TY - JOUR

T1 - An introduction to Bent Jørgensen's ideas

AU - Cordeiro, Gauss M.

AU - Labouriau, Rodrigo

AU - Botter, Denise

PY - 2021/2

Y1 - 2021/2

N2 - We briefly expose some key aspects of the theory and use of dispersion models, for which Bent Jørgensen played a crucial role as a driving force and an inspiration source. Starting with the general notion of dispersion models, built using minimalistic mathematical assumptions, we specialize in two classes of families of distributions with different statistical flavors: exponential dispersion and proper dispersion models. The construction of dispersion models involves the solution of integral equations that are, in general, untractable. These difficulties disappear when more mathematical structure is assumed: it reduces to the calculation of a moment generating function or of a Riemann-Stieltjes integral for the exponential dispersion and the proper dispersion models, respectively. A new technique for constructing dispersion models based on characteristic functions is introduced turning the integral equations above into a tractable convolution equation and yielding examples of dispersion models that are neither proper dispersion nor exponential dispersion models. A corollary is that the cardinality of regular and non-regular dispersion models are both large. Some selected applications are discussed including exponential families non-linear models (for which generalized linear models are particular cases) and several models for clustered and dependent data based on a latent Lévy process.

AB - We briefly expose some key aspects of the theory and use of dispersion models, for which Bent Jørgensen played a crucial role as a driving force and an inspiration source. Starting with the general notion of dispersion models, built using minimalistic mathematical assumptions, we specialize in two classes of families of distributions with different statistical flavors: exponential dispersion and proper dispersion models. The construction of dispersion models involves the solution of integral equations that are, in general, untractable. These difficulties disappear when more mathematical structure is assumed: it reduces to the calculation of a moment generating function or of a Riemann-Stieltjes integral for the exponential dispersion and the proper dispersion models, respectively. A new technique for constructing dispersion models based on characteristic functions is introduced turning the integral equations above into a tractable convolution equation and yielding examples of dispersion models that are neither proper dispersion nor exponential dispersion models. A corollary is that the cardinality of regular and non-regular dispersion models are both large. Some selected applications are discussed including exponential families non-linear models (for which generalized linear models are particular cases) and several models for clustered and dependent data based on a latent Lévy process.

KW - Dispersion Models

KW - Exponential Dispersion Models

KW - Generalised Linear Models

KW - Exponential dispersion models

KW - Dispersion models

KW - Proper dispersion models

KW - Non-linear models

KW - Exponential family

KW - Saddlepoint approximations

U2 - 10.1214/19-BJPS458

DO - 10.1214/19-BJPS458

M3 - Journal article

VL - 35

SP - 2

EP - 20

JO - Brazilian Journal of Probability and Statistics

JF - Brazilian Journal of Probability and Statistics

SN - 0103-0752

IS - 1

ER -