Problem 6
Over the centuries natural disasters in a particular country have occurred as a Poisson process of rate \(\lambda(t)\). Any disaster at time \(t\) is known to have occurred only with probability \(\pi(t)\), due to the patchiness of historical records. If records of different disasters are preserved independently, show that the point process of known disasters is Poisson with intensity \(\lambda(t) \pi(t)\)
Problem 9
Show that the MA(1) models \(Y_{t}=\varepsilon_{t}+\beta \varepsilon_{t-1}\) and \(Y_{t}=\varepsilon_{t}+\beta^{-1} \varepsilon_{t-1}\) have the same correlations and deduce that they are indistinguishable from their correlograms alone. If \(Y_{t}=(1+\beta B) \varepsilon_{t}\) in terms of the backshift operator \(B\), show that \(\varepsilon_{t}\) may be expressed as a linear combination of \(Y_{t}, Y_{t-1}, \ldots\) in which the infinite past has no effect only if \(|\beta|<1\). The ARMA process \(a(B) Y_{t}=b(B) \varepsilon_{t}\) is said to be invertible if the zeros of the polynomial \(b(z)\) all lie outside the unit disk. Show that the MA(1) process is invertible only if \(|\beta|<1\) Compare this with the condition for stationarity of the AR(1) model. Discuss.
Problem 10
Show that strict stationarity of a time series \(\left\\{Y_{j}\right\\}\) means that for any \(r\) we have $$ \operatorname{cum}\left(Y_{j_{1}}, \ldots, Y_{j_{r}}\right)=\operatorname{cum}\left(Y_{0}, \ldots, Y_{j_{r}-j_{1}}\right)=\kappa^{j_{2}-j_{1}, \ldots, j_{r}-j_{1}} $$ say. Suppose that \(\left\\{Y_{j}\right\\}\) is stationary with mean zero and that for each \(r\) it is true that \(\sum_{u}\left|\kappa^{u_{1}, \ldots, u_{r-1}}\right|=c_{r}<\infty\) The \(r\) th cumulant of \(T=n^{-1 / 2}\left(Y_{1}+\cdots+Y_{n}\right)\) is $$ \begin{aligned} \operatorname{cum}\left\\{n^{-1 / 2}\left(Y_{1}+\cdots+Y_{n}\right)\right\\} &=n^{-r / 2} \sum_{j_{1}, \ldots, j_{r}} \operatorname{cum}\left(Y_{j_{1}}, \ldots, Y_{j_{r}}\right) \\ &=n^{-r / 2} \sum_{j_{1}=1}^{n} \sum_{j_{2}, \ldots, j_{r}} \kappa^{j_{2}-j_{1}, \ldots, j_{r}-j_{1}} \\ &=n \times n^{-r / 2} \sum_{j_{2}, \ldots, j_{r}} \kappa^{j_{2}-j_{1}, \ldots, j_{r}-j_{1}} \\ & \leq n^{1-r / 2} \sum_{j_{2}, \ldots, j_{r}}\left|\kappa^{j_{2}-j_{1}, \ldots, j_{r}-j_{1}}\right| \leq n^{1-r / 2} c_{r} \end{aligned} $$ Justify this reasoning, and explain why it suggests that \(T\) has a limiting normal distribution as \(n \rightarrow \infty\), despite the dependence among the \(Y_{j}\). Obtain the cumulants of \(T\) for the MA(1) model, and convince yourself that your argument extends to the \(\mathrm{MA}(q)\) model. Can you extend the argument to arbitrary linear combinations of the \(Y_{j} ?\)
Problem 11
Let \(X_{1}, \ldots, X_{n}\) be independent exponential variables with rates \(\lambda_{j} .\) Show that \(Y=\) \(\min \left(X_{1}, \ldots, X_{n}\right)\) is also exponential, with rate \(\lambda_{1}+\cdots+\lambda_{n}\), and that \(\operatorname{Pr}\left(Y=X_{j}\right)=\) \(\lambda_{j} /\left(\lambda_{1}+\cdots+\lambda_{n}\right)\). Hence write down an algorithm to simulate data from a continuoustime Markov chain with finite state space, using exponential and multinomial random number generators.
Problem 14
A Poisson process of rate \(\lambda(t)\) on the set \(\mathcal{S} \subset \mathbb{R}^{k}\) is a collection of random points with the following properties (among others): \- the number of points \(N_{\mathcal{A}}\) in a subset \(\mathcal{A}\) of \(\mathcal{S}\) has the Poisson distribution with mean \(\Lambda(\mathcal{A})=\int_{\mathcal{A}} \lambda(t) d t\) \- given \(N_{\mathcal{A}}=n\), the positions of the points are sampled randomly from the density \(\lambda(t) / \int_{\mathcal{A}} \lambda(s) d s, t \in \mathcal{A}\) (a) Assuming that you have reliable generators of \(U(0,1)\) and Poisson variables, show how to generate the points of a Poisson process of constant rate \(\lambda\) on the interval \(\left[0, t_{0}\right]\). (b) Let \(t=(x, y) \in \mathbb{R}^{2}, \eta, \xi \in \mathbb{R}, \tau>0, \lambda(x, y)=\tau^{-1}\\{1+\xi(y-\eta) / \tau\\}^{-1 / \xi-1}\). Give an algorithm to generate realisations from the Poisson process with rate \(\lambda(x, y)\) on $$ \mathcal{S}=\\{(x, y): 0 \leq x \leq 1, y \geq u, \lambda(x, y)>0\\}. $$ $$ \begin{array}{rrrrrrrrrrrrrrrrr} \hline 9 & 12 & 11 & 4 & 7 & 2 & 5 & 8 & 5 & 7 & 1 & 6 & 1 & 9 & 4 & 1 & 3 \\ 3 & 6 & 1 & 11 & 33 & 7 & 91 & 2 & 1 & 87 & 47 & 12 & 9 & 135 & 258 & 16 & 35 \\\ \hline \end{array} $$
