# Name of mean- or median-like values?

Consider the data points $$xi∈Rx_i \in \mathbb R$$ for $$i=1,…,ni=1,\ldots,n$$ as well as following definition:

$$ˆxc:=argminr∈R∑i|r−xi|c\hat x_c := \underset{{r\in \mathbb R}}{\operatorname{argmin}}\sum_i \vert r - x_i \vert^c$$

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

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

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

You can observe that for $$c→∞c\to \infty$$ the value of $$ˆxc\hat x_c$$ will be more and more influenced by the outliers and converge to the mid-range $$min\frac{\min x_i + \max x_i }{2}$$. (Which intuitively makes sense when you compare it to the behaviour of $$pp$$-norms.) You can also generalize this simple definition to multiple dimensions ($$x_i \in \mathbb R^dx_i \in \mathbb R^d$$) by replacing the absolute value $$\vert \cdot \vert\vert \cdot \vert$$ by a suitable norm $$\Vert \cdot \Vert\Vert \cdot \Vert$$.

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

A Fréchet mean has the form

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

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

$$d(x, y) = | x – y |^c d(x, y) = | x - y |^c$$

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