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Deviance is twice the difference between the maximum achievable log likelihood and that attained under the fitted model.

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Is there an integral form of the deviance residual? I've always seen the deviance residual written as $$ d_i = 2w_i\Big(y_i\big(\tilde{\theta}_i - \hat{\theta}_i\big) - \big(b(\tilde{\theta}_i) - b … (\hat{\theta}_i)\big) \Big)$$ While studying for an actuarial exam, I've run into the deviance residual in the following form $$d_i = 2w_i \int_{\mu_i}^{y_i} \dfrac{y_i - \xi}{\operatorname{Var …
asked Mar 24 by Demetri Pananos
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I've been studying some GLM theory, and I ran into deviance recently. Seems like the deviance has properties similar to a metric. Namely: $ D(y, y)=0$ Non-negativity $D(y, \mu) \geq 0$ I … hesitate to add subadditivity to the list, but Deviance has something like it. Furthermore, when we fit a GLM, we maximize the log liklihood function, which in turn should minimize the deviance, no? So does deviance have an interpretation as a distance? …
asked Feb 7 '18 by Demetri Pananos