RobustTRegression

RobustTRegression is an R package for performing robust regression using the t-distribution. It provides tools to fit regression models that are resistant to outliers and heavy-tailed errors, improving the reliability of parameter estimates in such scenarios.

Overview

We utilize two-stage procedure under the student’s-t regression framework as follows: we first estimate the degrees of freedom parameter ($\nu$), and then use this estimate to infer the regression coefficients ($\beta$). We provide several frequentist and bayesian methods, including:

• Profile likelihood
• Adjusted profile likelihood
• Independent Jeffrey's
• Marginalized Jeffrey's
• Marginalized Fisher

Functions

Simulate data

Generate synthetic data from a linear regression model where the error terms follow a Student's t-distribution with a specified degrees of freedom.

Example:

simulate_t_error_data(n = 300, p = 1, beta = rep(0,1), sigma = 1, nu = 2, seed = NULL)

Estimate $\nu$

$\nu$ estimation objective functions:

• Profile likelihood
• Adjusted profile likelihood
• Likelihood with sigma inverse prior, constant $\beta$ prior and $\nu$ Independent Jeffrey's prior
• Profile likelihood with $\nu$ Jeffrey's prior
• Profile likelihood with $\nu$ Fisher prior ($\nu$ block in Fisher information)

For each of the 5 $\nu$ estimation approaches, we provide functions to compute the 5 objective functions estimator for the degrees of freedom parameter \eqn{\nu} via optimization over \eqn{\omega = 1 / \nu}. Users need to specify the initial value in $\omega$ scale for BFGS optimization. By default, it is set as 0.5 (i.e., $\nu = 2$).

Example:

estimate_nu_profile(y, x, omega_init = 1/2)

We also provide a function to conduct estimations on all 5 approaches all at once. Users need to specify the initial value in $\omega$ scale for BFGS optimization.

Example:

run_all_estimators(y, x, omega_init = 1/2)

Estimate $\beta$

We provide functions to first compute a $\nu$ estimator and then proceed to optimize the t-likelihood to obtain $\beta$ estimators. Users can choose one of the following $\nu$-estimation approaches:

• Profile likelihood
• Adjusted profile likelihood
• Independent Jeffrey's
• Marginalized Jeffrey's
• Marginalized Fisher

Users need to specify the initial value in $\omega$ scale for BFGS optimization. By default, it is set as 0.5 (i.e., $\nu = 2$).

Additionally, users can also opt for non $\nu$-estimation approaches, which prodcue $\beta$ estimators directly:

• Fixed $\nu$ (specify method as a positive integer)
• OLS
• Huber

Example:

estimate_beta(y, x, method = "Profile", omega_init = 0.5)

Installation

You can install the package directly from GitHub using remotes:

# Install remotes if not already installed
if (!requireNamespace("remotes", quietly = TRUE)) {
  install.packages("remotes")
}

# Install RobustTRegression from GitHub
remotes::install_github("amanda-ng518/RobustTRegression")