The sequence (Tn) is called a doubly stochastic Poisson process. Special cases of this process are the mixed Poisson process where n(t) = t and the non homogeneous Poisson process where P(Y = yo) = 1.

The following types belong to the most common models for rj'-

2. log-linear type- n(t) = t ■ eatY (a > 0, 7 > 0)

3. logistic type- n(t) = t ■ [1 + ln(1 + atY)] (a > 0, y> -1) .

The frailty variable Y is a nonnegative random variable which can be specified, too-

1. Y is rectangular distributed in [a, b] with 0 < a < b. Then we get from equation (2) for n(t) > 0 by partial integration and with the convention 00 := 1

2. Let Y — y 0 be Gamma distributed with parameters c > 0 and b > 0 and pdf

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