LMDT: Local Manifold Dispersion Test for Heteroskedasticity

LMDT implements the Local Manifold Dispersion Test for detecting complex, non-linear, and high-dimensional heteroskedasticity in linear regression models.

Unlike classical tests (Breusch–Pagan, White) that rely on restrictive linear specifications or suffer from degrees-of-freedom explosion when $p > 15$, LMDT measures the graph-spectral smoothness (Dirichlet energy) of robust log-dispersion residuals over a self-tuning $k$-nearest neighbors normalized Laplacian matrix.

Key Features

Asymptotic Kurtosis Invariance: The zero-diagonal structure of the affinity operator asymptotically eliminates fourth-order error moments from the limiting variance at rate $O(1/n)$, providing robust size calibration under heavy-tailed (e.g., Student-$t$) error distributions.
Fast Computational Complexity: Evaluates in $O(k \cdot n \log n)$ time, scaling easily to thousands of observations.
High-Dimensional Scaling ($p > 50$): Combines unsupervised Gavish–Donoho optimal singular-value thresholding and supervised metric learning to prevent noise dilution.
S3 Object-Oriented Interface: Works seamlessly with lm() fit objects, symbolic formulas (y ~ x1 + x2), and numeric matrices.
Diagnostic Visualizations: Native S3 plot() method displays the null distribution and log-dispersion profiles.

Installation

From GitHub (Development Version)

# install.packages("remotes")
remotes::install_github("amjed-droid/LMDT")

From CRAN (Upcoming)

install.packages("LMDT")

Quick Start

1. Basic Usage with `lm()`

library(LMDT)

# Simulate regression data with non-linear heteroskedasticity
set.seed(42)
n <- 200
x1 <- runif(n, -2, 2)
x2 <- runif(n, -2, 2)
r <- sqrt(x1^2 + x2^2)
sigma <- 0.3 + 3.0 * exp(-(r - 1.2)^2 / (2 * 0.3^2))
y <- 2 * x1 - x2 + rnorm(n, sd = sigma)

# Fit linear model
fit <- lm(y ~ x1 + x2)

# Perform LMDT test
res <- lmdt_test(fit)
print(res)

# Plot diagnostic visualizations
plot(res)

2. High-Dimensional Regression ($p = 60$)

# Regressors with ambient noise dimensions
X <- matrix(rnorm(n * 60), n, 60)
y <- X[, 1] + rnorm(n, sd = 1 + abs(X[, 1]))

# Use supervised metric learning or adaptive PCA denoising
res_hd <- lmdt_test(X, residuals(lm(y ~ X)), denoise = "metric")
summary(res_hd)

Calibration Methods

method = "wild_bootstrap" (Default): Recommended for finite samples and non-Gaussian errors.
method = "permutation": Exact non-parametric permutation calibration under $H_0$.
method = "asymptotic": Fast Gaussian approximation via de Jong's Central Limit Theorem for degenerate quadratic forms.

Citation

To cite LMDT in publications, please use:

@article{jabbar2026lmdt,
  title={The Local Manifold Dispersion Test: A Nonparametric Graph-Spectral Approach for Detecting Complex and High-Dimensional Heteroskedasticity},
  author={Jabbar, Ahmed Sattar},
  journal={Working Paper, Department of Statistics, Mustansiriyah University},
  year={2026}
}

Author & Maintainer

Ahmed Sattar Jabbar
Department of Statistics, College of Administration and Economics,
Mustansiriyah University, Baghdad, Iraq.
Email: ahmed.state.me@gmail.com