Wasserstein projection estimators for circular distributions

TL;DR

This work addresses parameter estimation for circular distributions by projecting the empirical circular distribution onto a parametric circle model under the $L^p$ Wasserstein distance. It derives circle-specific optimal transport formulas, develops two computational schemes (general $p\ge 1$ and a fast $p=1$ method), and proves consistency of the estimators. Empirical results on the von Mises, wrapped Cauchy, and sine-skewed von Mises distributions show the Wasserstein projection estimators achieve accuracy comparable to the MLE, with the $L^1$ variant offering enhanced robustness to noise. The methods provide a practical, robust alternative for circular data analysis and extend Wasserstein projection ideas from the real line to circular geometry.

Abstract

For statistical models on circles, we investigate performance of estimators defined as the projections of the empirical distribution with respect to the Wasserstein distance. We develop algorithms for computing the Wasserstein projection estimators based on a formula of the Wasserstein distances on circles. Numerical results on the von Mises, wrapped Cauchy, and sine-skewed von Mises distributions show that the accuracy of the Wasserstein projection estimators is comparable to the maximum likelihood estimator. In addition, the $L^1$-Wasserstein projection estimator is found to be robust against noise contamination.

Figures (8)

• Figure 1: Ratio of mean squared error of the $L^1$ and $L^2$ Wasserstein projection estimators to that of the maximum likelihood estimator for the von Mises distribution ($\mu=0.3,\kappa=2$). Left: $\mu$, Right: $\kappa$.
• Figure 2: Ratio of mean squared error of the $L^1$ and $L^2$ Wasserstein projection estimators to that of the maximum likelihood estimator for the von Mises distribution ($\mu=0.3,n=10^5$). Left: $\mu$, Right: $\kappa$.
• Figure 3: Ratio of mean squared error of the $L^1$ and $L^2$ Wasserstein projection estimators to that of the maximum likelihood estimator for the wrapped Cauchy distribution ($\mu=\pi/8,\rho=0.4$). Left: $\mu$, Right: $\rho$.
• Figure 4: Ratio of mean squared error of the $L^1$ and $L^2$ Wasserstein projection estimators to that of the maximum likelihood estimator for the wrapped Cauchy distribution ($\mu=\pi/8,n=10^5$). Left: $\mu$, Right: $\rho$.
• Figure 5: Mean squared error of the $L^1$ Wasserstein projection estimator and maximum likelihood estimator for the sine-skewed von Mises distribution ($\mu=0,\kappa=1,\lambda=0.7$). Upper left: $\mu$, Upper right: $\kappa$, Lower: $\lambda$.

Theorems & Definitions (2)

• Theorem 1
• proof