Python - Tukey-Lambda Distribution in Statistics

Introduction

In probability theory and statistics, the Tukey-Lambda distribution is a family of probability distributions. This distribution can be used to model a variety of random variables, including those with heavy tails, such as income, rents, and stock market returns.

The Tukey-Lambda distribution has an additional parameter, λ, which can be used to adjust the shape of the distribution. When λ = 0, the distribution is equivalent to the exponential distribution. When λ = 1, the distribution is equivalent to the Gumbel distribution. As λ moves away from these values, the distribution becomes more skewed and has heavier tails.

Implementation in Python

The Tukey-Lambda distribution is not currently implemented in the standard Python libraries. However, it can be easily implemented using the SciPy library, which provides a way to define and fit custom probability distributions.

Here's a code snippet that demonstrates how to define and fit a Tukey-Lambda distribution using the SciPy library:

import numpy as np
from scipy.stats import tukeylambda

# Generate random data from a Tukey-Lambda distribution with λ = 0.5
data = tukeylambda.rvs(lambda_=0.5, size=1000)

# Fit a Tukey-Lambda distribution to the data
params = tukeylambda.fit(data)

# Compute the PDF of the fitted distribution
x = np.linspace(tukeylambda.ppf(0.01, *params),
                tukeylambda.ppf(0.99, *params), 100)
pdf = tukeylambda.pdf(x, *params)

# Plot the histogram of the data and the PDF of the fitted distribution
import matplotlib.pyplot as plt
count, bins, ignored = plt.hist(data, 30, density=True)
plt.plot(x, pdf, linewidth=2, color='r')
plt.show()

The tukeylambda.rvs() function generates random data from a Tukey-Lambda distribution with a given value of λ. The tukeylambda.fit() function fits a Tukey-Lambda distribution to the data and returns the parameters of the fitted distribution. The tukeylambda.pdf() function computes the probability density function (PDF) of the fitted distribution for a given value of x.

The resulting plot shows the histogram of the generated data and the PDF of the fitted Tukey-Lambda distribution.

Conclusion

The Tukey-Lambda distribution is a useful tool for modeling random variables with heavy tails. In Python, the SciPy library provides a way to define and fit Tukey-Lambda distributions to data. This allows programmers to accurately model a variety of real-world phenomena, leading to better statistical analysis and prediction.