Threshold selection in univariate extreme value analysis

Threshold selection plays a key role in various aspects of statistical inference of rare events. In this work, two new threshold selection methods are introduced. The first approach measures the fit of the exponential approximation above a threshold and achieves good performance in small samples. The second method smoothly estimates the asymptotic mean squared error of the Hill estimator and performs consistently well over a wide range of processes. Both methods are analyzed theoretically, compared to existing procedures in an extensive simulation study and applied to a dataset of financial losses, where the underlying extreme value index is assumed to vary over time.

Distributed Inference for Extreme Value Index

Summary This paper investigates a divide-and-conquer algorithm for estimating the extreme value index when data are stored in multiple machines. The oracle property of such an algorithm based on extreme value methods is not guaranteed by the general theory of distributed inference. We propose a distributed Hill estimator and establish its asymptotic theories. We consider various cases where the number of observations involved in each machine can be either homogeneous or heterogeneous, and either fixed or varying according to the total sample size. In each case, we provide sufficient, sometimes also necessary, condition, under which the oracle property holds. Some key words: Extreme value index, Distributed inference, Distributed Hill estimator

Extremes, Extremal Index Estimation, Records, Moment Problem for the Pseudo-Lindley Distribution and Applications

The pseudo-Lindley distribution which was introduced in Zeghdoudi and Nedjar (2016) is studied with regards to it upper tail. In that regard, and when the underlying distribution function follows the Pseudo-Lindley law, we investigate the the behavior of its values, the asymptotic normality of the Hill estimator and the double-indexed generalized Hill statistic process (Ngom and Lo, 2016), the asymptotic normality of the records values and the the moment problem.

An MFDFA Study to find Herd Behaviour and Information Asymmetry during Demonetization

The paper attempts to identify how the Indian stock market reacts to an unusual event like demonetization through the observation of herd behavior. The data set considered is the NIFTY 50 index collected on 9th November 2016. This method could become a failure if giant investors are well aware of the massive proceeding as stock markets are prone to information asymmetry. Thus, the existence of the same is checked using Hill estimator. The sectoral herding behavior is also captured for three selective sectors namely PSU banks, Energy sector, and Automobile sector as each sector may pose a different response towards the event. The volatility index is examined for a time period of 10 years from 2008 to 2018.

Threshold selection and trimming in extremes

We consider removing lower order statistics from the classical Hill estimator in extreme value statistics, and compensating for it by rescaling the remaining terms. Trajectories of these trimmed statistics as a function of the extent of trimming turn out to be quite flat near the optimal threshold value. For the regularly varying case, the classical threshold selection problem in tail estimation is then revisited, both visually via trimmed Hill plots and, for the Hall class, also mathematically via minimizing the expected empirical variance. This leads to a simple threshold selection procedure for the classical Hill estimator which circumvents the estimation of some of the tail characteristics, a problem which is usually the bottleneck in threshold selection. As a by-product, we derive an alternative estimator of the tail index, which assigns more weight to large observations, and works particularly well for relatively lighter tails. A simple ratio statistic routine is suggested to evaluate the goodness of the implied selection of the threshold. We illustrate the favourable performance and the potential of the proposed method with simulation studies and real insurance data.

Power law in tails of bourse volatility – evidence from India

Inverse cubic law has been an established Econophysics law. However, it has been only carried out on the distribution tails of the log returns of different asset classes (stocks, commodities, etc.). Financial Reynolds number, an Econophysics proxy for bourse volatility has been tested here with Hill estimator to find similar outcome. The Tail exponent or α ≈ 3, is found to be well outside the Levy regime (0 < α < 2). This confirms that asymptotic decay pattern for the cumulative distribution in fat tails following inverse cubic law. Hence, volatility like stock returns also follow inverse cubic law, thus stay way outside the Levy regime. This piece of work finds the volatility proxy (econophysical) to be following asymptotic decay with tail exponent or α ≈ 3, or, in simple terms, ‘inverse cubic law’. Risk (volatility proxy) and return (log returns) being two inseparable components of quantitative finance have been found to follow the similar law as well. Hence, inverse cubic law truly becomes universal in quantitative finance.