Estimation of a multivariate von Mises distribution for contaminated torus data

Abstract

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.

Figures (29)

• Figure 1: Sample of size $n=250$ from a bivariate von Mises distribution with $\boldsymbol{\kappa}=(10,20)$ and $\lambda=5$. Fitted density contours obtained using approximate maximum likelihood estimates superimposed.
• Figure 2: Genuine data. Sample of size $n=250$ from a bivariate von Mises distribution with $\boldsymbol{\kappa}=(10,20)$ and $\lambda=5$. Data are displayed as point on $\mathbb{T}^2$ from different perspectives.
• Figure 3: Contaminated data. Two sets of 50 outliers (blue points) are added to the $n = 250$ genuine data points (see Fig. \ref{['fig: torus-genuine']}). The top row shows the case where the outliers are clustered, while the bottom row shows the case where they are scattered. The data are displayed as points on $\mathbb{T}^2$ from different viewing angles.
• Figure 4: Fitted density contours for the two contaminated datasets. Contours based on approximate maximum likelihood estimates are superimposed on the data. Left: the outliers are clustered. Right: outliers are scattered.
• Figure 5: Fitted marginal densities. Sample from the bivariate von Mises distribution with clustered outliers. Marginal densities for the component $\theta_2$ are shown for different bandwidths. The true model is indicated by the red line.