Name of mean- or median-like values?

Consider the data points xiR for i=1,,n as well as following definition:

ˆxc:=argminrRi|rxi|c

This definition includes the median for c=1 as well as the mean for c=2.

Is there a name for this general class of mean/median/estimators?

The following image shows an example of some data and ˆxc for various values of c. The red line shows the values of ˆxc (horizontal axis) depending on the values of c (red vertical axis).

You can observe that for c the value of ˆxc will be more and more influenced by the outliers and converge to the mid-range min. (Which intuitively makes sense when you compare it to the behaviour of p-norms.) You can also generalize this simple definition to multiple dimensions (x_i \in \mathbb R^d) by replacing the absolute value \vert \cdot \vert by a suitable norm \Vert \cdot \Vert.

Answer

This is only a partial answer for c \in (0,1]:

A Fréchet mean has the form

\Psi(x) = \operatorname{argmin}_r \sum_i d(r, x_i)

where (M,d) is a metric space and x \in M^k and d is a metric. In our case

d(x, y) = | x – y |^c

is indeed a metric on M = \mathbb R, but only for c \in (0,1]. In this case we can indeed call it a Fréchet mean with the given metric d.

Attribution
Source : Link , Question Author : flawr , Answer Author : flawr

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