A note on reversed hazard rate of order statistics and record values

On the reversed hazard rate of sequential order statistics

“…The DHR assumption on F in Corollary 11 cannot be replaced by the DRHR property. This follows from Counterexample 3.1 in Kundu et al (2009).…”

“…Proof: If γ 1 = γ 2 holds, then the result follows from Theorem 3.2 in Kundu et al (2009). Let 2γ 1 ≥ γ 2 and γ 1 = γ 2 .…”

“…In Kundu et al (2009) and Wang and Zhao (2010) the transmission of the DRHR property is studied for k-records and m-generalized order statistics, in particular usual order statistics. Theorem 14 extends Theorem 3.1 in Kundu et al (2009) in the case n = 2, k ∈ N, and Theorem 2.3(i) in Wang and Zhao (2010) in the case r = 2 to the model of SOS.…”

“…We will present some results on the reversed hazard rate ordering and its associated ageing notion, the decreasing reversed hazard rate (DRHR) property (see, e.g., Block et al (1998); Sengupta and Nanda (1999); Chandra and Roy (2001); Nanda and Shaked (2001); Finkelstein (2002); Nanda et al (2003); Ahmad and Kayid (2005); Marshall and Olkin (2007); Shaked and Shanthikumar (2007)). Recent results on the DRHR property of some ordered random variables are given in Kundu et al (2009) and Wang and Zhao (2010).…”

Further Results on Quantile Entropy in the Past Lifetime

“…It should be noted that as t → ∞, H t (X) become the well-known Shannon entropy (Shannon [35]) given by H(X) = − ∞ 0 f (x) log f (x) dx. Readers can refer to Di Crescenzo and Longobardi [12,13], Nanda and Paul [29,30], Kundu, Nanda, and Hu [23] and Kundu, Nanda, and Maiti [24] for more works on the past entropy. By means of r, we can rewrite (1.1) as…”