The Pareto Distribution: From Intuition to Implementation in Python

Updated on April 24, 2025 5 minutes read

1 Why care? (A quick visual)

“Roughly 80% of GitHub stars sit on 20% of projects.”

That claim feels right—open-source fame is lopsided. But how lopsided, exactly? Plot a histogram of star counts and the right tail stretches far beyond the median. Heavy-tailed data like this are commonplace in wealth, city sizes, neural-network gradients, and website traffic. The simplest continuous model for such behaviour is the Pareto (Type I) distribution.

2 Pareto principle ≠ Pareto distribution

• Pareto principle (80/20 rule) – an empirical heuristic: in many settings, 80% of outcomes stem from 20% of causes.
• Pareto distribution – a parametric power-law family with PDF:

$$ f(x; \alpha, x_m) = \begin{cases} \displaystyle \frac{\alpha,x_m^{\alpha}}{x^{\alpha+1}}, & x \ge x_m, \[6pt] 0, & x < x_m. \end{cases} $$

Only when the shape parameter

$$ \alpha = \log_{4} 5 \approx 1.16 $$

does the distribution reproduce the exact 80/20 split.

3 Mathematical essentials

4 Simulating a Pareto in Python

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import pareto   # SciPy ≥1.11

alpha, x_m = 2.5, 1          # shape, scale
n = 50_000
# inverse-CDF sampling
samples = x_m * (1 - np.random.rand(n))**(-1/alpha)

# Plot histogram and theoretical PDF
plt.hist(samples, bins=200, density=True, alpha=0.4)
x = np.linspace(x_m, samples.max(), 400)
plt.plot(x, pareto.pdf(x, alpha, scale=x_m), lw=2)
plt.xscale('log')
plt.yscale('log')
plt.xlabel('x')
plt.ylabel('PDF (log-log)')
plt.title('Simulated Pareto(α=2.5)')
plt.show()

scipy.stats.pareto offers the same functionality, but the one-liner above emphasizes the inverse-transform trick.

5 Fitting the distribution

5.1 Maximum-likelihood (SciPy)

alpha_hat, loc_hat, scale_hat = pareto.fit(data, floc=0)  # force x_m > 0

alpha_hat is the MLE of the tail index. For a true Pareto it is asymptotically efficient.

5.2 Hill estimator (tail-only)

def hill(x, k):
    x = np.sort(x)[::-1]          # descending
    return k / np.sum(np.log(x[:k] / x[k]))

alpha_hat = hill(data, k=1000)

Use the largest (k) order statistics; choose (k) via stability plots.

5.3 Bootstrap confidence interval

import random
import numpy as np
from scipy.stats import pareto

alphas = []
for _ in range(999):
    resample = random.choices(data, k=len(data))
    alphas.append(pareto.fit(resample, floc=0)[0])

ci = np.percentile(alphas, [2.5, 97.5])
print(f"95% CI for α: {ci[0]:.3f}–{ci[1]:.3f}")

6 Model diagnostics

1. Q–Q plot of log-data vs fitted distribution.
2. Kolmogorov–Smirnov test (scipy.stats.kstest).
3. Tail comparison – overlay Pareto and (e.g.) lognormal PDFs on double-log axes; genuine power laws appear as straight lines.

from scipy.stats import kstest, lognorm

ks_p = kstest(data, 'pareto', args=(alpha_hat, 0, scale_hat)).pvalue
print(f'KS p-value: {ks_p:.3f}')

If a lognormal gives higher log-likelihood, your data might only look heavy-tailed.

7 Case Study: GitHub Stars

Download a public snapshot of the most-starred repositories (CSV, ~20 k repos).

import pandas as pd
import seaborn as sns
from scipy.stats import pareto
import matplotlib.pyplot as plt
import numpy as np

# Load data
stars = pd.read_csv('most_starred_repos.csv')['Stars'].values

# Filter and rescale
xmin = 50                       # only repos with ≥50 stars
data = stars[stars >= xmin] / xmin   # re-scale so x_m = 1

# Fit Pareto via MLE
alpha_hat, loc_hat, scale_hat = pareto.fit(data, floc=0, fscale=1)
print(f'α = {alpha_hat:.2f}')

# Visual check
sns.ecdfplot(data=data, log_scale=True)
plt.plot(np.sort(data),
         pareto.cdf(np.sort(data), alpha_hat, scale=1))
plt.xlabel('Stars / 50')
plt.ylabel('CDF')
plt.show()

A fitted alpha_hat of ~1.3 confirms a fat right tail—close to the fabled 80/20 but not identical.

8 When Not to Use Pareto

• Finite upper bound (e.g., percentages) – tails cannot be infinite.
• Multi-modal data – a single power law misses clusters.
• Variance matters – if $\alpha \le 2$, variance is infinite, so variance-based metrics fail.
• Regulatory/actuarial constraints – heavy tails violate solvency requirements unless tamed.

9 Beyond Classical Statistics: Pareto in ML

• Stochastic-gradient magnitudes in deep nets follow power-law spectra, motivating adaptive learning-rate schedules.
• Attention-weight distributions in large-language models show Pareto-like sparsity, inspiring mixed-precision storage.
• Neural scaling laws: empirical exponents for loss vs. compute can be linked to heavy-tailed weight updates.

10 One-Page Cheat Sheet

11 Key Takeaways

1. Pareto ≠ 80/20: the principle is a rule of thumb, while the distribution is a model with tunable α.
2. Check the tail: log–log plots and survival functions reveal true power laws.
3. Fit responsibly: use MLE/Hill estimators, bootstrap confidence intervals, and compare against lognormal.
4. Mind the moments: for α less than 2, variance is infinite—plan analyses accordingly.
5. Heavy tails are everywhere—understanding Pareto equips you for real-world, messy data.

Further Reading

• SciPy rv_continuous documentation for pareto.
• Wikipedia pages on Pareto principle and Pareto distribution for historical context.

Happy heavy-tail modelling!

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