2

I find it fascinating that the mean and median both minimizing the a measure of error of a point estimate. The median $m_1$ is any (non-unique) $m \in \mathbb R$ which minimizes the $L_1$ norm $\int |f - m|$. Likewise, the mean $m_2$ minimizes the $L_2$ norm $(\int |f - m|^2)^\frac12$.

Are there analogous concepts for general $L_p$ norms? The main reason I ask is that I find a lot of statistics, at least in typical practice, make a lot of arbitrary choices about distances and errors. I interpret the above as saying the choice of central location is determined what norm best quantifies errors, and this in turn determines how to measure other statistical notions. For example, the standard deviation is $L_2$ norm of the best $L_2$ estimate, the mean. (For $L_1$, I suppose this is the median error of the median?)

user71216
  • 21
  • 1
  • Good question. Take a look at https://en.wikipedia.org/wiki/M-estimator. – whuber Sep 13 '18 at 15:13
  • Thanks... That's a good starting point. Do you have any recommendations for detailed references, especially for L_p spaces? – user71216 Sep 15 '18 at 16:14

0 Answers0