Asymptotic statistics in insurance risk theory (Q2074417)

The author presents a family of results establishing the consistency and asymptotic normality of estimators for the ruin probability and the Gerber-Shiu function for the Lévy surplus model. The book focuses on theory, numerical examples illustrating the workings of the results in practice are not given. In Chapter 1, ruin probability and the Gerber-Shiu function are introduced and calculated for the diffusion perturbation of the Cramér-Lundberg model \[ X_{t} = u + c\cdot t + \sigma \cdot W_{t} - \sum_{i=1}^{N_{t}}U_{i} \] where \(u\) is the initial reserve, \(c\) is the premium rate, \(N_{t}\) is a Poisson process with intensity \(\lambda\), \(\sigma \in \mathbb{R}\), \(W_{t}\) is a Wiener process, and \(U_{i}\) are the independent, identically distributed insurance claim sizes. The main results presented in the book are for the case where the ruin probability \[ \psi(u) = \mathbb{P}[\inf\{t > 0\colon X_{t} < 0\} < \infty|X_{0} = u] \] satisfies the Lundberg approximation \(\psi(u) \sim e^{-\gamma \cdot u}\) as \(u \rightarrow \infty\) where \(\gamma\) is the adjustment coefficient, i.e.\ the positive root of \[ \lambda \cdot (m_{U}(\gamma)-1) - c \cdot \gamma + \frac{\sigma^{2}}{2}\cdot \gamma^{2} = 0 \] with \(m_{U}\) denoting the moment generating function of \(U_{i}\). Chapter 2 introduces Lévi processes, and extends the results of Chapter 1 on analytic formulas for the ruin probability and the Gerber-Shiu function to the Lévy surplus model \[ X_{t} = u + c\cdot t + Z_{t} \] where \(Z_{t}\) is a Lévy process. Chapter 3 discusses the concepts of statistical inference which are used later in the book: e.g., parametric and nonparametric estimators, weak and strong consistency, asymptotic normality, asymptotic confidence bounds, \(M\)- and \(Z\)-estimators, and the Delta Method. In Chapter 4 it is shown -- under some regularity conditions -- that an asymptotically normal parametric estimator for the claim distribution can be used to construct an asymptotically normal estimator for the ruin probability function \(\psi(u)\). An important technical step in the argument is to understand the asymptotic properties of the first and second derivatives of \(\psi(u)\), which is achieved by establishing the corresponding renewal equations. In addition, asymptotic confidence intervals are provided for the ruin probability. Chapter 5 discusses inference for the Gerber-Shiu function \[ \phi(u) = \mathbb{E}\left[\left. e^{-\delta\cdot \tau}\cdot w(X_{\tau-},|X_{\tau}|)\cdot \mathbf{1}_{[\tau < \infty]}\right|X_{0} = u\right] \] where \(\delta > 0\), \(w\) is a penalty function, and \(\tau = \inf\{t > 0\colon X_{t} < 0\}\). A nonparametric estimator for \(\phi\) is constructed by approximating it with \[ \phi_{m}(u) = \frac{1}{2\pi}\int^{m\cdot \pi}_{-m\cdot \pi}e^{-i\cdot s \cdot u}[\mathcal{F}(\phi)](s)\mathrm{d}s \] where \(\mathcal{F}\) stands for the Fourier transform, and by using an empirical estimator for \(\mathcal{F}(\phi)\) obtained from the empirical estimator \[ \hat \Psi(s) = \frac{\frac{1}{n}\sum_{k=1}^{n}e^{i\cdot s\cdot(c\cdot h - X_{k\cdot h}+X_{(k-1)\cdot h})}-1}{h} \] of the characteristic exponent \[ \Psi(s) = \frac{1}{t}\log\mathbb{E}\left[e^{i \cdot s\cdot X_{t}}\right] \] where \(h = h(n)\) is carefully chosen. The main result of the section is an upper bound for \(\mathbb{E}\left[\left|\left|\hat \phi_{m} - \phi\right|\right|^{2}\right]\).