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Multivariate Exponential Families: A Concise Guide to Statistical Inference

Part of the SpringerBriefs in Statistics Book Series

DOI: 10.1007/978-3-030-81900-2

ISBN: 978-3-030-81899-9

Publisher: Springer International Publishing - Springer Journals

Oct 8, 2021

Available

Introduction: Aims and Outline

Bedbur, Stefan; Kamps, Udo

Pages: 1 –3

[Parametric families of probability distributions and their properties are extensively studied in the literature on statistical modeling and inference. Exponential families of distributions comprise density functions of a particular form, which enables general assertions and leads to nice...
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Parametrizations and Basic Properties

Bedbur, Stefan; Kamps, Udo

Pages: 5 –42

[A formal definition of an exponential family is the starting point of a widely self-contained, mathematically rigorous and concise course on exponential families in a systematic structure. The definition is followed by a variety of detailed examples of one- and multiparameter, uni- and...
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Distributional and Statistical Properties

Bedbur, Stefan; Kamps, Udo

Pages: 43 –64

[With a focus on statistical inference, distributional and statistical properties of multivariate and multiparameter exponential families are studied, such as generating functions, marginal and conditional distributions, and product measures. Sufficiency and completeness of statistics in...
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Parameter Estimation

Bedbur, Stefan; Kamps, Udo

Pages: 65 –91

[Parameter estimation is discussed when the underlying parametric model forms an exponential family. For maximum likelihood estimates, the topics existence, uniqueness, representation, and properties are addressed, and constrained maximum likelihood estimation is examined as well. Efficient...
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Hypotheses Testing

Bedbur, Stefan; Kamps, Udo

Pages: 93 –115

[In a one-parameter exponential family, uniformly most powerful tests are stated for particular, but important hypotheses. Moreover, uniformly most powerful unbiased tests are shown in other cases and in multiparameter exponential families. Statistical tests for several parameters simultaneously...
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Exemplary Multivariate Applications

Bedbur, Stefan; Kamps, Udo

Pages: 117 –135

[The usefulness of results for exponential families is demonstrated by means of three multivariate examples of different kind. As a discrete probability distribution, the family of negative multinomial distributions is considered, and results in parameter estimation and statistical testing are...

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