Let the $k$-power mean of two numbers $x$ and $y$ be defined as $M^k(x,y) = \left(\frac{x^k+y^k}{2}\right)^{1/k}$.

  • For the case $k=1$, we have that if $X,Y$ are independently normally distributed, then so is $M^1(X,Y)$.
  • For the case $k\rightarrow \infty$, we have that if $X,Y$ are independently Frechet (Gumbel/Extreme Value) distributed, then so is $M^{\infty}(X,Y)$.
  • For the case $k\rightarrow - \infty$, we have that if $X,Y$ are independently negative Frechet distributed, then so is $M^{-\infty}(X,Y)$.

What do we know about all the intermediate cases?

My own ultimate purpose is that I would like for analytical convenience, for a given $k$, a one-parameter family of distributions that is closed under the operation "take the $k$-power mean of two iid variables".

I suspect, since I have not heard of such families of distributions, that they either do not exist, or that they do not yield much tractability in practice.

More explicitly (perhaps this will clarify what I am thinking about): Let $X_1,X_2\sim F$, where $F$ has support on $[0,\infty)$. What is the distribution of $\left( \frac{X_1^k+X_2^k}{2}\right)^{1/k}$?

Denote the distribution by $G$. $$G(x) = P\left[ (\frac{X_1^k+X_2^k}{2})^{1/k} \le x\right] = P\left[ X_1^k \le 2x^k-X_2^k\right] = \int P(X_1^k < 2x^k-x_2^k|x_2) dF(x_2) = \int F((2x^k-x_2^k)^{1/k}f(x_2)dx_2.$$ Taking the derivative, we get $$g(x) = x^{k-1}\int (2x^k-x_2^k)^{1/k-1}f\left((2x^k-x_2^k)^{1/k}\right)f(x_2)dx_2.$$

Now my question can be phrased: Is there an (ideally analytically convenient) family of density functions such that the function (that operates on densities): $$ f \mapsto \left\{x \mapsto x^{k-1}\int (2x^k-y^k)^{1/k-1}f\left((2x^k-y^k)^{1/k}\right)f(y)dy\right\}$$ maps the family to itself?

In the particular case when $k=1$, the function (again, operating on densities) reduces to convolution. Gamma distributions are mapped to gamma distributions and normal distributions are mapped to normal distributions. When (say) $k=2$, I do not know which distributions are preserved under the mapping. One example of a (ideally familiar) distributional family preserved under the mapping would be great.

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    $\begingroup$ Could you be more specific? This question seems wide open and could have many conceivable answers, depending on how it is to be interpreted. Exactly which "intermediate" case do you have in mind and what property are you inquiring about? $\endgroup$ – whuber Apr 10 '17 at 15:05
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    $\begingroup$ What is the CLT tag doing here? The number of random variables is fixed at $2$. We nay have asymptotics as regards the value of the power, but this does not relate to CLT. $\endgroup$ – Alecos Papadopoulos Apr 10 '17 at 16:24
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    $\begingroup$ After taking the $k^{th}$ powers of all distributions, your modified question is equivalent to "what one-parameter distribution families are closed under averaging"? In other words, apart from determining whether you can take a $k^{th}$ power in an invertible way (which will problematic for certain values of $k$ when the distribution has a positive probability of negative values), you may assume $k=1$. $\endgroup$ – whuber Apr 10 '17 at 16:43
  • $\begingroup$ Agreed wrt the CLT tag. I had in mind that the normal distribution is a fixed point of $X\mapsto \frac{X_1+X_2}{\sqrt{2}}$, but indeed that is not the CLT. $\endgroup$ – Har Apr 10 '17 at 17:02
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    $\begingroup$ The analysis in your question does not match the question itself, which explicitly asks about "one-parameter" families. Take, for instance, the statement "Gamma distributions are mapped to gamma distributions." Although true, that supposes the family has more than one parameter: you need both a shape and scale parameter for this statement to be true. What are you really trying to ask? $\endgroup$ – whuber Apr 10 '17 at 18:09

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