「平均」の一般化のために、中央値は平均の一種ですか？

「平均」の概念は、従来の算術平均よりもはるかに広く歩き回ります。中央値を含むまで伸びますか？類推により、

$$\text{raw data} \overset{\text{id}}{\longrightarrow} \text{raw data} \overset{\text{mean}}{\longrightarrow} \text{raw mean} \overset{\text{id}^{-1}}{\longrightarrow} \text{arithmetic mean} \\ \text{raw data} \overset{\text{recip}}{\longrightarrow} \text{reciprocals} \overset{\text{mean}}{\longrightarrow} \text{mean reciprocal} \overset{\text{recip}^{-1}}{\longrightarrow} \text{harmonic mean} \\ \text{raw data} \overset{\text{log}}{\longrightarrow} \text{logs} \overset{\text{mean}}{\longrightarrow} \text{mean log} \overset{\text{log}^{-1}}{\longrightarrow} \text{geometric mean} \\ \text{raw data} \overset{\text{square}}{\longrightarrow} \text{squares} \overset{\text{mean}}{\longrightarrow} \text{mean square} \overset{\text{square}^{-1}}{\longrightarrow} \text{root mean square} \\ \text{raw data} \overset{\text{rank}}{\longrightarrow} \text{ranks} \overset{\text{mean}}{\longrightarrow} \text{mean rank} \overset{\text{rank}^{-1}}{\longrightarrow} \text{median}$$

私が描いているアナロジーは、次によって与えられる準算術平均です。

$$M_f(x_1,\dots,x_n)=f^{-1}\left(\frac{1}{n}\sum_{i=1}^{n}f(x_i) \right)$$

比較のために、5項目のデータセットの中央値が3番目の項目に等しいと言うと、データを1から5にランク付けすることと同等であることがわかります（関数示す場合があります$f$）。変換されたデータの平均を取る（3）; ランク3（一種$f^{-1}$）のデータ項目の値を読み返します。

幾何平均、調和平均、RMSの例では、$f$は固定された関数であり、任意の数に単独で適用できます。対照的に、ランクを割り当てるか、ランクから元のデータに戻る（必要に応じて補間する）には、データセット全体の知識が必要です。さらに、私が準算術平均を読んだ定義では、$f$は連続している必要があります。中央値は準算術平均の特別な場合と見なされますか？その場合、はどのように$f$定義されますか？または、中央値はこれまでに「平均」の他のより広い概念のインスタンスとして記述されていますか？使用できる一般化は、準算術平均だけではありません。

問題の一部は用語です（とにかく「平均」とはどういう意味ですか、特に「中央傾向」または「平均」とは対照的ですか？）。たとえば、ファジー制御システムに関する文献では、集約関数は、F （a 、a ）= aおよびF （b 、b ）= b ; 集約関数$F:[a,b] \times [a,b] \to [a,b]$$F(a,a)=a$$F(b,b)=b$のすべてのためのx 、yは∈ [ 、bが】（一般的な意味で） "平均"と呼ばれます。言うまでもなく、このような定義は信じられないほど広いです！そして、この文脈では、中央値は確かに一種の平均と呼ばれます。[ 1 ]しかし、平均のあまり広範ではない特徴が、中央値、いわゆる一般化平均を包含するのに十分に拡張できるかどうか興味があります。$\min(x,y) \leq F(x,y) \leq \max(x,y)$$x,y \in [a,b]$$^{[1]}$（これは「パワー平均」として説明する方が良いかもしれません）、レーマー平均はそうではありませんが、他の人はそうかもしれません。価値のあるものとして、ウィキペディアには「その他の手段」のリストに「中央値」が含まれていますが、さらなるコメントや引用はありません。

：3つ以上の入力に適切に拡張されたこのような平均の広い定義は、ファジィ制御の分野では標準的であり、インターネット検索中に中央値として記述されている中央値のインスタンスで何度も現れます。例えば、Fodor、JC、＆Rudas、IJ（2009）、「移行性のある集約関数のいくつかのクラスについて」、IFSA / EUSFLAT Conf。（pp。653-656）。なお、「平均」という用語の最も初期のユーザーの一人（のはこの紙のノートmoyenneが）だったコーシーで、クールドールはドゥエコールロワイヤルポリテクニック、1èrePARTIEを分析。algébrique（1821）を分析します。後にアチェール、チシーニ、$[1]$コルモゴロフとデ・フィネッティは、コーシーよりも「平均」のより一般的な概念を開発しており、フォーダー、J。、およびルーベンス、M。（1995）、「手段の意味について」、Journal of Computational and Applied Mathematics、64（1） 、103-115。

中央値を「一般的な種類の平均」とみなす方法の1つは、次のとおりです。まず、順序統計の観点から通常の算術平均を慎重に定義します。

$$\bar{x} = \sum_i w_i x_{(i)},\qquad w_i=\frac{_1}{^n}\,.$$

次に、順序統計の通常の平均を他の重み関数で置き換えることにより、順序を説明する「一般化平均」の概念が得られます。

In that case, a host of potential measures of center become "generalized sorts of means". In the case of the median, for odd $n$, $w_{(n+1)/2}=1$ and all others are 0, and for even $n$, $w_{\frac{n}{2}}=w_{\frac{n}{2}+1}=\frac{1}{2}$.

Similarly, if we look at M-estimation, location estimates might also be thought of as a generalization of the arithmetic mean (where for the mean, $\rho$ is quadratic, $\psi$ is linear, or the weight-function is flat), and the median falls also into this class of generalizations. This is a somewhat different generalization than the previous one.

There are a variety of other ways we might extend the notion of 'mean' that could include median.

If you think of the mean as the point minimizing the quadratic loss function SSE, then the median is the point minimizing the linear loss function MAD, and the mode is the point minimizing some 0-1 loss function. No transformations required.

So the median is an example of a Fréchet mean.

One easy but fruitful generalization is to weighted means, $\sum_{i=1}^n w_i x_i / \sum_{i=1}^n w_i,$ where $\sum_{i=1}^n w_i = 1$. Clearly the common or garden mean is the simplest special case with equal weights $w_i = 1/n$.

Letting the weights depend on the order of values in magnitude, from smallest to largest, points to various other special cases, notably the idea of a trimmed mean, which is known by other names too.

To avoid excessive use of notation where it is not needed or especially helpful, imagine for example ignoring the smallest and largest values and taking the (equally weighted) mean of the others. Or imagine ignoring the two smallest and two largest and taking the mean of the others; and so forth. The most vigorous trimming would ignore all but the one or two middle values in order, depending on whether the number of values was odd or even, which is naturally just the familiar median. Nothing in the idea of trimming commits you to ignoring equal numbers in each tail of a sample, but saying more about asymmetric trimming would take us further away from the main idea in this thread.

In short, means (unqualified) and medians are extreme limiting cases of the family of (symmetric) trimmed means. The overall idea is to allow compromises between one ideal of using all the information in the data and another ideal of protecting oneself from extreme data points, which may be unreliable outliers.

See the reference here for one fairly recent review.

The question invites us to characterize the concept of "mean" in a sufficiently broad sense to encompass all the usual means--power means, $L^p$ means, medians, trimmed means--but not so broadly that it becomes almost useless for data analysis. This reply discusses some of the axiomatic properties that any reasonably useful definition of "mean" should have.

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Basic Axioms

A usefully broad definition of "mean" for the purpose of data analysis would be any sequence of well-defined, deterministic functions $f_n:A^n\to A$ for $A\subset\mathbb{R}$ and $n=1, 2, \ldots$ such that

(1) $\newcommand{\x}{\mathrm{x}} \newcommand{\min}{\text{min}}\min (\x)\le f_n(\x)\le \max(\x)$ for all $\x = (x_1, x_2, \ldots, x_n)\in A^n$ (a mean lies between the extremes),

(2) $f_n$ is invariant under permutations of its arguments (means do not care about the order of the data), and

(3) each $f_n$ is nondecreasing in each of its arguments (as the numbers increase, their mean cannot decrease).

We must allow for $A$ to be a proper subset of real numbers (such as all positive numbers) because plenty of means, such as geometric means, are defined only on such subsets.

We might also want to add that

(1') there exists at least some $\x\in A$ for which $\min(\x)\ne f_n(\x)\ne \max(\x)$ (means are not extremes). (We cannot require that this always hold. For instance, the median of $(0,0,\ldots,0,1)$ equals $0$, which is the minimum.)

These properties seem to capture the idea behind a "mean" being some kind of "middle value" of a set of (unordered) data.

Consistency axioms

I am further tempted to stipulate the rather less obvious consistency criterion

(4.a) The range of $f_{n+1}(t, x_1, x_2, \ldots, x_n)$ as $t$ varies throughout the interval $[\min(\x), \max(\x)]$ includes $f_n(\x)$. In other words, it is always possible to leave the mean unchanged by adjoining an appropriate value $t$ to a dataset. In conjunction with (3), it implies that adjoining extreme values to a dataset will pull the mean towards those extremes.

If we wish to apply the concept of mean to a distribution or "infinite population", then one way would be to obtain it in the limit of arbitrarily large random samples. Of course the limit might not always exist (it does not exist for the arithmetic mean when the distribution has no expectation, for instance). Therefore I do not want to impose any additional axioms to guarantee the existence of such limits, but the following seems natural and useful:

(4.b) Whenever $A$ is bounded and $\x_n$ is a sequence of samples from a distribution $F$ supported on $A$, then the limit of $f_n(\x_n)$ almost surely exists. This prevents the mean from forever "bouncing around" within $A$ even as sample sizes get larger and larger.

Along the same lines, we could further narrow the idea of a mean to insist that it become a better estimator of "location" as sample sizes increase:

(4.c) Whenever $A$ is bounded, then the variance of the sampling distribution of $f_n(X^{(n)})$ for a random sample $X^{(n)} = (X_1, X_2, \ldots, X_n)$ of $F$ is nondecreasing in $n$.

Continuity axiom

We might consider asking means to vary "nicely" with the data:

(5) $f_n$ is separately continuous in each argument (a small change in the data values should not induce a sudden jump in their mean).

This requirement might eliminate some strange generalizations, but it does not rule out any well-known mean. It will rule out some aggregation functions.

An invariance axiom

We can conceive of means as applying to either interval or ratio data (in Stevens' well-known sense). We cannot demand they be invariant under shifts of location (the geometric mean is not), but we can require

(6) $f_n(\lambda \x) = \lambda f_n(\x)$ for all $\x \in A^n$ and all $\lambda \gt 0$ for which $\lambda \x \in A^n$. This says only that we are free to compute $f_n$ using any units of measurement we like.

All the means mentioned in the question satisfy this axiom except for some aggregation functions.

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Discussion

General aggregation functions $f_2$, as described in the question, do not necessarily satisfy axioms (1'), (2), (3), (5), or (6). Whether they satisfy any consistency axioms may depend on how they are extended to $n\gt 2$.

The usual sample median enjoys all these axiomatic properties.

We could augment the consistency axioms to include

(4.d) $f_{2n}(\x;\x) = f_n(\x)$ for all $\x \in A^n.$

This implies that when all elements of a dataset are repeated equally often, the mean does not change. This may be too strong, though: the Winsorized mean does not have this property (except asymptotically). The purpose of Winsorizing at the $100\alpha\%$ level is to provide resistance against changes in at least $100\alpha\%$ of the data at either extreme. For instance, the 10% Winsorized mean of $(1,2,3,6)$ is the arithmetic mean of $(2,2,3,3)$, equal to $2.5$, but the 10% Winsorized mean of $(1,1,2,2,3,3,6,6)$ is $3.5$.

I do not know which of the consistency axioms (4.a), (4.b), or (4.c) would be most desirable or useful. They appear to be independent: I don't think any two of them imply the third.

I think the median can be considered a type of a generalization of the arithmetic mean. Specifically, the arithmetic mean and the median (among others) can be unified as special cases of the Chisini mean. If you are going to perform some operation over a set of values, the Chisini mean is a number that you can substitute for all of the original values in the set and still get the same result. For example, if you want to sum your values, replacing all the values with the arithmetic mean will yield the same sum. The idea is that a certain value is representative of the numbers in the set in the context of a certain operation over those numbers. (An interesting implication of this way of thinking is that a given value—the arithmetic mean—can only be considered representative under the assumption that you are doing certain things with those numbers.)

This is less obvious for the median (and I note that the median is not listed as one of the Chisini means on Wolfram or Wikipedia), but if you were to allow operations over ranks, the median could fit within the same idea.

The question is not well defined. If we agree on the common "street" definition of mean as the sum of n numbers divided by n then we have a stake in the ground. Further If we would look at measures of central tendency we could say both Mean and Median are generealization but not of each other. Part of my background is in non parametrics so I like the median and the robustness it provides, invariance to monotonic transformation and more. but each measure has it's place depending on objective.