Contents
Is Poisson distribution exponential family?
The normal, exponential, log-normal, gamma, chi-squared, beta, Dirichlet, Bernoulli, categorical, Poisson, geometric, inverse Gaussian, von Mises and von Mises-Fisher distributions are all exponential families.
Is exponential distribution sub-Gaussian?
Hence, all elements of this exponential family are sub-gaussian, and consequentially sub-exponential (according to definition 1 below).
What is sub exponential distribution?
Subexponential distributions are a special class of heavy{tailed distributions. The name arises from one of their properties, that their tails decrease more slowly than any exponential tail; see (1.4).
Is Gaussian a Subgaussian?
Sub Gaussian random variables exist, for example the Gaussian random variable is subgaussian. Hoeffding’s Lemma (1963) asserts that bounded random variables are also sub Gaussian. . We can integrate two times using ψ(0) = log(1) = 0 and ψ (0) = E [X]=0.
What are the statistics of an exponential family?
Exponential families of distributions provides a general framework for selecting a possible alternative parameterisation of a parametric family of distributions, in terms of natural parameters, and for defining useful sample statistics, called the natural sufficient statistics of the family.
When did Kahane introduce the subgaussian random variable?
To the best of the author’s knowledge, subgaussian random variables were introduced by Kahane in [3], where they played a role to establish a sucient condition for the almost-sure uniform convergence of certain random series of functions.
Where did the name subgaussian random variables come from?
The name \\subgaussian” is the English counterpart of the French \\sous-gaussienne” coined by Kahane in [3]. Subsequent works have studied subgaussian random variables and processes either per se or in connection with various other subjects.
How to convert an exponential family to a canonical form?
By defining a transformed parameter η = η ( θ ), it is always possible to convert an exponential family to canonical form. The canonical form is non-unique, since η ( θ) can be multiplied by any nonzero constant, provided that T ( x) is multiplied by that constant’s reciprocal, or a constant c can be added to η ( θ) and h ( x) multiplied by