive Exploration 

# Generate interactive plot
library(plotly)
curves <- simulate_mills_curves(
  x_range = c(0.1, 10),
  distributions = c("normal", "t3", "t10", "t30"),
  log_scale = TRUE
)

p <- plot_mills_curves(curves, log_y = TRUE, interactive = TRUE)
p  # Opens interactive plotly visualization

Example 3: Monte Carlo Verification 

# Verify Mills ratio empirically
set.seed(123)
mc_result <- monte_carlo_mills(
  n_sim = 10000,
  x_val = 2,
  distribution = "normal"
)

cat("Empirical Mills ratio:", round(mc_result$empirical_mills, 4), "\n")
cat("True Mills ratio:", round(mc_result$true_mills, 4), "\n")
cat("Relative error:", round(mc_result$relative_error, 4), "\n")

Testing

Run the test suite:

devtools::test()

Check package:

devtools::check()

Reproducible Pipeline

This package uses targets to ensure all examples are tested:

# Run the pipeline to test all examples
targets::tar_make()

# View the pipeline
targets::tar_visnetwork()

License

MIT License

Author

John Gavin

Implementation support: Claude Assistant

References

  • Cook, J. D. (2026). “Mills Ratio and Tail Thickness.” Blog post
  • Mills, J. P. (1926). “Table of the ratio: Area to bounding ordinate, for any portion of normal curve.” Biometrika, 18(3/4), 395-400.

Project Structure

Citation

If you use this package in your research, please cite:

@software{millsratio2026,
  author = {Gavin, John},
  title = {millsratio: Interactive Analysis of Mills Ratios and Tail Thickness},
  year = {2026},
  url = {https://github.com/yourusername/millsratio}
}