On some generalizations of mixtures of exponential distributions. (Ukrainian, English) Zbl 1026.60022

Drozdenko, M. O.; Zhegrij, T. I.; Yadrenko, M. I.

Teor. Jmovirn. Mat. Stat. 65, 39-45 (2001); translation in Theory Probab. Math. Stat. 65, 45-52 (2002).

A distribution of the form \(F(t)=\int_0^{\infty}(1-e^{-xt})dH(x)\), where \(H(x)\) is a distribution function, is called a mixture of exponential distributions. This distribution may be considered as distribution with the random intensity \(\eta\) that has the distribution function \(H(x)\). The hazard rate function \(\lambda(t)=\left(\int_0^{\infty}xe^{-xt} dH(x)\right)/ \left(\int_0^{\infty}e^{-xt} dH(x)\right)\) is decreasing in this case. The authors propose a wider class of distributions which contains mixtures of exponential distributions and distributions with increasing and non-monotone hazard rate functions. The proposed distributions have hazard rate functions of the form \(\lambda(t)=\xi+g(t)\) or \(\lambda(t)=\xi g(t)\), where \(\xi\) is a random variable with the distribution function \(H(x)\) and \(g(t)\) is a nonnegative function. Examples of such distributions are presented.

Reviewer: Mikhail Moklyachuk (Kyïv)

MSC:

60E99

Distribution theory

Keywords:

mixture; exponential distribution; hazard rate function