Further results on closure properties of LPQE order

Further Results on Quantile Entropy in the Past Lifetime

“…It follows from (4.4) that X ≤ hr (≥ hr )X w and hence X ≤ disp (≥ disp )X w . Thus, the desired results are proved by Theorem 2.2 of Kang [21].…”

“…Using H F −1 (u) (X), Sunoj, Sankaran, and Nanda [36] not only introduced the definitions of decreasing (increasing) quantile entropy in the past lifetime [DPQE (IPQE)] class of life distributions and less quantile entropy in the past lifetime (LPQE) order, but also explored some properties of them. Recently, after giving an equivalent definition of the LPQE order, Kang [21] surveyed some closure and reversed closure properties of the LPQE order under several stochastic models. The objective of this paper is to explore further properties of DPQE (IPQE) classes of life distributions and LPQE order.…”

“…Sunoj, Sankaran, and Nanda [36] proved that the LPQE order is closed under the linear transformations. Kang [21] proved that the LPQE order is closed under non-linear transformations and other stochastic models. In this section, we first give sufficient conditions for a function of a random variable to have more (less) quantile entropy in the past lifetime than itself.…”

Further Results on the IDCPE Class of Life Distributions

“…It was introduced by Shannon [2] and Wiener [3], and developed subsequently by Ebrahimi and Pellerey [4], Ebrahimi [5], Ebrahimi and Kirmani [6], Crescenzo and Longobardi [7], Navarro et al [8], etc. Furthermore, some generalizations of H X have been proposed, see, for example, Di Crescenzo and Longobardi [9,10], Nanda and Paul [11][12][13], Abbasnejad et al [14], Kundu et al [15], Kumar and Taneja [16], Khorashadizadeh et al [17], Nanda et al [18], Kayal [19], Vineshkumar [20], Kang [21], Kang and Yan [22], Yan and Kang [23], and others.…”

On the Generalized Decreasing Mean Time to Failure or Replaced Ordering

“…To do these things, some refined stochastic orders were defined in statistics. For more details on stochastic orders, one may refer to the studies by Maria Fernandez-Ponce et al [ 3 ], Belzunce [ 4 ], Müller and Stoyan [ 5 ], Li and Yam [ 6 ], Shaked and Shanthikumar [ 7 , 8 ], Zhao and Balakrishnan [ 9 ], Sunoj et al [ 10 ], Kang [ 11 , 12 ], Yan [ 13 ], Kang and Yan [ 14 ], Vineshkumar [ 15 ], and the references therein. However, sometimes, we need to compare two risk assets by the aid of the third referred system, such as when we compare two risk assets [ 16 ], their values are changeable with the settlement time or with the kind of valuation currency [ 17 ].…”