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Define a function $f(X; p)$, which measures a statistic on a set of points $X = {x_1, ..., x_n}$ (and $x_i \in (0, 1)$), with respect to a parameter $p$. I am looking for ways to define the function $f(.)$, such that on one extreme it becomes arithmetic mean, and on the other extreme it becomes a geometric mean:

  • if $p=0$ then $ f(X; p=0) = \frac{1}{n}\sum_i x_i $
  • if $p=1$ then $ f(X; p=0) = (\prod_i x_i)^{1/n} $

Note: I am not looking for an "interpolation"; but rather a function that could be deformed into either "means" (similar to how an $L_p$ norm can be multiple different operations, depending on the value of $p$).

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$$f(X;p) = M_{1-p}(x_1, \ldots, x_n)=\left( \frac1n \sum_{i=1}^n x_i^{1-p}\right)^\frac1{1-p}$$

When $p=0$, we clearly have the AM.

Also, $M_p$ also satisfies the condition that $\lim\limits_{p \to 1}M_{1-p}(x_1, \ldots, x_n)=\lim\limits_{p \to 0}M_p(x_1, \ldots, x_n)$ to be equal to the geometric mean.

This is known as the generalized mean.

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    $\begingroup$ Probably it will be better to use the standard convention for $M_p$ to prevent confusion, and define $f(X;p)$ in terms of $M_p$ ? $\endgroup$ – lisyarus Feb 25 at 14:50
  • $\begingroup$ Good idea, thanks. $\endgroup$ – Siong Thye Goh Feb 25 at 14:52

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