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The probabilistic model for linear regression is $$\begin{align} \mu &= \beta_0 + \beta_1 X_1 + \dots + \beta_k X_k \\ y &\sim \mathcal{N}(\mu, \sigma) \end{align}$$ which is the same as saying $$ y - \mu = \varepsilon \sim \mathcal{N}(0, \sigma) $$ This follows from the properties of a normal distribution if $Y \sim \mathcal{N}(\mu, \sigma)$, then $...


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I found the answer in the package code of the rugarch model (in r). Seems like the authors divide the BIC by the number of observations. I assume that this allows them to compare to the GARCH(1,1) that divide the BIC by the number of observations. Amusingly enough, despite floudering with this question for a few hours, I found the answer to it only minutes ...


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The connection you are looking for is that as shown below: $$ \begin{aligned} \theta_{\min KL} &= \arg\min_{\theta} DK_L\left(P_D||P_\theta\right)\\ &= \arg\min_{\theta} \Big(\mathbb{E}[P_D\log P_D] - \mathbb{E}[P_D\log P_\theta]\Big)\\ &=\arg\min_{\theta} \Big(P_D\log P_D - P_D\mathbb{E}[\log P_\theta]\Big)\quad\text{Since}~P_D \perp\theta\\ &...


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The "two models" you are referring to are mathematically equivalent representations of the same model. Recall that $\log(x y) = \log(x) + \log(y)$. The second formula uses sum of logs because this is more numerically stable. Using log transformation doesn't change the relative ordering of the values, if $x > y$ then $\log(x) > \log(y)$, so ...


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Point 1. Quoting from Section 20.10 of Frank Harrell's Regression Modeling Strategies: The c index [concordance] is the proportion of all pairs of subjects whose survival time can be ordered such that the subject with the higher predicted survival is the one who survived longer. So a concordance of 0.5 is what you get if a model can't distinguish survival ...


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You can compute the likelihood ratio as a function of $p$ $$\Lambda_{LR} = \frac{\mathcal{L}(p\vert \text{H} = 50, \text{T} = 50)}{ \mathcal{L}(0.5\vert \text{H} = 50, \text{T} = 50)} = 2^{100} p^{50} (1-p)^{50}$$ In the context of a likelihood ratio test where the alternative hypothesis is a composite hypothesis you choose the value of $p$ for which the ...


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