I consider a weighted sum of $n$ correlated and identically-distributed random variables. The weights in the sum, $w_i$ for $i=1, 2,...,n$, are non-negative and sum to 1. I am investigating solutions that minimise the kurtosis of this sum over the $w_i$.
The kurtosis of the sum is the ratio of the fourth moment to the square of the variance as follows: $K=\frac{u}{v}$ where $u(w)=\sum_{i,j,k,l=1}^{n} w_i w_j w_k w_l \mu_{ijkl}$ and $v(w)=(\sum_{i,j}^{n} w_i w_j \sigma_{ij})^2$, and $\mu_{ijkl}$ is the centralised fourth co-moment and $\sigma_{ij}$ the centralised second co-moment i.e. the covariance.
In addition to $u(w)>=v(w)$, $u$ and $v$ have some special properties worth noting:
- Both are non-negative convex functions that can be written as a sum-of-squares.
- Both are polynomials in the $w_i$ that contain all, and only, monomials of order 4.
- Both are homogeneous functions of order 4, which implies e.g. $\sum_{i=1}^{n} w_i \frac{\delta u}{\delta w_i}=4u$.
$K$ is therefore a homogeneous function of order $0$ and, though $u$ and $v$ are convex, $K$ is neither a convex nor a quasi-convex function. However, based on looking at many numerical examples I believe that in the case of correlations all being non-negative and $<1$, there is only one minimum that must therefore be global. My question is whether I can prove this to be the case (or construct a counterexample) but it seems difficult when the function itself is not convex. I believe that algebraic geometry or intersection theory could be the best route to an answer if someone could kindly help? Below is what I have done so far.
Non-negative correlations imply that all of the $\mu_{ijkl}$ and $\sigma_{ij}$ are non-negative too, so that $u$ and $v$ are convex polynomials in the $w_i$ with no negative coefficients. The first order Kuhn-Tucker conditions for the optimisation yield, after some simplification:
- $w_i>=0$
- $\sum_{i=1}^{n} w_i = 1$
- $w_i \frac{\delta u}{\delta w_i}/u=w_i \frac{\delta v}{\delta w_i}/v \implies v*w_i \frac{\delta u}{\delta w_i}=u*w_i \frac{\delta v}{\delta w_i}$
- $\frac{\delta u}{\delta w_i}/u>=\frac{\delta v}{\delta w_i}/v \implies v*\frac{\delta u}{\delta w_i}>=u*\frac{\delta v}{\delta w_i}$.
Items 2 and 3 represent a system of polynomial equations over domains 1 and 4 and also the intersection of a hyperplane and $n$ related polynomials. To explore potential solutions I am using Mathematica. It struggles to solve 1-4 simultaneously, but easily solves 1-3. Filtering results that satisfy 4 so far produces a unique solution for many hundreds of examples. (For completeness I do have a separate optimiser which is guaranteed to find the global minimum of the original problem).
I wondered if there are any known pieces of work or theorems that can speak to the expected number of solutions to 1-3 or 1-4?
Thanks.
PS I have more colour on what solutions of 1-3 look like and in practice I use a different formulation in Mathemtica for simplicity but I do not believe it affects my question.