ORDER STATISTICS WITH DISCRETE DISTRIBUTIONS AND PROBABILITY THAT TIES OCCUR SOMEWHERE

Let X_<1:n> <___- X_<2:n> <___- … <___- X_<n:n> denote the order statistics of a random sample of size n with an arbitrary distribution. First, we obtain general expressions of the joint distribution of k order statistics represented by X_<i_1:n>, X_<i_2:n>, … X_<i_k:n>, where 1 <___- i_1 <___- i_2 <___- … <___-i_k <___- n. From these expressions, the joint distribution of k order statistics with a discrete distribution can be obtained exactly by means of computer algebra. We also obtain the probability that ties occur somewhere when the distribution is discrete. Finally with attention to bootstrap sample having a finite population, we obtain some examples for a bootstrap sample by using our results.