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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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