Exponential Probability Distribution Family

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Am Exponential Probability Distribution Family is a probability distribution family that is restricted to exponential functions (composed of exponential probability functions).

References

2013

1. Andersen, Erling (September 1970). "Sufficiency and Exponential Families for Discrete Sample Spaces". Journal of the American Statistical Association (Journal of the American Statistical Association, Vol. 65, No. 331) 65 (331): 1248–1255. doi:10.2307/2284291. JSTOR 2284291. MR268992.
2. Pitman, E.; Wishart, J. (1936). "Sufficient statistics and intrinsic accuracy". Mathematical Proceedings of the Cambridge Philosophical Society 32 (4): 567–579. doi:10.1017/S0305004100019307.
3. Darmois, G. (1935). "Sur les lois de probabilites a estimation exhaustive" (in French). C.R. Acad. Sci. Paris 200: 1265–1266.
4. Koopman, B (1936). "On distribution admitting a sufficient statistic". Transactions of the American Mathematical Society (Transactions of the American Mathematical Society, Vol. 39, No. 3) 39 (3): 399–409. doi:10.2307/1989758. JSTOR 1989758. MR1501854.
5. Kupperman, M. (1958) "Probabilities of Hypotheses and Information-Statistics in Sampling from Exponential-Class Populations", Annals of Mathematical Statistics, 9 (2), 571–575 Template:JSTOR

2000

• http://turing.une.edu.au/~stat354/notes/node57.html
• QUOTE: The exponential family of distributions is a one-parameter family that can be written in the form $f(x;\theta)=B(\theta)h(x)e^{[p(\theta)K(x)]}, \, a\lt x \lt b, \text{ (7.3)}$ where $\gamma\lt \theta\lt \delta$. If, in addition,
• (a) neither $a$ nor $b$ depends on $\theta$,
• (b) $p(\theta)$ is a non-trivial continuous function of $\theta$,
• (c) each of $K'(x) \not\equiv 0$ and $h(x)$ is a continuous function of $x$, $a\lt x\lt b$,
• we say that we have a regular case of the exponential family.

Most of the well-known distributions can be put into this form, for example, binomial, Poisson, geometric, gamma and normal