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