On Tue, Nov 13, 2012 at 11:59 AM, Rolf Turner <rolf.turner@xtra.co.nz>wrote:

>
> My apologies for returning to this issue after such a considerable
> length of time ... but I wanted to check the result in Cramer's book,
> and only yesterday managed to get myself organised to go the
> library and  check it out.
>
> What bothers me is what happens when f(Q.p) = 0.  The formula
> that you give --- which is exactly the same as that which appears
> in Cramer, page 369, would appear to imply that the variance is
> infinite when f(Q.p) = 0.  This doesn't feel right to me.
>

That analysis is only first order, so really all it says is that n *
variance goes to infinity rather than to a finite constant as when
f(Q.p)>0.  I expect that if you looked at different sample sizes you'd find
that variance eventually decreases slower than 1/n, perhaps n^(-2/3) or
something

For p=0.51, the asymptotics probably aren't going to work until n is large
enough that Q.50 is well out in the tails of the normal distribution
centered around Q.51


> I did a wee experiment with f(x) = 15*x^2*(1-x^2)/4 for -1 <= x <= 1,
> which makes f(Q.50) = f(0) = 0.
>

With this density you can write down the density of the median or other
order statistic and thus write down an integral that gives the exact
variance.  Better still, it's a polynomial, so you could evaluate the
integral exactly.


    -thomas

-- 
Thomas Lumley
Professor of Biostatistics
University of Auckland

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