Suppose Y is a mixture distribution defined as follows:
If are beta distributed random variables (making Y a mixture of three beta distributions); and
are the 95th percentiles of the densities of Y and X1 respectively (similarly for X2 and X3), is it necessarily true that:
And does this hold for any rv's X1, X2, X3 that are defined over a positive range? Intuitively it makes sense that it does, but there may be something I haven't considered.
Thanks.
In general, NO.
Consider the simpler case of two different (independent) beta variables -- say a beta(1,2) and a beta(2,1). In that particular case a 50-50 mixture is standard uniform, which has 0.95 quantile equal to 0.95.
The weighted average of the two quantiles of the components is smaller than that (it's about 0.8755):
> qbeta(.95,1,2)
[1] 0.7763932
> qbeta(.95,2,1)
[1] 0.9746794
(This was the first example I tried, but knowing that quantiles are not linear in general, I expected almost any random example should fail unless it was a special case.)
[whuber's example in comments is an especially simple and compelling one -- but I wanted to show that it was still clearly the case for beta random variables - which overlap completely, since they're all on the unit interval]
