Index: The Book of Statistical ProofsModel Selection ▷ Classical information criteria ▷ Akaike information criterion ▷ Corrected AIC and uncorrected AIC

Theorem: In the infinite data limit, the corrected Akaike information criterion converges to the uncorrected Akaike information criterion

$\label{eq:aicc-aic} \lim_{n \to \infty} \mathrm{AIC}_\mathrm{c}(m) = \mathrm{AIC}(m) \; .$

Proof: The corrected Akaike information criterion is defined as

$\label{eq:aicc} \mathrm{AIC}_\mathrm{c}(m) = \mathrm{AIC}(m) + \frac{2k^2 + 2k}{n-k-1} \; .$

Note that the number of free model parameters $k$ is finite. Thus, we have:

$\label{eq:aicc-aic-qed} \begin{split} \lim_{n \to \infty} \mathrm{AIC}_\mathrm{c}(m) &= \lim_{n \to \infty} \left[ \mathrm{AIC}(m) + \frac{2k^2 + 2k}{n-k-1} \right] \\ &= \lim_{n \to \infty} \mathrm{AIC}(m) + \lim_{n \to \infty} \frac{2k^2 + 2k}{n-k-1} \\ &= \mathrm{AIC}(m) + 0 \\ &= \mathrm{AIC}(m) \; . \end{split}$
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Metadata: ID: P316 | shortcut: aicc-aic | author: JoramSoch | date: 2022-03-18, 17:00.