By Lennart Bondesson
Generalized Gamma convolutions have been brought by means of Olof Thorin in 1977 and have been utilized by him to teach that, particularly, the Lognormal distribution is infinitely divisible. After that a huge variety of papers quickly seemed with new leads to a a little random order. the various papers seemed within the Scandinavian Actuarial magazine. This paintings is an try to current the most effects in this classification of chance distributions and similar periods in a slightly logical order. The aim has been to be on a degree that isn't too complex. even though, because the box is quite technical, such a lot readers will locate tough passages within the textual content. those that don't need to go to a mysterious land positioned among the land of likelihood thought and information and the land of classical research aren't examine this paintings. while a few years in the past I submitted a survey to a magazine it was once recommended via the editor, ok. Krickeberg, that it may be improved to a booklet. in spite of the fact that, at the moment i used to be particularly reluctant to take action when you consider that there remained such a lot of difficulties to be solved or to be solved in a smoother approach than prior to. in addition, there has been at the moment a few loss of probabilistic interpretations and functions. a few of the difficulties at the moment are solved yet nonetheless it's felt that extra purposes than these awarded within the paintings can be found.
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Additional info for Generalized Gamma Convolutions and Related Classes of Distributions and Densities
Sample text
We have an exponential family of GGC's. 6 (Size-biased SIImpling). xf(x), f E ff, corresponding to sampling from a GGC with probability proportional to size x. It is assumed that f xf(x)dx < 00 but if that integral is not finite, f(x) could be multiplied by e- lJx. Is g(x) also a GGC? lP{s). tp'(s)/lP{s) = ~. i . J1 + a-s Hence, in this case g equals the convolution of f with two Gamma densities and is therefore a GGC. This conclusion holds also for many other GGC's but not for all. However, g is always ID since C·tp'(s)/lP{s) = f t~·¥ U(dt) is the mgf of an MED; cf.
4. xDe-lJxf(x) is a GGC for every n be identified in Chapter 5. 7 (More size-biased samplingj a complicated emmple). Consider the pdf f(x) = C· (e-lx - e-X), x > 0, 0 < A < 1. It corresponds to the convolution of two Exponential distributions and is hence a GGC. Let f(x) = C·x-7f(x), 0 < '"( < 2, corresponding to size-biased sampling from f(x) with weight-4unction x-7. A surprising fact is that f is also a GGC as we shall see. The mgf of f can be shown to be given by ! (A-s)C - (1-s)C, cp(s) IX -log(A-s) + log(I-s), (1-s)C - (A-s)C, c = r-l, '"( < 1 '"( = 1 c = r-l, '"( > 1 The constant of proportionality is irrelevant for what follows and is set to be 1.
E. the derivative of an increasing function. 1, FE 9' iff (skcp'(s)/cp(s))(k) is absolutely monotone for all k E lNo. For n = 0 a well-known characterization of the ID distributions appears if the theorem is read correctly. For n = 1 a characterization of the sel£-decomposable distributions appears. Proof. 3). We proceed by induction on n. For n = 0 the theorem holds. 9) hold for k ~ n-1. 4) S = osn + sn. :n-l (sv)i )-(-l)nr n (y) dy, J. j =0 where now (_l)nr(n)(y)dy may be just a nonnegative measure on (0,00).