Score Function

The score function  is the partial derivativeof the log-likelihood function , where  is the standard likelihood function.

Defining the likelihood function

(1)

shows that

(2)

and thus that

			(3)
			(4)
			(5)

Using the above formulation of , one can easily compute various statistical measurements associated with . For example, the mean  can be shown to equal zero while the variance is precisely the Fisher information matrix. The score function has extensive uses in many areas of mathematics, both pure and applied, and is a key component of the field of likelihood theory.

REFERENCES:

Rodriguez, G. "Lecture Notes on Generalized Linear Models." 2007. https://data.princeton.edu/wws509/notes/.

Sun, D. and Xiao, F. "Likelihood Theory with Score Function." 2013. https://www.stats.uwo.ca/faculty/bellhouse/Likelihood_Theory_with_Score_Function.pdf