Pub Date : 2021-02-05DOI: 10.1080/24754269.2021.1877961
Duo Zhang, Liucang Wu, K. Ye, Min Wang
Semiparametric mixed-effects double regression models have been used for analysis of longitudinal data in a variety of applications, as they allow researchers to jointly model the mean and variance of the mixed-effects as a function of predictors. However, these models are commonly estimated based on the normality assumption for the errors and the results may thus be sensitive to outliers and/or heavy-tailed data. Quantile regression is an ideal alternative to deal with these problems, as it is insensitive to heteroscedasticity and outliers and can make statistical analysis more robust. In this paper, we consider Bayesian quantile regression analysis for semiparametric mixed-effects double regression models based on the asymmetric Laplace distribution for the errors. We construct a Bayesian hierarchical model and then develop an efficient Markov chain Monte Carlo sampling algorithm to generate posterior samples from the full posterior distributions to conduct the posterior inference. The performance of the proposed procedure is evaluated through simulation studies and a real data application.
{"title":"Bayesian quantile semiparametric mixed-effects double regression models","authors":"Duo Zhang, Liucang Wu, K. Ye, Min Wang","doi":"10.1080/24754269.2021.1877961","DOIUrl":"https://doi.org/10.1080/24754269.2021.1877961","url":null,"abstract":"Semiparametric mixed-effects double regression models have been used for analysis of longitudinal data in a variety of applications, as they allow researchers to jointly model the mean and variance of the mixed-effects as a function of predictors. However, these models are commonly estimated based on the normality assumption for the errors and the results may thus be sensitive to outliers and/or heavy-tailed data. Quantile regression is an ideal alternative to deal with these problems, as it is insensitive to heteroscedasticity and outliers and can make statistical analysis more robust. In this paper, we consider Bayesian quantile regression analysis for semiparametric mixed-effects double regression models based on the asymmetric Laplace distribution for the errors. We construct a Bayesian hierarchical model and then develop an efficient Markov chain Monte Carlo sampling algorithm to generate posterior samples from the full posterior distributions to conduct the posterior inference. The performance of the proposed procedure is evaluated through simulation studies and a real data application.","PeriodicalId":22070,"journal":{"name":"Statistical Theory and Related Fields","volume":"5 1","pages":"303 - 315"},"PeriodicalIF":0.5,"publicationDate":"2021-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/24754269.2021.1877961","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42904985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-31DOI: 10.1080/24754269.2020.1867795
Anqi Chen, Cheng-Yu Sun, Boxin Tang
ABSTRACT This article considers the problem of selecting two-level designs under the baseline parameterisation when some two-factor interactions are important. We propose a minimum aberration criterion, which minimises the bias caused by the non-negligible effects. Using this criterion, a class of optimal designs can be further distinguished from one another, and we present an algorithm to find the minimum aberration designs among the D-optimal designs. Sixteen-run and twenty-run designs are summarised for practical use.
{"title":"Selecting baseline designs using a minimum aberration criterion when some two-factor interactions are important","authors":"Anqi Chen, Cheng-Yu Sun, Boxin Tang","doi":"10.1080/24754269.2020.1867795","DOIUrl":"https://doi.org/10.1080/24754269.2020.1867795","url":null,"abstract":"ABSTRACT This article considers the problem of selecting two-level designs under the baseline parameterisation when some two-factor interactions are important. We propose a minimum aberration criterion, which minimises the bias caused by the non-negligible effects. Using this criterion, a class of optimal designs can be further distinguished from one another, and we present an algorithm to find the minimum aberration designs among the D-optimal designs. Sixteen-run and twenty-run designs are summarised for practical use.","PeriodicalId":22070,"journal":{"name":"Statistical Theory and Related Fields","volume":"5 1","pages":"95 - 101"},"PeriodicalIF":0.5,"publicationDate":"2021-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/24754269.2020.1867795","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47930848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-20DOI: 10.1080/24754269.2021.1871708
K. Ye, Xiaobin Yang, Y. Ji, Min Wang
Toxicity study, especially in determining the maximum tolerated dose (MTD) in phase I clinical trial, is an important step in developing new life-saving drugs. In practice, toxicity levels may be categorised as binary grades, multiple grades, or in a more generalised case, continuous grades. In this study, we propose an overall MTD framework that includes all the aforementioned cases for a single toxicity outcome (response). The mechanism of determining MTD involves a function that is predetermined by user. Analytic properties of such a system are investigated and simulation studies are performed for various scenarios. The concept of the continual reassessment method (CRM) is also implied in the framework and Bayesian analysis, including Markov chain Monte Carlo (MCMC) methods are used in estimating the model parameters.
{"title":"A system for determining maximum tolerated dose in clinical trial","authors":"K. Ye, Xiaobin Yang, Y. Ji, Min Wang","doi":"10.1080/24754269.2021.1871708","DOIUrl":"https://doi.org/10.1080/24754269.2021.1871708","url":null,"abstract":"Toxicity study, especially in determining the maximum tolerated dose (MTD) in phase I clinical trial, is an important step in developing new life-saving drugs. In practice, toxicity levels may be categorised as binary grades, multiple grades, or in a more generalised case, continuous grades. In this study, we propose an overall MTD framework that includes all the aforementioned cases for a single toxicity outcome (response). The mechanism of determining MTD involves a function that is predetermined by user. Analytic properties of such a system are investigated and simulation studies are performed for various scenarios. The concept of the continual reassessment method (CRM) is also implied in the framework and Bayesian analysis, including Markov chain Monte Carlo (MCMC) methods are used in estimating the model parameters.","PeriodicalId":22070,"journal":{"name":"Statistical Theory and Related Fields","volume":"5 1","pages":"288 - 302"},"PeriodicalIF":0.5,"publicationDate":"2021-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/24754269.2021.1871708","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45763211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-18DOI: 10.1080/24754269.2021.1871873
J. Shao
Covariate-adaptive randomisation has a more than 45 years of history of applications in clinical trials, in order to balance treatment assignments across prognostic factors that may have influence on the outcomes of interest. However, almost no theory had been developed for covariate-adaptive randomisation until a paper on the theory of testing hypotheses published in 2010. In this article, we review aspects of methodology and theory developed in the last decade for statistical inference under covariate-adaptive randomisation. We focus on issues such as whether a conventional procedure valid under the assumption that treatments are assigned completely at random is still valid or conservative when the actual randomisation is covariate-adaptive, how a valid inference procedure can be obtained by modifying a conventional method or directly constructed by stratifying the covariates used in randomisation, whether inference procedures have different properties when covariate-adaptive randomisation schemes have different degrees of balancing assignments, and how to further adjust covariates in the inference procedures to gain more efficiency. Recommendations are made during the review and further research problems are discussed.
{"title":"Inference after covariate-adaptive randomisation: aspects of methodology and theory","authors":"J. Shao","doi":"10.1080/24754269.2021.1871873","DOIUrl":"https://doi.org/10.1080/24754269.2021.1871873","url":null,"abstract":"Covariate-adaptive randomisation has a more than 45 years of history of applications in clinical trials, in order to balance treatment assignments across prognostic factors that may have influence on the outcomes of interest. However, almost no theory had been developed for covariate-adaptive randomisation until a paper on the theory of testing hypotheses published in 2010. In this article, we review aspects of methodology and theory developed in the last decade for statistical inference under covariate-adaptive randomisation. We focus on issues such as whether a conventional procedure valid under the assumption that treatments are assigned completely at random is still valid or conservative when the actual randomisation is covariate-adaptive, how a valid inference procedure can be obtained by modifying a conventional method or directly constructed by stratifying the covariates used in randomisation, whether inference procedures have different properties when covariate-adaptive randomisation schemes have different degrees of balancing assignments, and how to further adjust covariates in the inference procedures to gain more efficiency. Recommendations are made during the review and further research problems are discussed.","PeriodicalId":22070,"journal":{"name":"Statistical Theory and Related Fields","volume":"5 1","pages":"172 - 186"},"PeriodicalIF":0.5,"publicationDate":"2021-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/24754269.2021.1871873","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41476425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-02DOI: 10.1080/24754269.2020.1862586
Jingyu Ji, Deyuan Li
We congratulate Professor Zhengjun Zhang for a topnotch contribution to the literature and thank the editor for the invitation to participate in the discussion of the excellent review paper. Zhang (2020) provides an informative summary ofmodelling systematic risk with nonlinear time series models and tail dependence measures based on extreme observations. In the current era, where risk management is becoming more and more important, this review paper is timely and will provide the impetus for future research in this and broader areas. Section 5.1 in Zhang (2020) proposes the autoregressive tail-index model to characterise and describe systematic risk and risk contagions. The autoregressive tail-index model is not only a good measure of systematic risk in the US stock market (Zhao et al., 2018), but can also be applied to the study of extreme climate (Deng et al., 2020) and other fields. In this discussion, we are more interested in whether the autoregressive tail-index model is also suitable to characterise the systematic risk in China’s stock market. We present an analysis of the stock negative log-returns of the Shanghai Composite Index (SSE Index), which is one of the most important indexes for China’s stock market. The data contains the daily closing prices of 180 components of SSE Index and is downloaded from WindFinancial Terminal from 01/05/2005 to 10/20/2020 with 3836 observations for each stock. Figure 1 plots the daily closing prices of SSE Index. Since SSE Index adjusts its component stocks every half year, we analyse 87 stocks that have always been included in the index from 01/05/2005 to 10/20/2020. For day t, we obtain 87 negative log-returns and calculate the maximaQ t = max1≤i≤87 ri,t , where ri,t is the daily negative log-return for stock i. Figure 2 shows the histogram of {Q t : t = 1, 2, . . . , 3836}. By Figure 2, we see that there are many daily maxima of negative log-returns clustered around 0.11, due to the Limit Up-Limit Down Rule of China’s stock market. This rule prohibits trading activity in exchangelisted securities at prices outside specified price bands. Motivated by Section 5.1 in Zhang (2020), we first fit a GARCH(1,1) model with normal distributed innovations to each individual negative log-retuens series. Using the negative log-returns series divided by the fitted volatilities, we obtain standardised negative log-returns series for each stock. Taking the maximum value of the 87 standardised negative log-returns each day, we obtain a time series {Qt : t = 1, 2, . . . , 3836}, see Figure 3. It is seen that there exist four possible peaks around June 2006, November 2008, January 2016 and September 2018. In fact, China’s stockmarket experienced substantial boom and burst during these five periods. On June 7, 2006, SSE Index plummeted 88.45 points. In 2008, the US subprime mortgage crisis spread to the world and triggered a financial tsunami. SSE Index dropped from the highest of 6124 in October 2007 to the lowest of
我们祝贺张正军教授对文献的卓越贡献，并感谢编辑邀请我们参与这篇优秀评论论文的讨论。张（2020）提供了一个关于用非线性时间序列模型和基于极端观测的尾部依赖性度量来建模系统风险的信息摘要。在当前风险管理变得越来越重要的时代，这篇综述论文是及时的，将为未来在这一领域和更广泛领域的研究提供动力。张（2020）第5.1节提出了自回归尾部指数模型来表征和描述系统风险和风险传染。自回归尾指数模型不仅是衡量美国股市系统风险的一个很好的指标（赵et al.，2018），还可以应用于极端气候的研究（Deng et al.，2020）等领域。在这场讨论中，我们更感兴趣的是自回归尾部指数模型是否也适用于描述中国股市的系统性风险。本文对中国股市最重要的指数之一上证指数的股票负对数收益率进行了分析。该数据包含上证指数180个组成部分的每日收盘价，从WindFinancial终端下载，时间为2005年5月1日至2020年10月20日，每只股票有3836个观察结果。图1描绘了上证指数的每日收盘价格。由于上证指数每半年调整一次成分股，我们分析了从2005年5月1日到2020年10月20日一直被纳入该指数的87只股票。对于第t天，我们获得87个负对数回报，并计算最大Q t=max1≤i≤87 ri，t，其中ri，t是股票i的每日负对数回报。图2显示了{Q t:t=1，2，…，3836}的直方图。通过图2，我们可以看到，由于中国股市的涨停-跌停规则，负对数回报率的日最大值聚集在0.11附近。该规则禁止以特定价格区间以外的价格进行交易所上市证券的交易活动。受张（2020）第5.1节的启发，我们首先将具有正态分布创新的GARCH（1,1）模型拟合到每个负对数回归序列。使用负对数收益率序列除以拟合的波动率，我们获得了每只股票的标准化负对数收益序列。取每天87个标准化负对数回报的最大值，我们得到了一个时间序列{Qt:t=1,2，…，3836}，见图3。可以看出，在2006年6月、2008年11月、2016年1月和2018年9月前后存在四个可能的峰值。事实上，中国股市在这五个时期经历了巨大的繁荣和破灭。2006年6月7日，上证指数暴跌88.45点。2008年，美国次贷危机波及全球，引发金融海啸。上证指数从2007年10月的最高点6124点跌至2008年10月最低点1664点。截至2015年底，上证指数上涨12.6%，反弹至3600点。2016年1月，中国股市经历了大幅抛售，在市场下跌7%后，于2016年1月份7日停牌。2016年1月末26日，上证指数跌破2015年8月的最低点，1月27日跌至2638点。上证指数在2018年10月11日跌破2016年最低点2638点后，于10月18日跌破2500点，10月19日跌至2449点，较2015年最高点暴跌逾一半。{Qt}序列的表现与中国股市的实证观察结果一致。图4显示了{Qt}序列的直方图，这表明标准化的负对数回报可能遵循Fréchet分布。我们通过自回归尾部指数模型拟合{Qt}，即Zhang（2020）中的模型（5.1）-（5.3）。拟合的参数值和标准偏差如表1所示。结果表明，所有参数都是显著的，这表明模型（5.1）-（5.3）适用于上证指数中87只股票的横截面最大值。图5显示了回收的尾部指数{αõt}。显然，当极端事件出现时，尾部指数往往会下降，反映出增加
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Pub Date : 2021-01-02DOI: 10.1080/24754269.2020.1862587
Ting Zhang
I congratulate and thank the author for providing a systematic and thorough review of both classical approaches and modern developments on the modelling of extremal events and tail dependence in th...
我祝贺并感谢作者对极端事件和尾部依赖模型的经典方法和现代发展进行了系统而彻底的回顾。。。
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Pub Date : 2021-01-02DOI: 10.1080/24754269.2021.1871710
Zhengjun Zhang
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)
我很高兴我的评论文章在概率论、理论统计、估计方法和应用方面激发了如此广泛和深思熟虑的讨论。讨论者提出了许多很好的观点。我感谢各位讨论者对所讨论的内容以及与极值研究相关的更广泛的理论和方法主题的兴趣。特别是，Ji和Li(2021)找到了一种方法，可以将所审查的模型之一扩展到研究中国股票市场的系统性风险。Qi(2021)指出广义极值分布中静态尾指数参数的估计还很不完善，然后讨论了Hall(1982)、Peng and Qi(2009)和F. Wang et al.(2019)的三种最大似然估计，以处理落在不同范围的尾指数。Smith(2021)对过去三十年来极值理论的发展提供了更笼统的看法。读者可以从阅读讨论和参考文献中受益。T. Wang and Yan(2021)不仅将讨论扩展到Resnick(2004)和Davis and Mikosch(2009)引入的两种极端依赖测度，还指出了在许多极端值应用中存在的一些实际问题。Xu和wang(2021)提出了一些有趣的想法，将尾商相关系数扩展到条件尾商相关系数，以实现条件尾独立性。他们还概述了将新的极值理论的极大值形式应用于高维推理的一些想法，例如多重检验问题。T. Zhang (2021a)关注时间序列极值，主张测量累积尾对抗效应，即序列尾依赖程度和T. Zhang (2021b)的期望极限定理。我的评论主要集中在用非线性时间序列模型和尾部依赖度量来研究极值和系统风险，当然，这并不是对所审查的主题和讨论者讨论的主题的最终结论，也不是对极值文献研究的许多其他广泛主题的最终结论。我期待着所有这些领域的未来发展。本文将进一步阐明每个已回顾的度量、模型、它们的应用及其进一步发展背后的一些基本思想。可解释性、可计算性和可测试性。一些基本属性，如可解释性、可计算性、可预测性、稳定性和可测试性，在统计应用程序中通常是需要的。一般来说，参数化模型能够满足这些性质，被广泛采用。例如，线性回归是日常最常用的模型，皮尔逊线性相关系数是两个随机变量之间最常用的依赖度量。另一方面，参数模型可能不够通用，其模型的假设可能不被满足。因此，非参数(半参数)模型、随机森林、深度学习模型和神经网络模型是首选。然而，这些通用和高级模型在实现上述部分或全部期望属性方面带来了一些困难。至于在实践中如何选择一种模式，这取决于许多因素。George Box说所有的模型都是错误的，但有些是有用的。在参数模型和非参数模型之间存在权衡。我们可能会说所有的模型都是有用的，但是每个人的优势是不同的。类似线性回归和皮尔逊线性相关系数在极值上下文中还没有很好地定义。T. Wang和Yan(2021)讨论的极端依赖度量和Ledford和Tawn(1996, 1997)最流行的尾相关系数度量η通常涉及非参数估计。Zhang(2008)引入了商相关系数(QCC)和尾商相关系数(TQCC)作为线性相关系数(LCC)的替代相关测度。从Z. Zhang(2020)的示例3.1和3.2可以看出，LCC是基于绝对误差的度量，而QCC/TQCC是基于相对误差的度量
{"title":"Rejoinder of “On studying extreme values and systematic risks with nonlinear time series models and tail dependence measures”","authors":"Zhengjun Zhang","doi":"10.1080/24754269.2021.1871710","DOIUrl":"https://doi.org/10.1080/24754269.2021.1871710","url":null,"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)","PeriodicalId":22070,"journal":{"name":"Statistical Theory and Related Fields","volume":"5 1","pages":"45 - 48"},"PeriodicalIF":0.5,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/24754269.2021.1871710","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44507983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-02DOI: 10.1080/24754269.2020.1862589
Y. Qi
I would like to take this opportunity to congratulate Zhengjun for his continuing contribution to extremevalue statistics in recent years. In this review paper, some fundamental theories on univariate extremes and multivariate extremes are introduced, and recent developments on extremes from some structured stochastic processes are also given. The results in the latter sections of the paper are largely due to Zhengjun and his coauthors. The paper provides some insights for future challenges on extremes and can help young researchers follow the contemporary research topics. Below I offer some comments onunivariate extremevalue statistics. Although the theory for univariate extremes is quite complete, the statistical methods such as the estimation and inference procedures are far from perfect. Set μ = 0 and σ = 1 in the definition (2.6) in the paper and write
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Pub Date : 2021-01-02DOI: 10.1080/24754269.2020.1869897
Tiandong Wang, Jun Yan
We congratulate Prof. Zhang for this timely review on recent advances in extreme value theory for heterogeneous populations and on time series models for extreme observations. This is a substantial effort. Not only does it give a summary of the state-of-the-art work in time series modelling of extremes but also suggests interesting methodological and applied research questions. Our discussion focuses on extremal dependence metrics and practical applications for the time series models.
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Pub Date : 2021-01-02DOI: 10.1080/24754269.2021.1871709
R. Smith
This discussion reviews the paper by Zhengjun Zhang in the context of broader research on multivariate extreme value theory and max-stable processes.
本文在更广泛地研究多元极值理论和极大稳定过程的背景下，回顾了张正军的论文。
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