CDF of Sample Quantile
On 14/02/2011 9:58 AM, Bentley Coffey wrote:
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?
I think the answer to your question is no, but I think it's the wrong question. Suppose Xm:n is the mth sample quantile in a sample of size n, and you want to calculate P(Xm:n > x). Let X be a draw from the underlying distribution, and suppose P(X > x) = p. Then the event Xm:n > x is the same as the event that out of n independent draws of X, at least n-m+1 are bigger than x: a binomial probability. And R can calculate binomial probabilities using pbinom(). Duncan Murdoch