TY - JOUR
T1 - Unifying the named natural exponential families and their relatives
AU - Morris, Carl N.
AU - Lock, Kari F.
N1 - Funding Information:
Carl N. Morris is Professor (E-mail: [email protected]) and Kari F. Lock is a Doctoral Student (E-mail: [email protected]), Department of Statistics, Harvard University, Cambridge, MA 02138. This work was supported by Harvard University’s Clark Fund. The authors thank Cindy Chris-tiansen for her contributions to a 1988 version of Figure 1, and Cindy and Joe Blitzstein for their valuable discussions.
PY - 2009
Y1 - 2009
N2 - Five of the six univariate natural exponential families (NEFs) with quadratic variance functions (QVFs), meaning that their variances are at most quadratic functions of their means, are the Normal, Poisson, Gamma, Binomial, and Negative Binomial distributions. The sixth is the NEF-CHS, the NEF generated from convolved Hyperbolic Secant distributions. These six NEF-QVFs and their relatives are unified in this article and in the main diagram via arrows that connect NEFs with many other named distributions. Relatives include all of Pearson's families of conjugate distributions (e.g., Inverted Gamma, Beta, F, and Skewed-t), conjugate mixtures (including two Polya urn schemes), and conditional distributions (including Hypergeometrics and Negative Hypergeometrics). Limit laws that also relate these distributions are indicated by solid arrows in Figure 1.
AB - Five of the six univariate natural exponential families (NEFs) with quadratic variance functions (QVFs), meaning that their variances are at most quadratic functions of their means, are the Normal, Poisson, Gamma, Binomial, and Negative Binomial distributions. The sixth is the NEF-CHS, the NEF generated from convolved Hyperbolic Secant distributions. These six NEF-QVFs and their relatives are unified in this article and in the main diagram via arrows that connect NEFs with many other named distributions. Relatives include all of Pearson's families of conjugate distributions (e.g., Inverted Gamma, Beta, F, and Skewed-t), conjugate mixtures (including two Polya urn schemes), and conditional distributions (including Hypergeometrics and Negative Hypergeometrics). Limit laws that also relate these distributions are indicated by solid arrows in Figure 1.
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U2 - 10.1198/tast.2009.08145
DO - 10.1198/tast.2009.08145
M3 - Article
AN - SCOPUS:77950656579
SN - 0003-1305
VL - 63
SP - 247
EP - 253
JO - American Statistician
JF - American Statistician
IS - 3
ER -