Rejoinder of “On studying extreme values and systematic risks with nonlinear time series models and tail dependence measures”

Abstract

I am pleased that my review article has stimulated such broader and thoughtful discussions in probability theory, theoretical statistics, estimation methods, and applications. The discussants have made many excellent points. I appreciate the discussants’ interest in the reviewed contents and much broader theoretical and methodological topics related to extreme value study. In particular, Ji and Li (2021), find a way that one of the reviewed models can be extended to study the systematic risks in the Chinese stock market. Qi (2021) points out that the estimation of the static tail index parameter in the generalised extreme value distribution is still far from perfect, and then discusses three maximum likelihood estimations from Hall (1982), Peng and Qi (2009), and F. Wang et al. (2019) to handle the tail index that falls in different ranges. Smith (2021) offers a much more general view of the development of extreme value theory over the last thirty years. Readers can benefit from reading the discussions and the references discussed therein. T. Wang and Yan (2021) not only extend discussions to two extreme dependence measures introduced by Resnick (2004) and Davis and Mikosch (2009) but also point out some practical issues existed in many extreme value applications. Xu andWang (2021) show some interesting ideas of extending the tail quotient correlation coefficient to the conditional tail quotient correlation coefficient for conditional tail independence. They also outline some ideas of applying the new extreme value theory formaxima of maxima for high-dimensional inference, e.g., multiple testing problems. T. Zhang (2021a) focuses on time series extremes and advocates measuring the cumulative tail adversarial effect, i.e., the degree of serial tail dependence and the desired limit theorem in T. Zhang (2021b). My review is focussing on studying extreme values and systematic risks with nonlinear time series models and tail dependence measures, and of course, it is not the final word on the reviewed topics and the topics discussed by the discussants, and many other broad topics researched by the extreme value literature. I look forward to future developments in all of these areas. This rejoinder will further clarify some basic ideas behind each reviewed measures, models, their applications, and their further developments. Interpretability, computability, and testability. Some basic properties, such as interpretability, computability, predictability, stability, and testability, are often desired in statistical applications. In general, parametric models can satisfy these properties and are widely adopted. For example, linear regressions are the most popular models used daily, and Pearson’s linear correlation coefficient is the most commonly used dependence measure between two random variables. On the other hand, parametric models may not be general enough, and their models’ assumptions may not be satisfied. As a result, nonparametric (semi-parametric) models, random forest, deep learning models, and neural network models are preferred. However, these general and advanced models bring some difficulties in achieving some or all of the aforementioned desired properties. As to how to choose a model in practice, it depends on many factors. George Box stated that all models are wrong, but some are useful. There is a tradeoff between parametricmodels and nonparametricmodels.Wemay say that all models are useful, but the strengths vary with each individual. Analogs to linear regression and Pearson’s linear correlation coefficient are not yet well defined in the extreme value context. The extreme dependence measures discussed in T. Wang and Yan (2021) and the most popular coefficient of tail dependence measure η by Ledford and Tawn (1996, 1997) often involve nonparametric estimations. The quotient correlation coefficient (QCC) and the tail quotient correlation coefficient (TQCC)were introduced in Z. Zhang (2008) as alternative correlation measures to the linear correlation coefficient (LCC). It can be seen from Examples 3.1 and 3.2 in Z. Zhang (2020) the LCC is an absolute error based measure while QCC/TQCC is a relative error based