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The Laplace integral method for a differential equation seeksDifferentiating under the integral and substituting intogivesSince $z{e}^{zt}={\partial }_t{e}^{zt}$, integration by parts yieldsIt is therefore enough to choose $f$ and $C$ so thatand the boundary term vanishes. Solving the first-order equation on $-1<t<1$ givesTake $C$ to be the interval $[-1,1]$. Although $f$ has integrable endpoint singularities,vanishes at both endpoints, so the boundary term is zero. ThusAt $z=0$, substituting $t=\sin \theta$ givesThe normalization $y(0)=1$ therefore sets $K=1/\pi$. By the defining regularity and normalization of the modified Bessel function, this proves the real integral representation of the modified Bessel function I0:

For a model with $k$ fitted parameters, the Akaike information criterion iswhere $\ell$ is the log-likelihood evaluated at the maximum-likelihood estimate.

In the normal linear model with known ${\sigma }^2$,The maximum-likelihood estimator is the ordinary least-squares estimator, with fitted mean $\hat{\mu }=HY$, whereSince only the $p$ components of $\beta$ are fitted,After multiplying by ${\sigma }^2$ and discarding the model-independent constant, this is exactly Mallows Cp:

To compare it with prediction error, write $Y=\mu +\epsilon$ and $Y^{\ast }=\mu +{\epsilon }^{\ast }$, where $\mu =X\beta$, the two errors are independent, and both have covariance ${\sigma }^{2}I$. Since $H\mu =\mu$, $H$ is an orthogonal projection of rank $p$, andThe cross term has zero expectation, soOn the other hand,Therefore the unbiased prediction-error identity for ordinary least squares gives