Scoring algorithm
In statistics, Fisher's Scoring algorithm is a form of Newton's method used to solve maximum likelihood equations numerically.
Sketch of Derivation
Let be random variables, independent and identically distributed with twice differentiable p.d.f. , and we wish to calculate the maximum likelihood estimator (M.L.E.) of . First, suppose we have a starting point for our algorithm , and consider a Taylor expansion of the score function, , about :
- ,
where
is the observed information matrix at . Now, setting , using that and rearranging gives us:
- .
We therefore use the algorithm
- ,
and under certain regularity conditions, it can be shown that .
Fisher Scoring
In practice, is usually replaced by , the Fisher information, thus giving us the Fisher Scoring Algorithm:
- .
File:Example.jpg'''Bold text''''Bold text''''==Application to Linear Models==
The method of Fisher Scoring is often used in the theory of linear models. Suppose we have a standard linear model
- ,
where Failed to parse (unknown function "\si"): {\displaystyle \varepsilon_i \sim N(0,\si#REDIRECT [[#REDIRECT [[Insert text]]#REDIRECT [[#REDIRECT [[Insert text]]#REDIRECT [[<br />Insert text<br /><br /><br /><sub> <gallery> Subscript text </gallery><blockquote> {| class="wikitable" |- Block quote |} </blockquote></sub>]]]]]]gma^2)} independently; now suppose we want to estimate β. It can be shown[1] that
where is the M.L.E. of β. It is therefore desirable to find and use it as an estimator of β.
References
- ^ A.C. Davidson Statistical Models. Cambridge University Press (2003).