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时变极值依赖结构的统计推断AI 解读

We analyze the extreme value dependence of independent, not necessarily identically distributed multivariate regularly varying random vectors. More specifically, we propose estimators of the spectral measure locally at some time point and of the spectral measures integrated over time. The uniform asymptotic normality of these estimators is proved under suitable nonparametric smoothness and regularity assumptions. We then use the process convergence of the integrated spectral…

尾过程推断及其在金融时间序列建模中的应用AI 解读

To draw inference on serial extremal dependence within heavy-tailed Markov chains, Drees et al., (2015) proposed nonparametric estimators of the spectral tail process. The methodology can be extended to the more general setting of a stationary, regularly varying time series. The large-sample distribution of the estimators is derived via empirical process theory for cluster functionals. The finite-sample performance of these estimators is evaluated via Monte Carlo…

簇泛函经验过程的极限定理AI 解读

Let (Xn,i)1≤i≤n,n∈ℕ be a triangular array of row-wise stationary ℝd-valued random variables. We use a “blocks method” to define clusters of extreme values: the rows of (Xn,i) are divided into mn blocks (Yn,j), and if a block contains at least one extreme value, the block is considered to contain a cluster. The cluster starts at the first extreme value in the block and ends at the last one. The main results are uniform central limit theorems for empirical processes $Z_{n}(f):=\frac{1}{\sqrt{nv_{n}}}\sum_{j=1}^{m_{n}}(f(Y_{n,j})-Ef(Y_{n,j}))$…

极值理论中的极小极大风险界AI 解读

Asymptotic minimax risk bounds for estimators of a positive extreme value index under zero-one loss are investigated in the classical i.i.d. setup. To this end, we prove the weak convergence of suitable local experiments with Pareto distributions as center of localization to a white noise model, which was previously studied in the context of nonparametric local density estimation and regression. From this result we derive upper and lower bounds on the asymptotic minimax risk…

如何绘制 Hill 图AI 解读

An abundance of high quality data sets requiring heavy tailed models necessitates reliable methods of estimating the shape parameter governing the degree of tail heaviness. The Hill estimator is a popular method for doing this but its practical use is encumbered by several difficulties. We show that an alternative method of plotting Hill estimator values is more revealing than the standard method unless the underlying data comes from a Pareto distribution.

极值指数估计的最优收敛速度AI 解读

Hall and Welsh established the best attainable rate of convergence for estimates of a positive extreme value index $\gamma$ under a certain second order condition implying that the distribution function of the maximum of n random variables converges at an algebraic rate to the pertaining extreme value distribution. As a first generalization, we obtain a surprisingly sharp bound on the estimation error if $\gamma$ is still assumed to be positive, but the rate of convergence…

极值指数的改进 Pickands 估计量AI 解读

Consider a distribution function that belongs to the weak domain of attraction of an extreme value distribution. The extreme value index $\beta$ will be estimated by mixtures of Pickands estimators, where the weights are generated by a probability measure which satisfies a certain integrability condition. We prove a functional limit theorem for a process of Pickands estimators and asymptotic normality of the refined Pickands estimator. For negative $\beta$ the new estimator…