Multivariate harmonic new better than used in expectation distributions (Q1115062)

The definition of multivariate harmonic new better than used in expectation (MHNBUE) life distributions is introduced. The definition is a multivariate generalization of the useful univariate aging property, harmonic new better than used in expectation (HNBUE), which is weaker than new better than used in expectation (NBUE). Various closure properties are proved. Unlike the previous extensions we also prove the very important result of closure under convolution. Some examples are given to illustrate the relationship between the proposed MHNBUE and the MHNBUE properties presented by the authors and \textit{B. Klefsjö} [Scand. J. Stat., Theory Appl. 10, 19-25 (1983; Zbl 0513.62055)]. We also study the dual property of multivariate harmonic new worse than used in expectation (MHNWUE).