Zixuan Zhao, Yanglei Song, Wenyu Jiang, Dongsheng Tu
Pocock and Simon's minimization method is a popular approach for covariate-adaptive randomization in clinical trials. Valid statistical inference with data collected under the minimization method requires the knowledge of the limiting covariance matrix of within-stratum imbalances, whose existence is only recently established. In this work, we propose a bootstrap-based estimator for this limit and establish its consistency, in particular, by Le Cam's third lemma. As an application, we consider in simulation studies adjustments to existing robust tests for treatment effects with survival data by the proposed estimator. It shows that the adjusted tests achieve a size close to the nominal level, and unlike other designs, the robust tests without adjustment may have an asymptotic size inflation issue under the minimization method.
波科克和西蒙的最小化方法是临床试验中一种常用的协方差自适应随机方法。使用最小化方法收集的数据进行有效的统计推断，需要知道层内不平衡的极限协方差矩阵，而该矩阵的存在最近才被证实。在这项工作中，我们提出了一种基于自举法的极限估计方法，并特别通过 Le Cam 的第三个 Lemma 建立了其一致性。作为一项应用，我们在模拟研究中考虑用提出的估计器调整现有的生存数据治疗效果稳健检验。结果表明，调整后的检验规模接近名义水平，与其他设计不同的是，在最小化方法下，未经调整的稳健检验可能存在规模膨胀的渐近问题。
{"title":"Consistent covariances estimation for stratum imbalances under minimization method for covariate-adaptive randomization","authors":"Zixuan Zhao, Yanglei Song, Wenyu Jiang, Dongsheng Tu","doi":"10.1111/sjos.12703","DOIUrl":"https://doi.org/10.1111/sjos.12703","url":null,"abstract":"Pocock and Simon's minimization method is a popular approach for covariate-adaptive randomization in clinical trials. Valid statistical inference with data collected under the minimization method requires the knowledge of the limiting covariance matrix of within-stratum imbalances, whose existence is only recently established. In this work, we propose a bootstrap-based estimator for this limit and establish its consistency, in particular, by Le Cam's third lemma. As an application, we consider in simulation studies adjustments to existing robust tests for treatment effects with survival data by the proposed estimator. It shows that the adjusted tests achieve a size close to the nominal level, and unlike other designs, the robust tests without adjustment may have an asymptotic size inflation issue under the minimization method.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"30 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139053744","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the construction of confidence bands for survival curves under the outcome-dependent stratified sampling. A main challenge of this design is that data are a biased dependent sample due to stratification and sampling without replacement. Most literature on regression approximates this design by Bernoulli sampling but variance is generally overestimated. Even with this approximation, the limiting distribution of the inverse probability weighted Kaplan-Meier estimator involves a general Gaussian process, and hence quantiles of its supremum is not analytically available. In this paper, we provide a rigorous asymptotic theory for the weighted Kaplan-Meier estimator accounting for dependence in the sample. We propose the novel hybrid method to both simulate and bootstrap parts of the limiting process to compute confidence bands with asymptotically correct coverage probability. Simulation study indicates that the proposed bands are appropriate for practical use. A Wilms tumor example is presented.
{"title":"Confidence Bands for Survival Curves from Outcome-Dependent Stratified Samples","authors":"Takumi Saegusa, Peter Nandori","doi":"10.1111/sjos.12700","DOIUrl":"https://doi.org/10.1111/sjos.12700","url":null,"abstract":"We consider the construction of confidence bands for survival curves under the outcome-dependent stratified sampling. A main challenge of this design is that data are a biased dependent sample due to stratification and sampling without replacement. Most literature on regression approximates this design by Bernoulli sampling but variance is generally overestimated. Even with this approximation, the limiting distribution of the inverse probability weighted Kaplan-Meier estimator involves a general Gaussian process, and hence quantiles of its supremum is not analytically available. In this paper, we provide a rigorous asymptotic theory for the weighted Kaplan-Meier estimator accounting for dependence in the sample. We propose the novel hybrid method to both simulate and bootstrap parts of the limiting process to compute confidence bands with asymptotically correct coverage probability. Simulation study indicates that the proposed bands are appropriate for practical use. A Wilms tumor example is presented.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"9 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138825587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Subhadra Dasgupta, Siuli Mukhopadhyay, Jonathan Keith
This work is focused on finding G -optimal designs theoretically for kriging models with two -dimensional inputs and separable exponential covariance structures. For design comparison, the notion of evenness of two-dimensional grid designs is developed. The mathematical relationship between the design and the supremum of the mean squared prediction error (SMSPE) function is studied and then optimal designs are explored for both prospective and retrospective design scenarios. In the case of prospective designs, the new design is developed before the experiment is conducted and the regularly spaced grid is shown to be the G -optimal design. Retrospective designs are constructed by adding or deleting points from an already existing design. Deterministic algorithms are developed to find the best possible retrospective designs (which minimizes the SMSPE). It is found that a more evenly spread design under the G -optimality criterion leads to the best possible retrospective design. For all the cases of finding the optimal prospective designs and the best possible retrospective designs, both frequentist and Bayesian frameworks have been considered. The proposed methodology for finding retrospective designs is illustrated with a spatio-temporal river water quality monitoring experiment.
这项工作的重点是为具有二维输入和可分离指数协方差结构的克里金模型从理论上找到 G 最佳设计。为了进行设计比较，提出了二维网格设计的均匀性概念。研究了设计与均方预测误差（SMSPE）函数上确界之间的数学关系，然后探讨了前瞻性设计和回顾性设计两种情况下的最优设计。在前瞻性设计的情况下，新设计是在实验进行之前开发的，而规则间隔的网格被证明是 G 最佳设计。回顾性设计是通过在已有设计中添加或删除点来构建的。我们开发了确定性算法来寻找最佳的回顾性设计（使 SMSPE 最小）。研究发现，在 G 最佳准则下，更均匀分布的设计会导致最佳的回顾性设计。在寻找最优前瞻性设计和最佳回顾性设计的所有情况下，都考虑了频繁主义和贝叶斯框架。我们用一个时空河流水质监测实验来说明所提出的寻找回溯设计的方法。
{"title":"G-optimal grid designs for kriging models","authors":"Subhadra Dasgupta, Siuli Mukhopadhyay, Jonathan Keith","doi":"10.1111/sjos.12699","DOIUrl":"https://doi.org/10.1111/sjos.12699","url":null,"abstract":"This work is focused on finding G -optimal designs theoretically for kriging models with two -dimensional inputs and separable exponential covariance structures. For design comparison, the notion of evenness of two-dimensional grid designs is developed. The mathematical relationship between the design and the supremum of the mean squared prediction error (<i>SMSPE</i>) function is studied and then optimal designs are explored for both prospective and retrospective design scenarios. In the case of prospective designs, the new design is developed before the experiment is conducted and the regularly spaced grid is shown to be the G -optimal design. Retrospective designs are constructed by adding or deleting points from an already existing design. Deterministic algorithms are developed to find the best possible retrospective designs (which minimizes the <i>SMSPE</i>). It is found that a more evenly spread design under the G -optimality criterion leads to the best possible retrospective design. For all the cases of finding the optimal prospective designs and the best possible retrospective designs, both frequentist and Bayesian frameworks have been considered. The proposed methodology for finding retrospective designs is illustrated with a spatio-temporal river water quality monitoring experiment.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138566680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yiming Liu, Guangming Pan, Guangren Yang, Wang Zhou
We propose a new test to investigate the conditional mean dependence between a response variable and the corresponding covariates in the high dimensional regimes. The test statistic is an extreme-type one built on the nonparametric method. The limiting null distribution of the proposed extreme type statistic under a mild mixing condition is established. Moreover, to make the test more powerful in general structures we propose a more general test statistic and develop its asymptotic properties. The power analysis of both methods is also considered. In real data analysis, we also propose a new way to conduct the feature screening based on our results. To evaluate the performance of our estimators and other methods, extensive simulations are conducted.
{"title":"Nonparametric conditional mean testing via an extreme-type statistic in high dimension","authors":"Yiming Liu, Guangming Pan, Guangren Yang, Wang Zhou","doi":"10.1111/sjos.12697","DOIUrl":"https://doi.org/10.1111/sjos.12697","url":null,"abstract":"We propose a new test to investigate the conditional mean dependence between a response variable and the corresponding covariates in the high dimensional regimes. The test statistic is an extreme-type one built on the nonparametric method. The limiting null distribution of the proposed extreme type statistic under a mild mixing condition is established. Moreover, to make the test more powerful in general structures we propose a more general test statistic and develop its asymptotic properties. The power analysis of both methods is also considered. In real data analysis, we also propose a new way to conduct the feature screening based on our results. To evaluate the performance of our estimators and other methods, extensive simulations are conducted.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"18 7","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multivariate extreme value distributions are a common choice for modelling multivariate extremes. In high dimensions, however, the construction of flexible and parsimonious models is challenging. We propose to combine bivariate max-stable distributions into a Markov random field with respect to a tree. Although in general not max-stable itself, this Markov tree is attracted by a multivariate max-stable distribution. The latter serves as a tree-based approximation to an unknown max-stable distribution with the given bivariate distributions as margins. Given data, we learn an appropriate tree structure by Prim's algorithm with estimated pairwise upper tail dependence coefficients as edge weights. The distributions of pairs of connected variables can be fitted in various ways. The resulting tree-structured max-stable distribution allows for inference on rare event probabilities, as illustrated on river discharge data from the upper Danube basin.
{"title":"Modelling multivariate extreme value distributions via Markov trees*","authors":"Shuang Hu, Zuoxiang Peng, Johan Segers","doi":"10.1111/sjos.12698","DOIUrl":"https://doi.org/10.1111/sjos.12698","url":null,"abstract":"Multivariate extreme value distributions are a common choice for modelling multivariate extremes. In high dimensions, however, the construction of flexible and parsimonious models is challenging. We propose to combine bivariate max-stable distributions into a Markov random field with respect to a tree. Although in general not max-stable itself, this Markov tree is attracted by a multivariate max-stable distribution. The latter serves as a tree-based approximation to an unknown max-stable distribution with the given bivariate distributions as margins. Given data, we learn an appropriate tree structure by Prim's algorithm with estimated pairwise upper tail dependence coefficients as edge weights. The distributions of pairs of connected variables can be fitted in various ways. The resulting tree-structured max-stable distribution allows for inference on rare event probabilities, as illustrated on river discharge data from the upper Danube basin.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"106 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive approximations to the bias and squared bias with errors of order where is the sample size. Our results hold for a large class of estimators, including quantiles, transformations of unbiased estimators, maximum likelihood estimators in (possibly) incorrectly specified models, and functions thereof. Furthermore, we use the approximations to derive estimators of the mean squared error (MSE) which are correct to order . Since the variance of many estimators is of order , this level of precision is needed for the MSE estimator to properly take the variance into account. We also formulate a new focused information criterion (FIC) for model selection based on the estimators of the squared bias. Lastly, we illustrate the methods on data containing the number of battle deaths in all major inter-state wars between 1823 and the present day. The application illustrates the potentially large impact of using a less-accurate estimator of the squared bias.
{"title":"Accurate bias estimation with applications to focused model selection","authors":"Ingrid Dæhlen, Nils Lid Hjort, Ingrid Hobæk Haff","doi":"10.1111/sjos.12696","DOIUrl":"https://doi.org/10.1111/sjos.12696","url":null,"abstract":"We derive approximations to the bias and squared bias with errors of order <math altimg=\"urn:x-wiley:sjos:media:sjos12696:sjos12696-math-0001\" display=\"inline\" location=\"graphic/sjos12696-math-0001.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>o</mi>\u0000<mo stretchy=\"false\">(</mo>\u0000<mn>1</mn>\u0000<mo stretchy=\"false\">/</mo>\u0000<mi>n</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000</mrow>\u0000$$ oleft(1/nright) $$</annotation>\u0000</semantics></math> where <math altimg=\"urn:x-wiley:sjos:media:sjos12696:sjos12696-math-0002\" display=\"inline\" location=\"graphic/sjos12696-math-0002.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>n</mi>\u0000</mrow>\u0000$$ n $$</annotation>\u0000</semantics></math> is the sample size. Our results hold for a large class of estimators, including quantiles, transformations of unbiased estimators, maximum likelihood estimators in (possibly) incorrectly specified models, and functions thereof. Furthermore, we use the approximations to derive estimators of the mean squared error (MSE) which are correct to order <math altimg=\"urn:x-wiley:sjos:media:sjos12696:sjos12696-math-0003\" display=\"inline\" location=\"graphic/sjos12696-math-0003.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>o</mi>\u0000<mo stretchy=\"false\">(</mo>\u0000<mn>1</mn>\u0000<mo stretchy=\"false\">/</mo>\u0000<mi>n</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000</mrow>\u0000$$ oleft(1/nright) $$</annotation>\u0000</semantics></math>. Since the variance of many estimators is of order <math altimg=\"urn:x-wiley:sjos:media:sjos12696:sjos12696-math-0004\" display=\"inline\" location=\"graphic/sjos12696-math-0004.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>O</mi>\u0000<mo stretchy=\"false\">(</mo>\u0000<mn>1</mn>\u0000<mo stretchy=\"false\">/</mo>\u0000<mi>n</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000</mrow>\u0000$$ Oleft(1/nright) $$</annotation>\u0000</semantics></math>, this level of precision is needed for the MSE estimator to properly take the variance into account. We also formulate a new focused information criterion (FIC) for model selection based on the estimators of the squared bias. Lastly, we illustrate the methods on data containing the number of battle deaths in all major inter-state wars between 1823 and the present day. The application illustrates the potentially large impact of using a less-accurate estimator of the squared bias.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"5 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jinyuan Liu, Xinlian Zhang, Tuo Lin, Ruohui Chen, Yuan Zhong, Tian Chen, Tsungchin Wu, Chenyu Liu, Anna Huang, Tanya T. Nguyen, Ellen E. Lee, Dilip V. Jeste, Xin M. Tu
Abstract This article proposes a distance‐based framework incentivized by the paradigm shift towards feature aggregation for high‐dimensional data, which does not rely on the sparse‐feature assumption or the permutation‐based inference. Focusing on distance‐based outcomes that preserve information without truncating any features, a class of semiparametric regression has been developed, which encapsulates multiple sources of high‐dimensional variables using pairwise outcomes of between‐subject attributes. Further, we propose a strategy to address the interlocking correlations among pairs via the U‐statistics‐based estimating equations (UGEE), which correspond to their unique efficient influence function (EIF). Hence, the resulting semiparametric estimators are robust to distributional misspecification while enjoying root‐n consistency and asymptotic optimality to facilitate inference. In essence, the proposed approach not only circumvents information loss due to feature selection but also improves the model's interpretability and computational feasibility. Simulation studies and applications to the human microbiome and wearables data are provided, where the feature dimensions are tens of thousands. This article is protected by copyright. All rights reserved.
{"title":"A New Paradigm for High‐dimensional Data: Distance‐Based Semiparametric Feature Aggregation Framework via Between‐Subject Attributes","authors":"Jinyuan Liu, Xinlian Zhang, Tuo Lin, Ruohui Chen, Yuan Zhong, Tian Chen, Tsungchin Wu, Chenyu Liu, Anna Huang, Tanya T. Nguyen, Ellen E. Lee, Dilip V. Jeste, Xin M. Tu","doi":"10.1111/sjos.12695","DOIUrl":"https://doi.org/10.1111/sjos.12695","url":null,"abstract":"Abstract This article proposes a distance‐based framework incentivized by the paradigm shift towards feature aggregation for high‐dimensional data, which does not rely on the sparse‐feature assumption or the permutation‐based inference. Focusing on distance‐based outcomes that preserve information without truncating any features, a class of semiparametric regression has been developed, which encapsulates multiple sources of high‐dimensional variables using pairwise outcomes of between‐subject attributes. Further, we propose a strategy to address the interlocking correlations among pairs via the U‐statistics‐based estimating equations (UGEE), which correspond to their unique efficient influence function (EIF). Hence, the resulting semiparametric estimators are robust to distributional misspecification while enjoying root‐n consistency and asymptotic optimality to facilitate inference. In essence, the proposed approach not only circumvents information loss due to feature selection but also improves the model's interpretability and computational feasibility. Simulation studies and applications to the human microbiome and wearables data are provided, where the feature dimensions are tens of thousands. This article is protected by copyright. All rights reserved.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"42 s195","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135342278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We give a thorough description of the asymptotic property of the maximum likelihood estimator (MLE) of the skewness parameter of a Skew Brownian Motion (SBM). Thanks to recent results on the Central Limit Theorem of the rate of convergence of estimators for the SBM, we prove a conjecture left open that the MLE has asymptotically a mixed normal distribution involving the local time with a rate of convergence of order . We also give a series expansion of the MLE and study the asymptotic behavior of the score and its derivatives, as well as their variation with the skewness parameter. In particular, we exhibit a specific behavior when the SBM is actually a Brownian motion, and quantify the explosion of the coefficients of the expansion when the skewness parameter is close to or 1.
{"title":"Maximum likelihood estimator for skew Brownian motion: the convergence rate","authors":"Antoine Lejay, Sara Mazzonetto","doi":"10.1111/sjos.12694","DOIUrl":"https://doi.org/10.1111/sjos.12694","url":null,"abstract":"Abstract We give a thorough description of the asymptotic property of the maximum likelihood estimator (MLE) of the skewness parameter of a Skew Brownian Motion (SBM). Thanks to recent results on the Central Limit Theorem of the rate of convergence of estimators for the SBM, we prove a conjecture left open that the MLE has asymptotically a mixed normal distribution involving the local time with a rate of convergence of order . We also give a series expansion of the MLE and study the asymptotic behavior of the score and its derivatives, as well as their variation with the skewness parameter. In particular, we exhibit a specific behavior when the SBM is actually a Brownian motion, and quantify the explosion of the coefficients of the expansion when the skewness parameter is close to or 1.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"11 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135874775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we modify the Bayes risk for the expectile, the so‐called variantile risk measure, to better capture extreme risks. The modified risk measure is called the adjusted standard‐deviatile. First, we derive the asymptotic expansions of the adjusted standard‐deviatile. Next, based on the first‐order asymptotic expansion, we propose two efficient estimation methods for the adjusted standard‐deviatile at intermediate and extreme levels. By using techniques from extreme value theory, the asymptotic normality is proved for both estimators for independent and identically distributed observations and for ‐mixing time series, respectively. Simulations and real data applications are conducted to examine the performance of the proposed estimators.
{"title":"Estimation of the Adjusted Standard‐deviatile for Extreme Risks","authors":"Haoyu Chen, Tiantian Mao, Fan Yang","doi":"10.1111/sjos.12693","DOIUrl":"https://doi.org/10.1111/sjos.12693","url":null,"abstract":"Abstract In this paper, we modify the Bayes risk for the expectile, the so‐called variantile risk measure, to better capture extreme risks. The modified risk measure is called the adjusted standard‐deviatile. First, we derive the asymptotic expansions of the adjusted standard‐deviatile. Next, based on the first‐order asymptotic expansion, we propose two efficient estimation methods for the adjusted standard‐deviatile at intermediate and extreme levels. By using techniques from extreme value theory, the asymptotic normality is proved for both estimators for independent and identically distributed observations and for ‐mixing time series, respectively. Simulations and real data applications are conducted to examine the performance of the proposed estimators.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"23 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135461959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper introduces a Nearly Unstable INteger‐valued AutoRegressive Conditional Heteroscedastic (NU‐INARCH) process for dealing with count time series data. It is proved that a proper normalization of the NU‐INARCH process weakly converges to a Cox–Ingersoll–Ross diffusion in the Skorohod topology. The asymptotic distribution of the conditional least squares estimator of the correlation parameter is established as a functional of certain stochastic integrals. Numerical experiments based on Monte Carlo simulations are provided to verify the behavior of the asymptotic distribution under finite samples. These simulations reveal that the nearly unstable approach provides satisfactory and better results than those based on the stationarity assumption even when the true process is not that close to nonstationarity. A unit root test is proposed and its Type‐I error and power are examined via Monte Carlo simulations. As an illustration, the proposed methodology is applied to the daily number of deaths due to COVID‐19 in the United Kingdom.
{"title":"Nearly Unstable Integer‐Valued ARCH Process and Unit Root Testing","authors":"Wagner Barreto-Souza, Ngai Hang Chan","doi":"10.1111/sjos.12689","DOIUrl":"https://doi.org/10.1111/sjos.12689","url":null,"abstract":"Abstract This paper introduces a Nearly Unstable INteger‐valued AutoRegressive Conditional Heteroscedastic (NU‐INARCH) process for dealing with count time series data. It is proved that a proper normalization of the NU‐INARCH process weakly converges to a Cox–Ingersoll–Ross diffusion in the Skorohod topology. The asymptotic distribution of the conditional least squares estimator of the correlation parameter is established as a functional of certain stochastic integrals. Numerical experiments based on Monte Carlo simulations are provided to verify the behavior of the asymptotic distribution under finite samples. These simulations reveal that the nearly unstable approach provides satisfactory and better results than those based on the stationarity assumption even when the true process is not that close to nonstationarity. A unit root test is proposed and its Type‐I error and power are examined via Monte Carlo simulations. As an illustration, the proposed methodology is applied to the daily number of deaths due to COVID‐19 in the United Kingdom.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135666555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：