Advances in exceedance statistics based on ordered random variables

It is clear that the random variable νm shows the number of the elements of the sample Y1, Y2, ..., Ym falling into random interval (f1(X1, X2, ..., Xn), f2(X1, X2, ..., Xn). We call νm the exceedance statistics in independent and identically distributed (iid) sequences of observations. We are interested in distributional properties of νm. The particular cases for the lower and upper thresholds being the r and s order statistics, Xr:n and Xs:n from the sample X1, X2, ..., Xn were investigated in connection with the theory of tolerance limits and invariant confidence intervals for future observations. See for instance Wilks(1941) , Robbins (1944), Gumbel and Shelling (1950), Epstein (1954), Sarkadi (1957), Siddigui (1970), David (1981), Bairamov and Petunin (1991).