finite mixture of order statistics

Question

Let $F(u)$ be a n-degree polynomial continuous distribution function in $[0,1]$, with $F(0)=0$, $F(1)=1$, that is $F(u)=\sum_{i=1}^{i=n} a_i u^i$. My question is: is that kind of distributions uniquely represented as a finite mixture of the n order statitics of n i.i.d uniform variables? The answer is yes for $n=2$ since we can always write: $F(u)=\alpha (2u-u^2) + (1-\alpha) u^2 = 2\alpha u + (1-2\alpha) u^2$ with $\alpha \in [0,1]$ and this representation generates all the possible cuadratic distribution functions in $[0,1]$. Note that $(2u-u^2)$ and $u^2$ are the distributions of the mininum and the maximum of $(u_1,u_2)$ respectively, where $u_1, u_2$ are i.i.d. uniform random variables.

Answer 1

No. Take $n=3$. The CDF's of the order statistics are $F_1(u) = u^3 - 3 u^2 + 3 u$, $F_2(u) = 3 u^2 - 2 u^3$, $F_3(u) = u^3$. The function $F(u) = 4 (u - 1/2)^3 + 1/2$ is a CDF, and its unique representation in terms of the basis functions $F_1, F_2, F_3$ is $F = F_1 - F_2 + F_3$.