Generalised Means

There are many notions of average in mathematics and statistics. Well-known are the mean, median and mode.

Also well-known, amongst the means, are the arithmetic mean (A), geometric mean (G), harmonic mean (H) and quadratic mean or  root mean squared (Q or RMS). Recently, I have become interested in the notion of a generalised mean (or generalized mean) over positive (non-negative and non-zero) real numbers \in \mathbb{R}^+.

Common Means

Consider a finite collection X=\{x_i | 0 \le i < n\} of values. As a special case, also consider functions taking a pair of values \{u,v\}.

A(X) = \dfrac{\sum x}{n}  i.e.  \dfrac{\sum_{x \in X} x}{n}

G(X) = \sqrt[n]{\prod x}

H(X) = \dfrac{n}{\sum \dfrac{1}{x}}

Q(X) = \sqrt{\dfrac{\sum x^2}{n}}


A(u,v) = \dfrac{u+v}{2}

G(u,v) = \sqrt{u v}

H(u,v) = \dfrac{2}{\dfrac{1}{u} + \dfrac{1}{v}} = \dfrac{2 u v}{u + v}

We might write some of these differently as:

n A(X) = \sum x

\dfrac{n}{H(X)} = \sum \dfrac{1}{x}

2 A(u,v) = u+v

\dfrac{2}{H(u,v)} = \dfrac{1}{u} + \dfrac{1}{v}

These functions satisfy

\min(X) \le H(X) \le G(X) \le A(X) \le Q(X) \le \max(X);

strictly so if X contains at least two distinct values.

(It also so happens, in the special case of only a pair of values, that A(u,v) \times H(u,v) = G^2(u,v) = (G(u,v))^2.)

Power Means

There is a generalised mean called a power mean, defined thus:

P_k(X) = \sqrt[k]{\dfrac{x^k}{n}}

Then, abusing this notation a little

P_{+\infty} = \lim_{k \rightarrow + \infty} P_k = \max

P_2 = Q

P_1 = A

P_0 = \lim_{k \rightarrow 0} P_k = G

P_{-1} = H

P_{-\infty} = \lim_{k \rightarrow - \infty} P_k = \min

Then: h \le k \Rightarrow P_h(X) \le P_k(X).

Lehmer Means

However, I also noticed that if we write

S_k(X) = \sum x^k

S_k(u,v) = u^k + v^k

then we also have:

A(X) = \dfrac{S_1(X)}{S_0(X)}

G(u,v) = \dfrac{S_{\frac12}(u,v)}{S_{-\frac12}(u,v)} — in the special case of a pair

H(X) = \dfrac{S_0(X)}{S_{-1} (X)}

which leads to an alternative generalised mean.

Unfortunately (for me), this idea is not new: it is called a Lehmer mean:

L_k(X) = \dfrac{S_k(X)}{S_{k-1}(X)} = \dfrac{\sum x^k}{\sum x^{k-1}}

Then, again abusing notation a little

L_{+\infty} = \lim_{k \rightarrow + \infty} L_k = \max

L_2 is the so-called contraharmonic mean

L_1 = A

L_{\frac12}(u,v) = G(u,v) — in the special case of a pair

L_0 = \lim_{k \rightarrow 0} L_k = H

L_{-\infty} = \lim_{k \rightarrow - \infty} L_k = \min

Similarly we have: h \le k \Rightarrow L_h(X) \le L_k(X).

(Many of these means also have weighted variants. It may also be possible to extend these definitions to cover infinite sets X [with suitable conditions], or even to continuous distributions.)


(or axiomatizations)

There are many attempts to axiomatise (or axiomatize) the notion of a mean. Examples of axioms for a mean M of two values are:

identity/idempotent: M(w,w) = w

symmetrical: M(u,v) = M(v,u)

linear: M(c u,c v) = c M(v,u)

left-monotonic: u \le v \Rightarrow M(u,w) \le M(v,w)

right-monotonic: u \le v \Rightarrow M(w,u) \le M(w,v)

bounded: \min\{u,v\} \le M(u,v) \le \max\{u,v\}

non-negative: 0 \le M(u,v)

positive: 0 < M(u,v)

left-continuous; right-continuous; …

These are not independent, and different subsets of these might be considered. It is perfectly possible that some reasonable axioms might not even be consistent with all of the above.

(Other algebraic properties might not be appropriate for a mean function [such as transitivity].)


If you know of other interesting generalisations of means or other notions of average, then please leave a comment.


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