Consider a joint angle $j$, such that $-180^{\circ} \lt j \lt 180^{\circ}$, with the true value unknown.
Given two estimates of $j$, I wish to combine them using a normalised weighted mean, such that the pair of weights sum to one.
The angle space wraps around ${\pm}180^{\circ}$, so the weightings are applied in a way that considers the smaller angular distance between the angle estimates. For example, if one estimate is $170^{\circ}$ and the other is $-160^{\circ}$, then, given an equal weighting ($0.5$ each), the weighted mean would be $-175^{\circ}$ and not $5^{\circ}$.
Based on this scenario, are the following statements correct?
1. The result of the weighted mean cannot be worse than *both* of the estimates. 2. The result of the weighted mean can only be better than the estimates if the true value lies between the estimates. (i.e. between the smaller angular region separating the estimates)
Now, if this was expanded to a set of joint angles, $J = \{j_1, \, j_2, \, \dots, \, j_n\}$, with two sets of estimates (i.e. two estimates per joint, as before) and still only two weightings (i.e. the elements within a set share the same weighting), then can the overall mean error of the weighted joint angle estimates be better than the mean errors of both sets? Can they be worse than that of both sets?
Can you please explain your answer with a simple anecdotal example.