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Current $$k$$-records and their use in distribution-free confidence intervals. (English) Zbl 1152.62026
Summary: In a sequence of independent and identically distributed (iid) random variables, the $$k$$ th 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 observations in the previous partial sample. We consider the $$k$$-record statistics at the time when the $$n$$-th $$k$$-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.

##### MSC:
 62G32 Statistics of extreme values; tail inference 62G15 Nonparametric tolerance and confidence regions 62G30 Order statistics; empirical distribution functions
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##### References:
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