Apakah median jenis rata-rata, untuk beberapa generalisasi "rata-rata"?

Konsep "berarti" menjelajah jauh lebih luas daripada rata-rata aritmatika tradisional; Apakah itu membentang sejauh termasuk median? Dengan analogi,

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

Analogi yang saya gambar adalah dengan mean kuasi-aritmatika , yang diberikan oleh:

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

Sebagai perbandingan, ketika kita mengatakan bahwa median dataset lima item sama dengan item ketiga, kita dapat melihat bahwa setara dengan pemeringkatan data dari satu menjadi lima (yang mungkin kita tunjukkan dengan fungsi $f$ ); mengambil rata-rata dari data yang diubah (yaitu tiga); dan membaca kembali nilai item data yang memiliki peringkat tiga (semacam$f^{-1}$ ).

Dalam contoh rata-rata geometrik, rata-rata harmonik dan RMS, $f$ adalah fungsi tetap yang dapat diterapkan pada sejumlah angka secara terpisah. Sebaliknya, baik untuk menetapkan peringkat, atau untuk bekerja kembali dari peringkat ke data asli (interpolasi jika diperlukan) membutuhkan pengetahuan dari seluruh kumpulan data. Terlebih lagi dalam definisi yang saya baca dari mean kuasi-aritmatika, $f$ diperlukan untuk kontinu. Apakah median pernah dianggap sebagai kasus khusus dari kuasi-aritmatika berarti, dan jika demikian, bagaimana $f$ didefinisikan? Atau apakah median yang pernah digambarkan sebagai contoh dari beberapa pengertian "rata-rata" yang lebih luas? Rata-rata kuasi-aritmatika tentu bukan satu-satunya generalisasi yang tersedia.

Bagian dari masalah ini adalah terminologis (apa sih artinya "berarti", terutama berbeda dengan "kecenderungan sentral" atau "rata-rata"?). Misalnya, dalam literatur untuk sistem kontrol fuzzy , fungsi agregasi adalah fungsi yang meningkat dengan F ( a , a ) = a dan F ( b , b mnt ( x , y $F:[a,b] \times [a,b] \to [a,b]$$F(a,a)=a$ ; fungsi agregasi yang$F(b,b)=b$ untuk semua x , y ∈ [ a , b ] disebut "rata-rata" (dalam arti umum). Definisi seperti itu, tentu saja, sangat luas! Dan dalam konteks ini median memang disebut sebagai jenis rata-rata. [ 1 ] Tetapi saya ingin tahu apakah karakterisasi yang kurang luas dari rerata masih dapat cukup jauh untuk mencakup median - yang disebutrerata umum$\min(x,y) \leq F(x,y) \leq \max(x,y)$$x,y \in [a,b]$$^{[1]}$(yang mungkin lebih baik digambarkan sebagai "kekuatan rata-rata") dan rata - rata Lehmer tidak, tetapi yang lain mungkin. Untuk apa nilainya, Wikipedia memasukkan "median" dalam daftar "cara lain" , tetapi tanpa komentar atau kutipan lebih lanjut.

: Definisi rata-rata yang luas, sesuai untuk lebih dari dua input, tampaknya standar di bidang kontrol fuzzy dan muncul berkali-kali selama pencarian di internet untuk contoh median yang digambarkan sebagai median; Saya akan mengutip misalnya Fodor, JC, & Rudas, IJ (2009), "Pada Beberapa Kelas Fungsi Agregasi yang Migratif",IFSA / EUSFLAT Conf. (hal. 653-656). Kebetulan, makalah ini mencatat bahwa salah satu pengguna awal dari istilah "berarti" (moyenne) adalahCauchy, di Cours d'analyse de l'École, politeknik royale, 1ère partie; Analisis algébrique (1821). Kemudian kontribusi dariAczél,Chisini,$[1]$ Kolmogorov dan de Finetti dalam mengembangkan konsep "rata-rata" yang lebih umum daripada yang diakui Cauchy di Fodor, J., dan Roubens , M. (1995), " Tentang kebermaknaan cara ", Jurnal Matematika Komputasi dan Terapan , 64 (1) , 103-115.

Inilah salah satu cara yang Anda anggap sebagai median sebagai "jenis rata-rata umum" - pertama, tentukan dengan hati-hati rata-rata aritmatika biasa dalam hal statistik pesanan:

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

Kemudian dengan mengganti rata-rata statistik pesanan biasa dengan beberapa fungsi bobot lainnya, kami mendapatkan gagasan tentang "rata-rata umum" yang bertanggung jawab atas pesanan.

Dalam hal itu, sejumlah langkah potensial dari pusat menjadi "sarana yang digeneralisasi". Dalam kasus median, untuk ganjil , w ( n + 1 ) / 2 = 1 dan yang lainnya adalah 0, dan untuk genap n , w $n$$w_{(n+1)/2}=1$$n$ .$w_{\frac{n}{2}}=w_{\frac{n}{2}+1}=\frac{1}{2}$

Demikian pula, jika kita melihat estimasi-M , estimasi lokasi mungkin juga dianggap sebagai generalisasi dari rata-rata aritmatika (di mana untuk mean, kuadratik, ψ linier, atau fungsi-beratnya datar), dan median termasuk dalam kelas generalisasi ini. Ini adalah generalisasi yang agak berbeda dari yang sebelumnya.$\rho$$\psi$

Ada berbagai cara lain yang dapat kita gunakan untuk memperluas pengertian 'rata-rata' yang dapat mencakup median.

Jika Anda menganggap mean sebagai titik yang meminimalkan fungsi kerugian kuadratik SSE, maka median adalah titik yang meminimalkan fungsi kerugian linear MAD, dan mode adalah titik yang meminimalkan beberapa fungsi kerugian 0-1. Tidak diperlukan transformasi.

Jadi median adalah contoh dari rata-rata Fréchet .

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.