Estimation of a multivariate von Mises distribution for contaminated torus data

The occurrence of atypical circular observations on the torus can badly affect parameter estimation of the multivariate von Mises distribution. This paper addresses the problem of robust fitting of the multivariate von Mises model using the weighted likelihood methodology. The key ingredients are non-parametric density estimation for multivariate circular data and the definition of appropriate weighted estimating equations. Some theoretical properties are discussed. The finite sample behavior of the proposed weighted likelihood estimator has been investigated by Monte Carlo numerical studies and empirical applications.

💡 Research Summary

The paper tackles the problem of robust estimation for the multivariate von Mises distribution when data on the torus are contaminated by atypical circular observations (outliers). Classical maximum‑likelihood estimation (MLE) is highly sensitive to such contamination, leading to biased location, concentration, and interaction parameter estimates. To mitigate this, the authors adopt a weighted‑likelihood (WL) framework that combines non‑parametric kernel density estimation for multivariate circular data with Pearson‑residual‑based weighting.

First, a kernel density estimator ˆfₙ(θ) is constructed using a product of one‑dimensional von Mises kernels, each governed by a concentration (bandwidth) parameter k*. The choice of k* controls the smoothness: large k* yields a sharply peaked kernel that reacts strongly to outliers, whereas small k* produces oversmoothing that may mask them. Next, the Pearson residual δ(θ)=ˆfₙ(θ)/m(θ;τ)−1 is computed, where m(θ;τ) is the parametric von Mises density with parameters τ=(µ,κ,Λ). A Residual Adjustment Function (RAF) A(·) – e.g., symmetric χ², power‑divergence, or generalized Kullback–Leibler – is applied to δ to produce a weight w(δ)=min{1,