International Monthly Seminar on Time Scales Analysis #6

In this work, we investigate the approximation of spatially nonstationary spatio-temporal GARCH (ST-GARCH) processes by spatially stationary counterparts at fixed locations. This approach enables a localized analysis of complex spatio-temporal volatility structures. Building upon the model's recursive formulation, we establish that the ST-GARCH process can be represented as a sum of random matrix products, allowing us to derive conditions under which the process admits a Lipschitz continuous approximation. We prove that, under mild regularity and continuity assumptions, the nonstationary process (X^2_t(s)) can be closely approximated by a spatially stationary process (X^2_{t,s_0}(s)) at a fixed point (s_0), with a convergence rate governed by the spatial distance (|s - s_0|_\infty). Furthermore, using a Taylor expansion and a derivative-based construction, we refine this approximation by including the first-order spatial derivative, yielding an improved representation as a linear combination of two spatially stationary processes. Our theoretical findings lay the groundwork for practical localized modeling and inference in real-world applications involving heterogeneous spatio-temporal data.

In extreme value theory (EVT), estimating the tail index of heavy-tailed distributions is crucial for understanding rare and extreme events. Traditional estimators such as the Hill and Maximum Likelihood Estimators (MLE) perform well with large samples but struggle
with small sample sizes due to increased bias and variance. In this paper, we introduce a novel estimation technique the Maximum Lq-Likelihood Estimator (MLqE), which incorporates a distortion parameter q, making it more robust to extreme observations and more accurate in small-sample scenarios. We demonstrate that the MLqE is consistent and asymptotically normal, outperforming the classical MLE in terms of mean squared error in
moderate and small sample sizes. Moreover, we present simulation results that highlight the superior performance of the MLqE, particularly when comparing it to the MLE in tail
index estimation. This method not only offers a significant improvement in the accuracy of heavy-tailed distribution parameter estimation but also provides a versatile tool for various
real-world applications, including finance, hydrology, and risk management.