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Actuarial statistics uses probability theory and statistical inference to quantify insurance claims, premiums, and the uncertainty of future losses. Aggregate claims model describes portfolios, reinsurance redistributes losses, and the Bühlmann model combines experience with population information.

Analysis of Boolean functions studies real-valued functions on a Boolean hypercube using Fourier analysis, probability theory, and combinatorics.

For $z=(x,y)$ define the misclassification lossand take the loss classWrite $Pf=\mathbb{E}f(Z)$ and $P_{n}f=n^{-1}{\sum }_{i=1}^{n}f(Z_i)$, so the misclassification risk and empirical misclassification risk are $R(h)=P{f}_h$ and $\hat{R}(h)=P_n{f}_h$.

Because $\hat{h}$ is an empirical risk minimizer, $P_n(f_{\hat{h}}-f_{h^{\ast }})\le 0$. Hence its excess risk obeysIntroduce an independent ghost sample and perform Rademacher symmetrization. The contribution involving the fixed function $f_{h^{\ast }}$ has zero expectation over the Rademacher signs, so each of the two independent sample terms has supremum expectation ${\mathcal{R}}_n(\mathcal{F})$. Therefore

Every difference $f_h-f_{h^{\ast }}$ takes values in $[-1,1]$. Replacing one observation can consequently change $G$ by at most $2/n$. Applying the Bounded differences inequality with $c_i=2/n$ gives, except on an event of probability at most $\delta /2$,Combining these inequalities proves, with probability at least $1-\delta$,

For a fixed sample $z_{1:n}$, its Empirical Rademacher complexity isIts sample expected value is ${\mathcal{R}}_n(\mathcal{F})$.

Because every $f\in \mathcal{F}$ takes values in $\{0,1\}$, replacing one sample point changes $\hat{\mathcal{R}}$ by at most $1/n$. PutThe lower-tail form of the Bounded differences inequality givesoutside an event of probability at most $\delta /3$.

For the excess-loss supremum $G$ from part (b), replacing one observation changes $G$ by at most $2/n$. A second application of the same inequality givesoutside another event of probability at most $\delta /3$. On the intersection of these two events, symmetrization and the union bound yieldThis event has probability at least $1-2\delta /3\ge 1-\delta$, and $4s=2\sqrt{2\log (3/\delta )/n}$. Thus

For a zero-one-valued loss class $\mathcal{F}$, the excess misclassification risk of an empirical risk minimizer $\hat{h}$ over a population minimizer $h^{\ast }$ satisfieswith probability at least $1-\delta$. The proof combines Rademacher symmetrization with the Bounded differences inequality for the supremum of the empirical excess-loss process.

The observable version isIt follows by another application of the Bounded differences inequality to the Empirical Rademacher complexity.