Current k-records and their use in distribution-free confidence intervals

In a sequence of independent and identically distributed (iid) random variables, the kth largest (smallest) observation in a partial sample is well-known as the upper (lower) k-record value, when its value is greater (smaller) than the corresponding observation in the previous partial sample. In this paper, we consider the k-record statistics at the time when the nth k-record of any kind (either an upper or lower) is observed, termed as current k-records. We derive a general expression for the joint probability density function (pdf) of these current k-records and use it to construct distribution-free confidence intervals for population quantiles. It is shown that the expected width of these confidence intervals is decreasing in k and increasing in n. We also discuss the construction of tolerance intervals and limits in terms of current k-records. Finally, a numerical example is presented to illustrate all the methods of inference developed here.