I need to calculate the probability that a sample quantile will exceed a
threshold given the size of the iid sample and the parameters describing the
distribution of each observation (normal, in my case). I can compute the
probability with brute force simulation: simulate a size N sample, apply R's
quantile() function on it, compare it to the threshold, replicate this MANY
times, and count the number of times the sample quantile exceeded the
threshold (dividing by the total number of replications yields the
probability of interest). The problem is that the number of replications
required to get sufficient precision (3 digits say) is so HUGE that this
takes FOREVER. I have to perform this task so much in my script (searching
over the sample size and repeated for several different distribution
parameters) that it takes too many hours to run.

I've searched for pre-existing code to do this in R and haven't found
anything. Perhaps I'm missing something. Is anyone aware of an R function to
compute this probability?

I've tried writing my own code using the fact that R's quantile() function
is a linear combination of 2 order statistics. Basically, I wrote down the
mathematical form of the joint pdf for the 2 order statistics (a function of
the sample size and the distribution parameters) then performed a
pseudo-Monte Carlo integration (i.e. using Halton Draws rather than R's
random draws) over the region where the sample quantile exceeds the
threshold. In theory, this should work and it takes about 1000 times fewer
clock cycles to compute than the Brute Force approach. My problem is that
there is a significant discrepancy between the results using Brute Force and
using this more efficient approach that I have coded up. I believe that the
problem is numerical error but it could be some programming bug; regardless,
I have been unable to locate the source of this problem and have spent over
20 hours trying to identify it this weekend. Please, somebody help!!!

So, again, my question: is there code in R for quickly evaluating the CDF of
a Sample Quantile given the sample size and the parameters governing the
distribution of each iid point in the sample?

Grateful for any help,

Bentley

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