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A note on reversed hazard rate of order statistics and record values. (English) Zbl 1153.62040
Summary: The reversed hazard rate is an important measure to study the lifetime random variable in reliability theory, survival analysis and stochastic modeling. We study the decreasing reversed hazard rate (DRHR) property of order statistics and record values. Some properties of order statistics related to the increasing uncertainty in the past life (IUPL) class have also been studied. We show that if \(X_{k:n}\) is DRHR (IUPL), so are \(X_{k-1:n},X_{k:n+1}\), and \(X_{k-1:n-1}\) where \(X_{k:n}\) denotes the \(k\)-th order statistic of a random sample of size \(n\). It is shown that if the \(n\)-th upper \(k\)-record \(R_n^{(k)}\) is the DRHR then so is \(R_{n-1}^{(k)}\). Further, we show that the DRHR property passes from the \(n\)-th upper record \(R_n\) to \(R_n^{(k)}\).

MSC:
62G30 Order statistics; empirical distribution functions
62G32 Statistics of extreme values; tail inference
62N05 Reliability and life testing
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