Comments on: Progressive censoring methodology: an appraisal

Abstract

Professor Balakrishnan provides a very clear and elaborate review of various theoretical results concerning progressively Type-II censored order statistics and their applications to inferential problems. An extensive list of references assists with further reading. Moreover, diverse open problems included in the text stimulate the reader to join the research on progressive censoring. My attention has been drawn specifically to Open Problem 2. In the literature different types of bounds for moments of usual order statistics and their functions are known. A class of such bounds was obtained by an application of the CauchySchwarz inequality. In the case, when this approach does not lead to an attainable bound, an improvement can be achieved through an elegant method of “greatest convex minorant” due to Moriguti (1953), see Balakrishnan (1993). As discussed briefly in Sect. 7.2 of Professor Balakrishnan’s review, Balakrishnan et al. (2001) generalized these bounds to the case of progressively Type-II right censored order statistics. In the same manner, but using the Holder inequality, Raqab (2003) derived upper bounds for moments of progressive Type-II censored order statistics in various scale units generated by central absolute moments. It should be mentioned here that Raqab’s (2003) bounds are special cases of more general results established earlier by Cramer et al. (2002) in the context of generalized order statistics. In his review Professor Balakrishnan poses a question, whether other bounds for moments of usual order statistics or their functions could be generalized to the case of progressive Type-II censoring situation. Yet, I have noticed that even the already known generalizations by Balakrishnan et al. (2001) and Raqab (2003) are not complete in the sense that bounds for