Spatial extremes: models for the stationary case. (English) Zbl 1104.60021

de Haan, Laurens; Pereira, Teresa T.

Ann. Stat. 34, No. 1, 146-168 (2006).

Let \(\varphi\) be a unimodal continuous probability density on \(\mathbb{R}\), let \(N(\cdot)=\sum_{j\geq 1}\varepsilon_{(X_j,Y_j)}(\cdot)\) be a homogeneous Poisson process on \(\mathbb{R}\times[0,\infty)\) and put \(Z(t):=\max_{j\geq 1}\varphi(X_j-t)/Y_j\), \(t\in\mathbb{R}\). Then \(Z\) is a simple max stable process, i.e., the distribution of the process \(Z\) coincides for any \(k\in\mathbb{N}\) with that of \(k^{-1}\max_{i\leq k}Z_i\), where \(Z_1,Z_2,\dots\) are independent copies of \(Z\). In particular we have for arbitrary \(t_1,\dots,t_d\in\mathbb{R}\) and \(x_1,\dots,x_d>0\) with \(d\in\mathbb{N}\) \[ P(Z(t_1)\leq x_1,\dots,Z(t_d)\leq x_d)=\exp\left(-\int_{\infty}^{\infty} \max_{i\leq d}\frac{\varphi(s-t_i)}{x_i}\,ds\right) \] and, thus, \(P(Z(t)\leq x)=\exp(-1/x)\), \(x>0\), \(t\in\mathbb{R}\). The first part of this paper focuses on the computation of the bivariate marginal distribution function \[ P(Z(t_1)\leq x_1,Z(t_2)\leq x_2)=\exp\left(-\int_{\infty}^{\infty} \max_{i=1,2}\frac{\varphi_{\beta}(s-t_i)}{x_i}\,ds\right), \] where \(\varphi_{\beta}(s)=\beta\varphi(\beta s)\) with a scale parameter \(\beta>0\), and \(\varphi\) is the standard normal, the double exponential and the \(t\)-density. The second part of this paper focuses on estimation of \(\beta\) based on \(n\) independent copies \(X_1,\dots,X_n\) of a stochastic process \(X\), which is in the domain of attraction of \(Z\), i.e., there are sequences of continuous functions \(a_n>0\), \(b_n\) on \(\mathbb{R}\) such that as \(n\to\infty\) \[ \left(\frac{\max_{i\leq n}X_i(t)-b_n(t)}{a_n(t)}\right)_{t\in\mathbb{R}} \to_{D}(Z(t))_{t\in\mathbb{R}} \] in \(C\)-space. Denote by \(F_t\) the distribution function of \(X(t)\). Estimation of \(\beta\) is based on the observation that \[ \frac{1}{u}P\left(\left(1-F_{t_i}(X(t_i))\right)\leq u,\, i\leq d \right)\to_{u\downarrow 0}=2\int_{\beta(\max_{i\leq d}t_i-\min_{i\leq d}t_i)/2}^{\infty}\varphi(s)\,ds. \] Consistency of the corresponding estimators is proved, and asymptotic normality is established under suitable conditions. The preceding considerations are extended to a homogeneous Poisson process \(\sum_{j\geq 1}\varepsilon_{(X_j,W_j,Y_j)}\) on \(\mathbb{R}^2\times[0,\infty)\) with \(Z(t_1,t_2):=\max_{j\geq 1} \varphi(X_j-t_1,W_j-t_2)/Y_j\) for \((t_1,t_2)\in\mathbb{R}^2\).

Reviewer: Michael Falk (Würzburg)

MSC:

60G70	Extreme value theory; extremal stochastic processes
62E20	Asymptotic distribution theory in statistics
60G10	Stationary stochastic processes
62M40	Random fields; image analysis
62H11	Directional data; spatial statistics
62G32	Statistics of extreme values; tail inference

Keywords:

Extreme-value theory; spatial tail dependence; max-stable processes; multivariate extremes; semiparametric estimation

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