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).




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  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


  • 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 [math]\displaystyle{ f(x;\theta)=B(\theta)h(x)e^{[p(\theta)K(x)]}, \, a\lt x \lt b, \text{ (7.3)} }[/math] where [math]\displaystyle{ \gamma\lt \theta\lt \delta }[/math]. If, in addition,
      • (a) neither [math]\displaystyle{ a }[/math] nor [math]\displaystyle{ b }[/math] depends on [math]\displaystyle{ \theta }[/math],
      • (b) [math]\displaystyle{ p(\theta) }[/math] is a non-trivial continuous function of [math]\displaystyle{ \theta }[/math],
      • (c) each of [math]\displaystyle{ K'(x) \not\equiv 0 }[/math] and [math]\displaystyle{ h(x) }[/math] is a continuous function of [math]\displaystyle{ x }[/math], [math]\displaystyle{ a\lt x\lt b }[/math],
    • 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