# Name of mean- or median-like values?

Consider the data points $$x_i \in \mathbb R$$ for $$i=1,\ldots,n$$ as well as following definition:

$$\hat x_c := \underset{{r\in \mathbb R}}{\operatorname{argmin}}\sum_i \vert r - x_i \vert^c$$

This definition includes the median for $$c=1$$ 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 $$\hat x_c$$ for various values of $$c$$. The red line shows the values of $$\hat x_c$$ (horizontal axis) depending on the values of $$c$$ (red vertical axis).

You can observe that for $$c\to \infty$$ the value of $$\hat x_c$$ will be more and more influenced by the outliers and converge to the mid-range $$\frac{\min x_i + \max x_i }{2}$$. (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 $$\Vert \cdot \Vert$$.

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 $$d$$.

The term "$$L_p$$-estimator" (i.e., L1, L2,..., where the $$p$$ in the literature is your $$c$$) is sometimes used to refer to such estimators. https://iopscience.iop.org/article/10.1088/0026-1394/43/3/004

However, I have hardly seen this term used for estimators of 1-dimensional location, more in regression (of which 1-d location is a special case).

More generally, M-estimators are defined by $$\sum_{i=1}^n \rho(x_i,\theta)=\min!$$.

A rather old paper (1917) Algebraic means by Fujisawa refers to them as power means although you have to take the $$c$$th root to get that. His article also discusses algebraic means which are related.

• I think this paper defines the power means as $P_k := \sqrt[k]{\frac{1}{n} \sum_i x_i^k}$, but I have the impression that these do not coincide with $\hat x_c$. (I don't even think we can find a "nice" expression $\hat x_c$ for $c>4$. Is this correct? Jan 13, 2019 at 17:33
• Algebraic means are means in between the arithmetic mean and the geometric mean. They are thus unrelated to the means in the question. Jan 14, 2019 at 19:34