Score Function

Definition

Let

• $X$ : Random variable, with
  • pdf $f(x;\theta )$, for $\theta \in \Omega$

Then the score function is defined as $S(x;\theta )=\frac{\partial }{\partial \theta }\ln f(x;\theta )$

Remark

Note

The score function determines the estimating equations for the MLE. That is, the MLE $\hat{\theta }$ solves:

$$\begin{array}{rl}\sum _{i=1}^{n}S(x_i;\theta )& =\sum _{i=1}^n\frac{\partial }{\partial \theta }\ln f(x_i;\theta )\\ & =0\end{array}$$