4
$\begingroup$

Suppose Y is a mixture distribution defined as follows:

1

If 2 are beta distributed random variables (making Y a mixture of three beta distributions); and 3 are the 95th percentiles of the densities of Y and X1 respectively (similarly for X2 and X3), is it necessarily true that:

4

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.

$\endgroup$
2
  • 2
    $\begingroup$ Consideration of cases where there are just two variables and one of them is certainly less than the other should quickly and clearly indicate why such a formula cannot possibly be correct. $\endgroup$
    – whuber
    Aug 8, 2017 at 17:05
  • $\begingroup$ Yes, you are of course correct (sorry, it's 3AM here) $\endgroup$
    – Anna Efron
    Aug 8, 2017 at 17:21

1 Answer 1

3
$\begingroup$

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]

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.