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Characterizations of the exponential distribution by geometric compound

Iwińska M., Popowska B.

Fasciculi Mathematici

2011

Nr 47

5--10

Under the reliability conditions IFRA/DFRA (NBU/ NWU), the exponential distribution is characterized by stochas- tic ordering properties which link the geometric compound with record values (spacing of record values). The index of record val- ues is random.

On the geometric compounding model with applications

Hu C. Y., Lin G. D.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 1

135--147

Under the geometric compounding model, we first investigate the relationship between the compound geometric distribution and the underlying distribution, including the preservation of the infinite divisibility property. An interesting upper bound for the tail probability of the compound geometric distribution is provided by using only the mean of the underlying distribution. Secondly, we apply the obtained results to understand better the L-class of life distributions. In particular, we strengthen a surprising result of Bhattacharjee and Sengupta [5] and show that there are life distributions FєL with the following properties: (i) the support of F consists of countably infinite points, (ii) (ii) the coefficient of variation of F is equal to one, and (iii) (iii) F is not in the HNBUE class (the harmonic new better than used in expectation class).Finally, we apply geometric compounds to characterize the semi-Mittag-Leffler distribution and extend a known result about the exponential distribution.