A Study of MLE and MAPE for the Isotropic Geodesic Normal Distribution under Varying Parameter Settings

Sammanfattning

Analyzing directional data on a sphere requires intrinsic statistical frameworks to respect the underlying geometry. While the maximum likelihood estimator (MLE) is commonly used, its practical reliability degrades in low-concentration settings. This thesis investigates the breakdown thresholds of the MLE and evaluates the stabilizing effects of a derived maximum a posteriori (MAP) estimator for the isotropic geodesic normal distribution (GND) on the 2-sphere. A Monte Carlo simulation study is conducted across varying sample sizes and concentration levels, comparing the MLE against MAP estimators with weakly, strongly, and moderately informative priors. The results demonstrate that the MLE loses estimation reliability at low concentrations, particularly for small sample sizes. In contrast, the MAP estimator effectively mitigates this breakdown; incorporating even a weak or offset prior significantly reduces the Mean Squared Error (MSE). Asymptotic consistency is confirmed for large samples, where the influence of the prior vanishes and the estimators converge. Finally, an application to the orbital planes of the nine planets illustrates the estimators' behavior on real-world data, confirming that the MAP estimator appropriately relies on the data for estimation when the data are highly concentrated.