APPROXIMATE DISTRIBUTION OF SAMPLE QUANTILES
The ith order statistic is the ith smallest observation in a random independent sample of size n. It is a natural estimator of the sampled distribution’s ith -quantile which we denote by where It is well known that the sampling distribution of this estimator becomes normal when n increases while p is kept fixed. We review formulas for computing the first few moments of this distribution (up to the kurtosis) which are then used to extend the normal approximation by including the and -proportional corrections. We then propose several transformations capable of substantially improving this approximation by eliminating its skewness term.
asymptotic distribution, Edgeworth expansion, quantiles.