4

According to Hougaard (1986), positive stable distribution on $\mathbb{R}^+$ belongs to exponential family, how about the case the support of stable distribution being less than zero?

The purpose of this question is to confirm whether MLE is equal to GLSE (General least square estimator) even with the error term following stable distribution not Normal distribution. (In A. Charnes et al's paper, MLE and GLSE are equvalent under iid and exponential family)

kjetil b halvorsen
  • 63,378
  • 26
  • 142
  • 467
kurtkim
  • 173
  • 9
  • From any random variable, whoxe distribution has an existing moment generating function, can generate an exponential family. But stable distributions (with index $\alpha < 2$ do not have mgf, so thst construction do not work. We could try to emulate the construction using the characteristic function, that "works" formally, but generates "density" functions which takes complex values, so d9 not really work. So I guess the answer to your question is NO. – kjetil b halvorsen Mar 29 '15 at 19:29
  • This paper? https://academic.oup.com/biomet/article-abstract/73/2/387/338958 – kjetil b halvorsen Sep 08 '19 at 22:30

1 Answers1

1

From any random variable, whose distribution has an existing moment generating function, we can generate an exponential family. Correspondingly, distributions belonging to an exponential family do have a moment generating function existing (in an open interval containing zero). But stable distribution (with index $\alpha < 2$) do not have mgf's, so they cannot be exponential families.

kjetil b halvorsen
  • 63,378
  • 26
  • 142
  • 467
  • 2
    maybe worth noting that this reasoning only precludes stable distributions from being natural exponential families; it is possible that they are exponential families with respect to some sufficient statistic $T(x) \neq x$ – πr8 Feb 10 '20 at 11:12
  • 1
    The Pareto distribution and Levy distribution can be described as an exponential family if you keep the location parameter fixed. Yet their moments are infinite. – Sextus Empiricus Jun 12 '20 at 21:24