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Consider a set of data: $$X = {x_1, x_2, ... x_n}$$

where $x_i$ is of the form: $$x_i = c + y_i$$ where c is a constant

The arithmetic mean over $X$ is: $$\frac{(\sum_{i=1}^n x_i)}{n}$$ The arithmetic mean is also: $$\frac{(\sum_{i=1}^n c)}{n} + > \frac{(\sum_{i=1}^n y_i)}{n}$$ And is also: $$c + \frac{(\sum_{i=1}^n > y_i)}{n}$$

To me, this only works with addition and multiplication for arithmetic means

My intuition says it works for multiplication with Geometric means, but I haven't gone through it yet. Could it work for a different operations as well?

Is there any operation where this works for Harmonic means or other power means?

Edit: Let me be more clear about what I am asking:

$$x_i = Operation(c, y_i)$$

where c is a constant

The arithmetic mean over $X$ is: $$\frac{(\sum_{i=1}^n x_i)}{n}$$ or $$\frac{(\sum_{i=1}^n Operation(c, y_i))}{n}$$ or $$Operation\left(\frac{(\sum_{i=1}^n c)}{n}, \frac{(\sum_{i=1}^n y_i)}{n}\right)$$ or $$Operation\left(c, \frac{(\sum_{i=1}^n y_i)}{n}\right)$$

For some definitions of $Operation(c,v)$

Generally, lets state this for arbitrary means: $$ Mean(X) = Mean(Operation(c, Y); \text{ }where\text{ } x_i = Operation(c, y_i)$$

$$ Mean(X) = Operation(c, Mean(Y))$$

Arithmetic mean appears to work for: $$Operation(c,v): c + v$$ $$Operation(c,v): c \times v$$

Geometric mean should work for: $$Operation(c,v): c \times v$$

Harmonic mean works for: $$Operation(c,v): c \times v$$ $$No\text{ }Clue...$$

Lets say $Operation(c, v)$ can be something like addition, multiplication, exponentiation, logarithm, and other "simple" things.

2 main questions here:

  1. I doubt my list of examples is conclusive, but are there other simple operators that this holds true for with each of the 3 means I have considered?
  2. What do you call these properties of the means? Would we say arithmetic mean is additively separable and geometric mean is multiplicatively separable? Is there a better way to describe this?
Mediocre_student1217
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  • Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. – Community Mar 16 '23 at 16:36
  • What do you want to work? Do you want to define an operator $O$, called "mean", such that if each $x_i = c + y_i,$ then $O(x_1, \ldots, x_n) = c + O(y_1, \ldots, y_n)$? – William M. Mar 16 '23 at 17:00
  • I have edited the question substantially to be more clear about what I am looking for. I'm wondering what operators work for arithmetic mean, what works for geometric mean, and what works for harmonic mean – Mediocre_student1217 Mar 16 '23 at 20:49

1 Answers1

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What you want is basically that arithmetic mean estimator is linear and geometric mean estimator is hmogeneous of degree 1. Home it helps !

NancyBoy
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  • Thanks for pointing me to the word homogeneous, that helped me narrow down my search a bit. From this site, I see that in fact arithmetic, geometric, and harmonic means are all homogeneous with degree 1. I anticipate this may extend to all variants of power means Homogeneity of degree 1 allows us to factor out a scalar multiple. Linear functions allow for separation of addition, is arithmetic mean the only mean with that property? What other operators can be used like this? – Mediocre_student1217 Mar 16 '23 at 20:55
  • Among the three mean you mentionned, only the arithmetic one is linear. As long as you have a power involved, it breaks the linear property. – NancyBoy Mar 16 '23 at 21:56
  • what about operations like exponents? where xi = yi^c. This isn't relying on linear but I think it should work for the geometric mean and other higher order power means. – Mediocre_student1217 Mar 17 '23 at 04:05
  • You shod try to prove it, but it won't work for geometric means... – NancyBoy Mar 17 '23 at 07:55