Weak Convergence of the Empirical Mean Excess Process with Application to Estimate the Negative Tail Index

Let Y i , 1 ≤ i ≤ n be i.i.d. random variables with the generalized Pareto distribution W γ,σ with γ < 0. We define the empirical mean excess process with respect to {Y i , 1 ≤ i ≤ n} as in Eq. 2.1 (see below) and investigate its weak convergence. As an application, two new estimators of the negative tail index γ are constructed based on the linear regression to the empirical mean excess function and their consistency and asymptotic normality are obtaine

On testing extreme value conditions

Applications of univariate extreme value theory rely on certain as- sumptions. Recently, two methods for testing these extreme value conditions are derived by [Dietrich, D., de Haan, L., Hüsler, J., Extremes 5: 71-85, (2002)] and [Drees, H., de Haan, L., Li, D., J. Stat. Plan. Inference, 136: 3498-3538, (2006)]. In this paper we compare the two tests by simulations and investigate the effect of a possible weight function by choosing a parameter, the test error and the power of each test. The conclusions are useful for extreme value application

Tail approximations to the density function in EVT

Let X 1, X 2, ...,X n be independent identically distributed random variables with common distribution function F, which is in the max domain of attraction of an extreme value distribution, i.e., there exist sequences a n > 0 and b n ∈ ℝ such that the limit of P(a_n^{-1}(\max_{1\leq i\leq n}X_i-b\!_n)\leq x) exists. Assume the density function f (of F) exists. We obtain an uniformly weighted approximation to the tail density function f, and an uniformly weighted approximation to the tail density function of P(a_n^{-1}(\max_{1\leq i\leq n}X_i-b\!_n)\leq x) under some second order conditio

Involvement of the JNK/FOXO3a/Bim Pathway in Neuronal Apoptosis after Hypoxic-Ischemic Brain Damage in Neonatal Rats.

c-Jun N-terminal kinase (JNK) plays a key role in the regulation of neuronal apoptosis. Previous studies have revealed that forkhead transcription factor (FOXO3a) is a critical effector of JNK-mediated tumor suppression. However, it is not clear whether the JNK/FOXO3a pathway is involved in neuronal apoptosis in the developing rat brain after hypoxia-ischemia (HI). In this study, we generated an HI model using postnatal day 7 rats. Fluorescence immunolabeling and Western blot assays were used to detect the distribution and expression of total and phosphorylated JNK and FOXO3a and the pro-apoptotic proteins Bim and CC3. We found that JNK phosphorylation was accompanied by FOXO3a dephosphorylation, which induced FOXO3a translocation into the nucleus, resulting in the upregulation of levels of Bim and CC3 proteins. Furthermore, we found that JNK inhibition by AS601245, a specific JNK inhibitor, significantly increased FOXO3a phosphorylation, which attenuated FOXO3a translocation into the nucleus after HI. Moreover, JNK inhibition downregulated levels of Bim and CC3 proteins, attenuated neuronal apoptosis and reduced brain infarct volume in the developing rat brain. Our findings suggest that the JNK/FOXO3a/Bim pathway is involved in neuronal apoptosis in the developing rat brain after HI. Agents targeting JNK may offer promise for rescuing neurons from HI-induced damage

Estimating Extreme Value Index by Subsampling for Massive Datasets with Heavy-Tailed Distributions

Modern statistical analyses often encounter datasets with massive sizes and heavy-tailed distributions. For datasets with massive sizes, traditional estimation methods can hardly be used to estimate the extreme value index directly. To address the issue, we propose here a subsampling-based method. Specifically, multiple subsamples are drawn from the whole dataset by using the technique of simple random subsampling with replacement. Based on each subsample, an approximate maximum likelihood estimator can be computed. The resulting estimators are then averaged to form a more accurate one. Under appropriate regularity conditions, we show theoretically that the proposed estimator is consistent and asymptotically normal. With the help of the estimated extreme value index, a normal range can be established for a heavy-tailed random variable. Observations that fall outside the range should be treated as suspected records and can be practically regarded as outliers. Extensive simulation experiments are provided to demonstrate the promising performance of our method. A real data analysis is also presented for illustration purpose

Distributed Inference for Tail Risk

For measuring tail risk with scarce extreme events, extreme value analysis is often invoked as the statistical tool to extrapolate to the tail of a distribution. The presence of large datasets benefits tail risk analysis by providing more observations for conducting extreme value analysis. However, large datasets can be stored distributedly preventing the possibility of directly analyzing them. In this paper, we introduce a comprehensive set of tools for examining the asymptotic behavior of tail empirical and quantile processes in the setting where data is distributed across multiple sources, for instance, when data are stored on multiple machines. Utilizing these tools, one can establish the oracle property for most distributed estimators in extreme value statistics in a straightforward way. The main theoretical challenge arises when the number of machines diverges to infinity. The number of machines resembles the role of dimensionality in high dimensional statistics. We provide various examples to demonstrate the practicality and value of our proposed toolkit

Adapting the Hill estimator to distributed inference:dealing with the bias

The distributed Hill estimator is a divide-and-conquer algorithm for estimating the extreme value index when data are stored in multiple machines. In applications, estimates based on the distributed Hill estimator can be sensitive to the choice of the number of the exceedance ratios used in each machine. Even when choosing the number at a low level, a high asymptotic bias may arise. We overcome this potential drawback by designing a bias correction procedure for the distributed Hill estimator, which adheres to the setup of distributed inference. The asymptotically unbiased distributed estimator we obtained, on the one hand, is applicable to distributed stored data, on the other hand, inherits all known advantages of bias correction methods in extreme value statistics