Summary We show that least squares cross-validation methods share a common structure which has an explicit asymptotic solution, when the chosen kernel is asymptotically separable in bandwidth and data. For density estimation with a multivariate Student t(ν) kernel, the cross-validation criterion becomes asymptotically equivalent to a polynomial of only three terms. Our bandwidth formulae are simple and noniterative thus leading to very fast computations, their integrated squared-error dominates traditional cross-validation implementations, they alleviate the notorious sample variability of cross-validation, and overcome its breakdown in the case of repeated observations. We illustrate our method with univariate and bivariate applications, of density estimation and nonparametric regressions, to a large dataset of Michigan State University academic wages and experience.
{"title":"Explicit solutions for the asymptotically-optimal bandwidth in cross-validation","authors":"Karim M Abadir, Michel Lubrano","doi":"10.1093/biomet/asae007","DOIUrl":"https://doi.org/10.1093/biomet/asae007","url":null,"abstract":"Summary We show that least squares cross-validation methods share a common structure which has an explicit asymptotic solution, when the chosen kernel is asymptotically separable in bandwidth and data. For density estimation with a multivariate Student t(ν) kernel, the cross-validation criterion becomes asymptotically equivalent to a polynomial of only three terms. Our bandwidth formulae are simple and noniterative thus leading to very fast computations, their integrated squared-error dominates traditional cross-validation implementations, they alleviate the notorious sample variability of cross-validation, and overcome its breakdown in the case of repeated observations. We illustrate our method with univariate and bivariate applications, of density estimation and nonparametric regressions, to a large dataset of Michigan State University academic wages and experience.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":"3 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary While researchers commonly use the bootstrap to quantify the uncertainty of an estimator, it has been noticed that the standard bootstrap, in general, does not work for Chatterjee's rank correlation. In this paper, we provide proof of this issue under an additional independence assumption, and complement our theory with simulation evidence for general settings. Chatterjee's rank correlation thus falls into a category of statistics that are asymptotically normal but bootstrap inconsistent. Valid inferential methods in this case are Chatterjee's original proposal for testing independence and Lin & Han (2022) 's analytic asymptotic variance estimator for more general purposes.
摘要 虽然研究人员通常使用引导法来量化估计器的不确定性，但人们注意到标准引导法一般不适用于查特吉秩相关。在本文中，我们在额外的独立性假设下证明了这一问题，并用一般情况下的模拟证据补充了我们的理论。因此，查特吉秩相关属于渐近正态但自举不一致的统计类别。在这种情况下，有效的推论方法是 Chatterjee 最初提出的用于检验独立性的方法，以及 Lin & Han (2022) 用于更一般目的的解析渐近方差估计器。
{"title":"On the failure of the bootstrap for Chatterjee's rank correlation","authors":"Zhexiao Lin, Fang Han","doi":"10.1093/biomet/asae004","DOIUrl":"https://doi.org/10.1093/biomet/asae004","url":null,"abstract":"Summary While researchers commonly use the bootstrap to quantify the uncertainty of an estimator, it has been noticed that the standard bootstrap, in general, does not work for Chatterjee's rank correlation. In this paper, we provide proof of this issue under an additional independence assumption, and complement our theory with simulation evidence for general settings. Chatterjee's rank correlation thus falls into a category of statistics that are asymptotically normal but bootstrap inconsistent. Valid inferential methods in this case are Chatterjee's original proposal for testing independence and Lin & Han (2022) 's analytic asymptotic variance estimator for more general purposes.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":"125 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary Estimation of the time-average variance constant is important for statistical analyses involving dependent data. This problem is difficult as it relies on a bandwidth parameter. Specifically, the optimal choices of the bandwidths of all existing estimators depend on the estimand itself and another unknown parameter which is very difficult to estimate. Thus, optimal variance estimation is unachievable. In this paper, we introduce a concept of converging flat-top kernels for constructing variance estimators whose optimal bandwidths are free of unknown parameters asymptotically and hence can be computed easily. We prove that the new estimator has an asymptotically constant risk and is locally asymptotically minimax.
{"title":"Asymptotically constant risk estimator of the time-average variance constant","authors":"K W Chan, C Y Yau","doi":"10.1093/biomet/asae003","DOIUrl":"https://doi.org/10.1093/biomet/asae003","url":null,"abstract":"Summary Estimation of the time-average variance constant is important for statistical analyses involving dependent data. This problem is difficult as it relies on a bandwidth parameter. Specifically, the optimal choices of the bandwidths of all existing estimators depend on the estimand itself and another unknown parameter which is very difficult to estimate. Thus, optimal variance estimation is unachievable. In this paper, we introduce a concept of converging flat-top kernels for constructing variance estimators whose optimal bandwidths are free of unknown parameters asymptotically and hence can be computed easily. We prove that the new estimator has an asymptotically constant risk and is locally asymptotically minimax.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":"16 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139678925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary We present a result according to which certain functions of covariance matrices are maximized at scalar multiples of the identity matrix. This is used to show that experimental designs that are optimal under an assumption of independent, homoscedastic responses can be minimax robust, in broad classes of alternate covariance structures. In particular it can justify the common practice of disregarding possible dependence, or heteroscedasticity, at the design stage of an experiment.
{"title":"A note on minimax robustness of designs against correlated or heteroscedastic responses","authors":"D P Wiens","doi":"10.1093/biomet/asae001","DOIUrl":"https://doi.org/10.1093/biomet/asae001","url":null,"abstract":"Summary We present a result according to which certain functions of covariance matrices are maximized at scalar multiples of the identity matrix. This is used to show that experimental designs that are optimal under an assumption of independent, homoscedastic responses can be minimax robust, in broad classes of alternate covariance structures. In particular it can justify the common practice of disregarding possible dependence, or heteroscedasticity, at the design stage of an experiment.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":"47 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139506157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A new efficient nonparametric estimator for Toeplitz covariance matrices is proposed. This estimator is based on a data transformation that translates the problem of Toeplitz covariance matrix estimation to the problem of mean estimation in an approximate Gaussian regression. The resulting Toeplitz covariance matrix estimator is positive definite by construction, fully data-driven and computationally very fast. Moreover, this estimator is shown to be minimax optimal under the spectral norm for a large class of Toeplitz matrices. These results are readily extended to estimation of inverses of Toeplitz covariance matrices. Also, an alternative version of the Whittle likelihood for the spectral density based on the discrete cosine transform is proposed. The method is implemented in the R package vstdct that accompanies the paper.
本文提出了一种新的高效托普利兹协方差矩阵非参数估计器。该估计器基于数据转换，将托普利兹协方差矩阵估计问题转化为近似高斯回归中的均值估计问题。由此产生的托普利兹协方差矩阵估计器在构造上是正定的，完全由数据驱动，计算速度非常快。此外，对于一大类 Toeplitz 矩阵，该估计器在谱规范下是最小最优的。这些结果很容易扩展到对托普利兹协方差矩阵逆的估计。此外，还提出了基于离散余弦变换的谱密度惠特尔似然法的替代版本。本文附带的 R 软件包 vstdct 实现了该方法。
{"title":"Efficient nonparametric estimation of Toeplitz covariance matrices","authors":"K Klockmann, T Krivobokova","doi":"10.1093/biomet/asae002","DOIUrl":"https://doi.org/10.1093/biomet/asae002","url":null,"abstract":"A new efficient nonparametric estimator for Toeplitz covariance matrices is proposed. This estimator is based on a data transformation that translates the problem of Toeplitz covariance matrix estimation to the problem of mean estimation in an approximate Gaussian regression. The resulting Toeplitz covariance matrix estimator is positive definite by construction, fully data-driven and computationally very fast. Moreover, this estimator is shown to be minimax optimal under the spectral norm for a large class of Toeplitz matrices. These results are readily extended to estimation of inverses of Toeplitz covariance matrices. Also, an alternative version of the Whittle likelihood for the spectral density based on the discrete cosine transform is proposed. The method is implemented in the R package vstdct that accompanies the paper.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":"13 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139506380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate a class of methods for selective inference that condition on a selection event. Such methods follow a two-stage process. First, a data-driven collection of hypotheses is chosen from some large universe of hypotheses. Subsequently, inference takes place within this data-driven collection, conditioned on the information that was used for the selection. Examples of such methods include basic data splitting, as well as modern data carving methods and post-selection inference methods for lasso coefficients based on the polyhedral lemma. In this paper, we adopt a holistic view on such methods, considering the selection, conditioning, and final error control steps together as a single method. From this perspective, we demonstrate that multiple testing methods defined directly on the full universe of hypotheses are always at least as powerful as selective inference methods based on selection and conditioning. This result holds true even when the universe is potentially infinite and only implicitly defined, such as in the case of data splitting. We give general theory and intuitions before investigating in detail several case studies where a shift to a non-selective or unconditional perspective can yield a power gain.
{"title":"On Selecting and Conditioning in Multiple Testing and Selective Inference","authors":"Jelle J Goeman, Aldo Solari","doi":"10.1093/biomet/asad078","DOIUrl":"https://doi.org/10.1093/biomet/asad078","url":null,"abstract":"We investigate a class of methods for selective inference that condition on a selection event. Such methods follow a two-stage process. First, a data-driven collection of hypotheses is chosen from some large universe of hypotheses. Subsequently, inference takes place within this data-driven collection, conditioned on the information that was used for the selection. Examples of such methods include basic data splitting, as well as modern data carving methods and post-selection inference methods for lasso coefficients based on the polyhedral lemma. In this paper, we adopt a holistic view on such methods, considering the selection, conditioning, and final error control steps together as a single method. From this perspective, we demonstrate that multiple testing methods defined directly on the full universe of hypotheses are always at least as powerful as selective inference methods based on selection and conditioning. This result holds true even when the universe is potentially infinite and only implicitly defined, such as in the case of data splitting. We give general theory and intuitions before investigating in detail several case studies where a shift to a non-selective or unconditional perspective can yield a power gain.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":"94 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139051164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary Subgraph counts, in particular the number of occurrences of small shapes such as triangles, characterize properties of random networks. As a result, they have seen wide use as network summary statistics. Subgraphs are typically counted globally, making existing approaches unable to describe vertex-specific characteristics. In contrast, rooted subgraphs focus on vertex neighbourhoods, and are fundamental descriptors of local network properties. We derive the asymptotic joint distribution of rooted subgraph counts in inhomogeneous random graphs, a model which generalizes most statistical network models. This result enables a shift in the statistical analysis of graphs, from estimating network summaries, to estimating models linking local network structure and vertex-specific covariates. As an example, we consider a school friendship network and show that gender and race are significant predictors of local friendship patterns.
{"title":"Central limit theorems for local network statistics","authors":"P A Maugis","doi":"10.1093/biomet/asad080","DOIUrl":"https://doi.org/10.1093/biomet/asad080","url":null,"abstract":"Summary Subgraph counts, in particular the number of occurrences of small shapes such as triangles, characterize properties of random networks. As a result, they have seen wide use as network summary statistics. Subgraphs are typically counted globally, making existing approaches unable to describe vertex-specific characteristics. In contrast, rooted subgraphs focus on vertex neighbourhoods, and are fundamental descriptors of local network properties. We derive the asymptotic joint distribution of rooted subgraph counts in inhomogeneous random graphs, a model which generalizes most statistical network models. This result enables a shift in the statistical analysis of graphs, from estimating network summaries, to estimating models linking local network structure and vertex-specific covariates. As an example, we consider a school friendship network and show that gender and race are significant predictors of local friendship patterns.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":"178 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139051073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Quality control charts aim at raising an alarm as soon as sequentially obtained observations of an underlying random process no longer seem to be within stochastic fluctuations prescribed by an ‘in-control’ scenario. Such random processes can often be modelled using the concept of stationarity, or even independence as in most classical works. An important out-of-control scenario is the changepoint alternative, for which the distribution of the process changes at an unknown point in time. In his seminal 1954 Biometrika paper, E. S. Page introduced the famous cumulative sum control charts for changepoint monitoring. Innovatively, decision rules based on cumulative sum procedures took the full history of the process into account, whereas previous procedures were based only on a fixed and typically small number of the most recent observations. The extreme case of using only the most recent observation, often referred to as the Shewhart chart, is more akin to serial outlier than changepoint detection. Page’s cumulative sum approach, introduced seven decades ago, is ubiquitous in modern changepoint analysis, and his original paper has led to a multitude of follow-up papers in different research communities. This review is focused on a particular subfield of this research, namely nonparametric sequential, or online, changepoint tests which are constructed to maintain a desired Type 1 error as opposed to the more traditional approach seeking to minimize the average run length of the procedures. Such tests have originated at the intersection of econometrics and statistics. We trace the development of these tests and highlight their properties, mostly using a simple location model for clarity of exposition, but also review more complex situations such as regression and time series models.
质量控制图的目的是，一旦连续获得的底层随机过程的观测结果似乎不再符合 "在控 "方案所规定的随机波动范围，就会发出警报。此类随机过程通常可以使用静止概念建模，甚至可以使用大多数经典著作中的独立概念建模。一个重要的失控情景是变化点替代方案，即过程的分布在一个未知的时间点发生变化。E. S. Page 在 1954 年发表的开创性论文《Biometrika》中，提出了著名的用于变化点监控的累积和控制图。创新性的是，基于累积总和程序的决策规则考虑到了整个过程的历史，而以前的程序仅基于固定的、通常为数不多的最新观测数据。仅使用最近观测值的极端情况通常被称为休哈特图表，它更类似于序列离群值，而非变化点检测。佩奇在七十年前提出的累积和方法在现代变化点分析中无处不在，他的原始论文在不同研究领域引发了大量后续论文。本综述的重点是这一研究的一个特殊子领域，即非参数序列或在线变化点检验，其构建目的是保持理想的 1 类误差，而不是寻求最小化程序平均运行长度的传统方法。这类检验起源于计量经济学和统计学的交叉学科。我们追溯了这些检验的发展历程，并强调了它们的特性，为了论述清晰，我们主要使用了简单的位置模型，但也回顾了回归和时间序列模型等更复杂的情况。
{"title":"The state of cumulative sum sequential change point testing seventy years after Page","authors":"Alexander Aue, Claudia Kirch","doi":"10.1093/biomet/asad079","DOIUrl":"https://doi.org/10.1093/biomet/asad079","url":null,"abstract":"\u0000 Quality control charts aim at raising an alarm as soon as sequentially obtained observations of an underlying random process no longer seem to be within stochastic fluctuations prescribed by an ‘in-control’ scenario. Such random processes can often be modelled using the concept of stationarity, or even independence as in most classical works. An important out-of-control scenario is the changepoint alternative, for which the distribution of the process changes at an unknown point in time. In his seminal 1954 Biometrika paper, E. S. Page introduced the famous cumulative sum control charts for changepoint monitoring. Innovatively, decision rules based on cumulative sum procedures took the full history of the process into account, whereas previous procedures were based only on a fixed and typically small number of the most recent observations. The extreme case of using only the most recent observation, often referred to as the Shewhart chart, is more akin to serial outlier than changepoint detection. Page’s cumulative sum approach, introduced seven decades ago, is ubiquitous in modern changepoint analysis, and his original paper has led to a multitude of follow-up papers in different research communities. This review is focused on a particular subfield of this research, namely nonparametric sequential, or online, changepoint tests which are constructed to maintain a desired Type 1 error as opposed to the more traditional approach seeking to minimize the average run length of the procedures. Such tests have originated at the intersection of econometrics and statistics. We trace the development of these tests and highlight their properties, mostly using a simple location model for clarity of exposition, but also review more complex situations such as regression and time series models.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":"12 3","pages":""},"PeriodicalIF":2.7,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138951837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Correction to: ‘A cross-validation-based statistical theory for point processes’","authors":"","doi":"10.1093/biomet/asad077","DOIUrl":"https://doi.org/10.1093/biomet/asad077","url":null,"abstract":"","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":"45 2","pages":""},"PeriodicalIF":2.7,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139169267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary Phylogenetic association analysis plays a crucial role in investigating the correlation between microbial compositions and specific outcomes of interest in microbiome studies. However, existing methods for testing such associations have limitations related to the assumption of a linear association in high-dimensional settings and the handling of confounding effects. Therefore, there is a need for methods capable of characterizing complex associations, including nonmonotonic relationships. This paper introduces a novel phylogenetic association analysis framework and associated tests to address these challenges by employing conditional rank correlation as a measure of association. These tests account for confounders in a fully nonparametric manner, ensuring robustness against outliers and the ability to detect diverse dependencies. The proposed framework aggregates conditional rank correlations for subtrees using a weighted sum and maximum approach to capture both dense and sparse signals. The significance level of the test statistics is determined by calibrating through a nearest neighbour bootstrapping method, which is straightforward to implement and can accommodate additional datasets when available. The practical advantages of the proposed framework are demonstrated through numerical experiments utilizing both simulated and real microbiome datasets.
{"title":"Phylogenetic Association Analysis with Conditional Rank Correlation","authors":"Shulei Wang, Bo Yuan, T Tony Cai, Hongzhe Li","doi":"10.1093/biomet/asad075","DOIUrl":"https://doi.org/10.1093/biomet/asad075","url":null,"abstract":"Summary Phylogenetic association analysis plays a crucial role in investigating the correlation between microbial compositions and specific outcomes of interest in microbiome studies. However, existing methods for testing such associations have limitations related to the assumption of a linear association in high-dimensional settings and the handling of confounding effects. Therefore, there is a need for methods capable of characterizing complex associations, including nonmonotonic relationships. This paper introduces a novel phylogenetic association analysis framework and associated tests to address these challenges by employing conditional rank correlation as a measure of association. These tests account for confounders in a fully nonparametric manner, ensuring robustness against outliers and the ability to detect diverse dependencies. The proposed framework aggregates conditional rank correlations for subtrees using a weighted sum and maximum approach to capture both dense and sparse signals. The significance level of the test statistics is determined by calibrating through a nearest neighbour bootstrapping method, which is straightforward to implement and can accommodate additional datasets when available. The practical advantages of the proposed framework are demonstrated through numerical experiments utilizing both simulated and real microbiome datasets.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":"15 2","pages":""},"PeriodicalIF":2.7,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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