Exponential Family of Distributions

Overview


Hardin Formulation


{% f(y; \theta, \phi) = exp[\frac{y \theta - b(\theta)}{a(\phi)} + c(y,\phi)] %}
(see Hardin chpt 2, Agresti)
Note that {% \theta %} is a function of the sample point, so it is written as {% \theta_i %} when we are given a point.

{% \theta %} is a function of a linear combination of the x's, here termed {% \eta %}
{% \eta = \beta_0 + \beta_1 x_1 + ... + \beta_m x_m %}
Or stated
{% \eta = X\vec{\beta} %}

Likelihoods


Given a dataset
{% {(\vec{x}_1, y_1), ... ,(\vec{x}_n, y_n)} %}
the Likelihood is given by
{% L(\theta) = \Pi_{i=1}^n exp(\frac{y_i\theta_i - b(\theta_i)}{a(\phi)} + c(y_i, \phi)) %}
and the log Likelihood is given by
{% \mathcal{L}(\theta) = \sum_{i=1}^n (\frac{y_i\theta_i - b(\theta_i)}{a(\phi)} + c(y_i, \phi)) %}

Topics


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