In this paper, a comparison between the life distribution of a new
unit with that of a used unit in the increasing convex order is made
leading to a new class of life distributions which we call “new better
than used in convex ordering of second order”. This class includes as
subclasses the NBU and the NBUC and is a subclass of the NBUCA
class. Preservation properties under convolution, random maxima,
mixing and formation of coherent structures are established.
Stochastic comparisons of the excess lifetime when the inter-arrival
times belong to the NBUC(2) class are developed. Some applications
of Poisson shock models and a test of exponentiality against
NBUC(2) alternative are presented.