More than one kind of information

(Jul 19, 2018)

This collection of blogs is called Information Matrix, and it is named after an important quantity in statistics.  If we are fitting a parametric model of the hazard rate, with log-likelihood:

\[ \ell( \alpha_1, \ldots, \alpha_n ) \]

as a function of \(n\) parameters \(\alpha_1, \ldots, \alpha_n\), then the information matrix is the matrix of second-order partial derivatives of \(\ell\). That is, the matrix \({\cal I}\) with \(ij\)th component:

\[ {\cal I}_{ij} = \frac{\partial^2 \ell}{\partial \alpha_i \partial \alpha_j}. \]

It is important because \(-{\cal I}^{-1}\) evaluated at the fitted maximum \((\hat{\alpha}_1, \ldots, \hat{\alpha}_n)\) approximates the variance-covariance matrix of…

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Tags: information, indicator process

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