Characterizations of natural exponential families with power variance functions by zero regression properties. (English) Zbl 0614.62009
Series of new characterizations by zero regression properties are derived for the distributions in the class of natural exponential families with power variance functions. Such a class of distributions has been introduced by the first author and P. Enis [Reproducibility and natural exponential families with power variance functions. Ann. Stat. 14, 1507-1522 (1986)] in the context of an investigation of reproducible exponential families. This class is broad and includes the following families: normal, Poisson-type, gamma, all families generated by stable distributions with characteristic exponent an element of the unit interval (among these are the inverse Gaussian, modified Bessel-type, and Whittaker-type distributions), and families of compound Poisson distributions generated by gamma variates.
The characterizations by zero regression properties are obtained in a unified approach and are based on certain relations which hold among the cumulants of the distributions in this class. Some remarks are made indicating how the techniques used here can be extended to obtain characterizations of general exponential families.
The characterizations by zero regression properties are obtained in a unified approach and are based on certain relations which hold among the cumulants of the distributions in this class. Some remarks are made indicating how the techniques used here can be extended to obtain characterizations of general exponential families.
MSC:
| 62E10 | Characterization and structure theory of statistical distributions |
| 60E05 | Probability distributions: general theory |
Keywords:
characterizations; zero regression properties; natural exponential families; power variance functions; reproducible exponential families; normal; Poisson-type; gamma; families generated by stable distributions; inverse Gaussian; modified Bessel-type; Whittaker-type distributions; compound Poisson; cumulantsReferences:
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