SOME CONVEXITY PROPERTIES OF THE DISTRIBUTION OF LOWER k-RECORD VALUES WITH EXTENSIONS

Abstract

Unimodality and strong unimodality of the distribution of ascendingly ordered random variables have been extensively studied in the literature, whereas these properties have not received much attention in the case of descendingly ordered random variates. In this paper, we show that log concavity of the reversed hazard rate implies that of the density function. Using this fundamental result, we establish some convexity properties of such random variables. To do this, we first provide a counterexample showing that a claim of Basak & Basak [7] about the lower record values is not valid. Then, we provide conditions under which unimodality properties of the distribution of lower k-record values would hold. Finally, some extensions to dual generalized order statistics in both univariate and multivariate cases are discussed.

Convolutions and generalization of logconcavity: Implications and applications

“…Using different methods of proof, Alimohammadi and Alamatsaz and Alimohammadi et al showed that the functions g r , , in (10) are logconcave for . In the following lemma, we extend this result.…”

“…Among the logconcavity results of Chen et al , Alimohammadi et al and those mentioned above, Theorem above is the most complete version. In fact, Theorem can also be used to examine logconcavity (SU) properties of the marginal pdf by letting instead of .…”

“…So, parts e, f, and g follow. Parts i and j were proved by Pellerey et al [49] and parts k and m were proved by Alimohammadi et al [11]. For a direct implication that a logconcave function is unimodal, part l, see Marshall and Olkin ([45], p. 99).…”

“…However, these findings can not be true for lifetime random variables because, in this case, we have . (iii) Pellerey et al proved that if the hazard rate h is logconcave, then it is increasing. Under the weaker condition of unimodality of the reversed hazard rate r , Alimohammadi et al proved that r has to be decreasing. (iv) Since the hazard rate can be written as , we have that h is increasing provided that r is increasing. However, from the results of Block et al , it follows that nonnegative random variables can not have IRHR distributions. (v) According to Fig.…”

“…(iii) Pellerey et al [49] proved that if the hazard rate h is logconcave, then it is increasing. Under the weaker condition of unimodality of the reversed hazard rate r, Alimohammadi et al [11] proved that r has to be decreasing. (iv) Since the hazard rate can be written as h = r · F /F , we have that h is increasing provided that r is increasing.…”