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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 $\Vert \cdot \Vert$.

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3 Answers 3

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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 $d$.

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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 $\sum_{i=1}^n \rho(x_i,\theta)=\min!$.

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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.

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    $\begingroup$ 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? $\endgroup$
    – flawr
    Jan 13, 2019 at 17:33
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    $\begingroup$ Algebraic means are means in between the arithmetic mean and the geometric mean. They are thus unrelated to the means in the question. $\endgroup$ Jan 14, 2019 at 19:34

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