Lecture Notes in Statistics

1. Introduction and Summary.- 1.1 Introduction.- 1.2 Summary.- 1.3 General remarks and notation.- 2. Basic Concepts and Facts.- 2.1 Some facts from probability theory.- 2.2 Particular probability distributions.- 2.3 Infinite divisibility.- 2.4 Pick functions and some applications.- 2.5 Some complementary results.- 3. Generalized Gamma Convolutions.- 3.1 Definition and basic theory.- 3.2 Examples of distributions in ?.- 3.3 A composition theorem.- 3.4 Two real characterizations of the mgf of a GGC.- 3.5 Widened GGC's.- 3.6 Bivariate GGC's.- 3.7 Complementary results and notes.- 4. Densities of Generalized Gamma Convolutions.- 4.1 GGC's as mixtures of Gamma distributions.- 4.2 A class of mixtures of Exponential distributions in ?.- 4.3 A fundamental class of densities in ?.- 4.4 A multiplicative property of GGC's.- 4.5 Complementary results and notes.- 5. Hyperbolically Completely Monotone Densities.- 5.1 The class b.- 5.2 The ?-class.- 5.3 A surprise and its consequences.- 5.4 Decreasing HCM-functions.- 5.5 Creation of new densities in ? by composition.- 5.6 Multiplication of ?-variables.- 5.7 ?-distributions with explicit Laplace transforms.- 5.8 Complementary results and notes.- 6. Back to ?.- 6.1 Real characterizations of some transforms.- 6.2 Multiplicative properties of ?.- 6.3 Some nested subclasses of ? and their properties.- 6.4 Summary and a new perspective.- 6.5 Complementary results and notes.- 7. Extended Generalized Gamma Convolutions.- 7.1 Definition and basic theory.- 7.2 Examples.- 7.3 EGGC's as variance mixtures of the Normal distribution.- 7.4 More on composition.- 7.5 Self-reciprocal densities.- 7.6 Complementary results and notes.- 8. Generalized Negative Binomial Convolutions.- 8.1 Definition and main properties.- 8.2 Examples of distributions in ?d.- 8.3 In search of a discrete ?-class.- 8.4 Complementary results and notes.- 9. The ?2-class of Distributions.- 9.1 Definition and main properties.- 9.2 Examples of ?2-distributions.- 9.3 First passage time distributions as ?2-distributions.- 9.4 On densities of distributions in ?2.- 9.5 Extensions.- 9.6 Notes.- 10. Shot-Noise Distributions.- 10.1 Shot-noise processes.- 10.2 Classes of shot-noise distributions on R+.- 10.3 Discrete shot-noise distributions.- 10.4 Notes.- Final Words.- Author Index.