standard error for quantile
Hi Ted
-----Original Message----- From: ted at deb [mailto:ted at deb] On Behalf Of Ted Harding Sent: Tuesday, October 30, 2012 6:41 PM To: r-help at r-project.org
<snip>
The general asymptotic result for the pth quantile (0<p<1) X.p of a sample of size n is that it is asymptotically Normally distributed with mean the pth quantile Q.p of the parent distribution and var(X.p) = p*(1-p)/(n*f(Q.p)^2) where f(x) is the probability density function of the parent distribution.
So if I understand correctly p*(1-p) is biggest when p=0.5 and decreases with smaller or bigger p. The var(X.p) then depends on ratio to parent distribution at this p probability. For lognorm distribution and 200 values the resulting var is
(0.5*(1-.5))/(200*qlnorm(.5, log(200), log(2))^2)
[1] 3.125e-08
(0.1*(1-.1))/(200*qlnorm(.1, log(200), log(2))^2)
[1] 6.648497e-08 so 0.1 var is slightly bigger than 0.5 var. For different distributions this can be reversed as Jim pointed out. Did I manage to understand? Thank you very much. Regards Petr
This is not necessarily very helpful for small sample sizes (depending on the parent distribution). However, it is possible to obtain a general result giving an exact confidence interval for Q.p given the entire ordered sample, though there is only a restricted set of confidence levels to which it applies. If you'd like more detail about the above, I could write up derivations and make the write-up available. Hoping this helps, Ted. ------------------------------------------------- E-Mail: (Ted Harding) <Ted.Harding at wlandres.net> Date: 30-Oct-2012 Time: 17:40:55 This message was sent by XFMail -------------------------------------------------