Exponential Family

[iuri_leite]

Dear All,

I have some questions that are not related to the SAS use.

I am reading about the distributions that belong to the exponential family. There is also a subgroup that belongs to the canonical exponential family. The Bernoulli is one of them, it belongs to the exponential family and the canonical exponential family. When writing the Bernoulli distribution in the canonical form we identify the canonical link which is the natural ling for a Generalized Linear Model for binary variable. In this case, the natural link is the logit. Still dealing with a GLM for binary variable, we have the probit model, in which the link function is The Inverse CDF of the normal distribution. It also maps the probability that assumes values between 0 and 1 into something that assumes values between minus infinity and plus infinity. However, the link function is not a natural one. Both are appropriate to work with dichotomous variable (associated to Bernoulli distribution), but in the canonical form the natural link is the logit. Is that correct?

The canonical link function refers to the natural one? The Inverse CDF of the normal distribution used in the probit is a link function, but not a canonical link function?

Thanks a a lot.

[KevinScott · Accepted solution]

In the canonical form, the natural link is the logit. Is that correct? Yes

Does the canonical link function refer to the natural one? Yes

The Inverse CDF of the normal distribution used in the probit is a link function but not a canonical one. Correct

The link is natural/canonical, If b(m) = q = Xb, where m is equal to the expectation of the exponentially distributed random variable:

b(q)=q, b is the identity function when density is written in canonical form. An advantage of canonical/natural links is that a minimal sufficient statistic for β exists, i.e., all the information about β is contained in a function of the data of the same dimensionality as β.