Skew-T Distribution (Fernandez & Steel)

This package provides a Python implementation of the skewed Student's t-distribution as proposed by Fernandez and Steel (1998). It is created as a scipy.stats compatible continuous random variable, making it easy to use for statistical modeling and analysis.

Key Features

• Easy to use: Implemented as a standard scipy.stats object.
• Flexible: Control both skewness (gamma) and tail thickness (df).
• Standard Methods: Includes pdf, cdf, ppf, rvs, and stats (mean/variance).

Installation

pip install skewt-fs

Mathematical Details

The implementation follows the original paper:

Fernández, C., & Steel, M. F. J. (1998). On Bayesian Modeling of Fat Tails and Skewness. Journal of the American Statistical Association, 93(441), 359-371.

Let $g_\nu(x)$ be the probability density function (PDF) of the standard symmetric Student's t-distribution with $\nu$ degrees of freedom. The PDF of the skewed Student's t-distribution $f(x | \nu, \gamma)$ is defined as:

$$ f(x | \nu, \gamma) = \frac{2}{\gamma + \frac{1}{\gamma}} \left[ g_\nu\left(\frac{x}{\gamma}\right)I(x \ge 0) + g_\nu(\gamma x)I(x < 0) \right] $$

where:

• $\nu$ is the degrees of freedom (df).
• $\gamma$ is the skewness parameter (gamma).
• $I(\cdot)$ is the indicator function.

Moments

The mean and variance of the distribution are:

Mean ($E[X]$), for $\nu > 1$: $$ E[X] = M_1 \left(\gamma - \frac{1}{\gamma}\right) $$

Variance ($Var(X)$), for $\nu > 2$: $$ Var(X) = (M_2 - M_1^2)\left(\gamma^2 + \frac{1}{\gamma^2}\right) + 2M_1^2 - M_2 $$

where $M_1 = E[|Z|]$ and $M_2 = E[Z^2]$ for a standard Student's t-distributed random variable $Z \sim g_\nu$.

How to Use the Package

The main object is skewt, which behaves like any other scipy.stats distribution object.

Importing

from skewt_fs import distribution
skewt = distribution.skewt

Basic Operations

You can use all the standard methods. The shape parameters df and gamma are passed as arguments.

# Define parameters: 5 degrees of freedom, right-skew (gamma > 1)
df = 5
gamma = 1.8

# Get theoretical mean and variance
mean, var = skewt.stats(df=df, gamma=gamma, moments='mv')
print(f"Theoretical Mean: {mean:.4f}")
print(f"Theoretical Variance: {var:.4f}")

# Evaluate the PDF at a point
pdf_val = skewt.pdf(x=1.0, df=df, gamma=gamma)
print(f"PDF at x=1: {pdf_val:.4f}")

# Evaluate the CDF at a point
cdf_val = skewt.cdf(x=1.0, df=df, gamma=gamma)
print(f"CDF at x=1: {cdf_val:.4f}")

# Find a percentile with the PPF (inverse CDF)
# For example, the 95th percentile
ppf_val = skewt.ppf(q=0.95, df=df, gamma=gamma)
print(f"95th percentile: {ppf_val:.4f}")

Generating Random Samples

Use the .rvs() method to generate random variates.

# Generate 1000 random samples
samples = skewt.rvs(df=df, gamma=gamma, size=1000)

# Verify sample moments
import numpy as np
print(f"Sample Mean: {np.mean(samples):.4f}")
print(f"Sample Variance: {np.var(samples):.4f}")

Visualization

You can easily plot the distribution to visualize the effect of the parameters.

# Set up the plot
import matplotlib.pyplot as plt
fig, ax = plt.subplots(figsize=(10, 6))
x = np.linspace(skewt.ppf(0.001, df, gamma), skewt.ppf(0.999, df, gamma), 200)

# Plot the PDF
ax.plot(x, skewt.pdf(x, df=df, gamma=gamma), 'r-', lw=3, alpha=0.8, label=f'skewt pdf (df={df}, γ={gamma})')

# Plot a histogram of the random samples
ax.hist(samples, bins=50, density=True, histtype='stepfilled', alpha=0.3, label='Sample Histogram')

# Add lines for mean and median
ax.axvline(mean, color='k', linestyle='--', label=f'Mean: {mean:.2f}')
ax.axvline(skewt.ppf(0.5, df, gamma), color='g', linestyle='-', label=f'Median: {skewt.ppf(0.5, df, gamma):.2f}')

ax.set_title("Fernandez-Steel Skewed Student's t-Distribution")
ax.set_xlabel("x")
ax.set_ylabel("Density")
ax.legend()
ax.grid(True, linestyle='--', alpha=0.6)
plt.show()

API Reference

skewt

An instance of the skewt_gen class. It is a continuous random variable object from scipy.stats.

Shape Parameters:

• df (float): Degrees of freedom, must be greater than 0.
• gamma (float): Skewness parameter, must be greater than 0.
  • gamma > 1: Right (positive) skew.
  • gamma < 1: Left (negative) skew.
  • gamma = 1: Symmetric (reverts to the standard Student's t-distribution).

Location and Scale: Like all scipy.stats objects, it also accepts loc (for mean/location) and scale parameters.

Contributing

Contributions are welcome! Please feel free to submit a pull request or open an issue on the project's GitHub repository.

License

This project is licensed under the MIT License.