[R] standard error for quantile

[PIKAL Petr]

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
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