# [R] standard error for quantile

(Ted Harding) Ted.Harding at wlandres.net
Tue Oct 30 18:41:02 CET 2012

```On 30-Oct-2012 13:46:17 PIKAL Petr wrote:
> Dear all
>
> I have a question about quantiles standard error, partly practical
> partly theoretical. I know that
>
> x<-rlnorm(100000, log(200), log(2))
> quantile(x, c(.10,.5,.99))
>
> computes quantiles but I would like to know if there is any function to
> find standard error (or any dispersion measure) of these estimated
> values.
>
> And here is a theoretical one. I feel that when I compute median from
> given set of values it will have lower standard error then 0.1 quantile
> computed from the same set of values.
>
> Is it true? If yes can you point me to some reasoning?
>
> Regards
> Petr
> ["PS" deleted]

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.

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

```