C18. Score and information

Topic: Introduction to generalised linear models · more questions on this topic
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What are the score vector and the expected Fisher information matrix for a log-likelihood?

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Definitions/notation. For log-likelihood \ell(\beta), the score is its gradient S(\beta)=\partial\ell/\partial\beta and expected Fisher information is I(\beta)=-\mathbb{E}_\beta[\partial^2\ell/\partial\beta\,\partial\beta^\top], with expectation under the model and covariates fixed. Under the usual differentiability and interchange conditions, \mathbb{E}_\beta[S(\beta)]=0 and I(\beta)=\operatorname{Var}_\beta(S(\beta)).