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Individualized Treatment Allocations with Distributional Welfare 个体化治疗分配与分配福利
arXiv - MATH - Statistics Theory Pub Date : 2023-11-27 DOI: arxiv-2311.15878
Yifan Cui, Sukjin Han
{"title":"Individualized Treatment Allocations with Distributional Welfare","authors":"Yifan Cui, Sukjin Han","doi":"arxiv-2311.15878","DOIUrl":"https://doi.org/arxiv-2311.15878","url":null,"abstract":"In this paper, we explore optimal treatment allocation policies that target\u0000distributional welfare. Most literature on treatment choice has considered\u0000utilitarian welfare based on the conditional average treatment effect (ATE).\u0000While average welfare is intuitive, it may yield undesirable allocations\u0000especially when individuals are heterogeneous (e.g., with outliers) - the very\u0000reason individualized treatments were introduced in the first place. This\u0000observation motivates us to propose an optimal policy that allocates the\u0000treatment based on the conditional emph{quantile of individual treatment\u0000effects} (QoTE). Depending on the choice of the quantile probability, this\u0000criterion can accommodate a policymaker who is either prudent or negligent. The\u0000challenge of identifying the QoTE lies in its requirement for knowledge of the\u0000joint distribution of the counterfactual outcomes, which is generally hard to\u0000recover even with experimental data. Therefore, we introduce minimax optimal\u0000policies that are robust to model uncertainty. We then propose a range of\u0000identifying assumptions under which we can point or partially identify the\u0000QoTE. We establish the asymptotic bound on the regret of implementing the\u0000proposed policies. We consider both stochastic and deterministic rules. In\u0000simulations and two empirical applications, we compare optimal decisions based\u0000on the QoTE with decisions based on other criteria.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526139","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}
引用次数: 0
Bayesian Approach to Linear Bayesian Networks 线性贝叶斯网络的贝叶斯方法
arXiv - MATH - Statistics Theory Pub Date : 2023-11-27 DOI: arxiv-2311.15610
Seyong Hwang, Kyoungjae Lee, Sunmin Oh, Gunwoong Park
{"title":"Bayesian Approach to Linear Bayesian Networks","authors":"Seyong Hwang, Kyoungjae Lee, Sunmin Oh, Gunwoong Park","doi":"arxiv-2311.15610","DOIUrl":"https://doi.org/arxiv-2311.15610","url":null,"abstract":"This study proposes the first Bayesian approach for learning high-dimensional\u0000linear Bayesian networks. The proposed approach iteratively estimates each\u0000element of the topological ordering from backward and its parent using the\u0000inverse of a partial covariance matrix. The proposed method successfully\u0000recovers the underlying structure when Bayesian regularization for the inverse\u0000covariance matrix with unequal shrinkage is applied. Specifically, it shows\u0000that the number of samples $n = Omega( d_M^2 log p)$ and $n = Omega(d_M^2\u0000p^{2/m})$ are sufficient for the proposed algorithm to learn linear Bayesian\u0000networks with sub-Gaussian and 4m-th bounded-moment error distributions,\u0000respectively, where $p$ is the number of nodes and $d_M$ is the maximum degree\u0000of the moralized graph. The theoretical findings are supported by extensive\u0000simulation studies including real data analysis. Furthermore the proposed\u0000method is demonstrated to outperform state-of-the-art frequentist approaches,\u0000such as the BHLSM, LISTEN, and TD algorithms in synthetic data.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"62 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526138","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}
引用次数: 0
Pseudo-likelihood Estimators for Graphical Models: Existence and Uniqueness 图模型的伪似然估计:存在性与唯一性
arXiv - MATH - Statistics Theory Pub Date : 2023-11-27 DOI: arxiv-2311.15528
Benjamin Roycraft, Bala Rajaratnam
{"title":"Pseudo-likelihood Estimators for Graphical Models: Existence and Uniqueness","authors":"Benjamin Roycraft, Bala Rajaratnam","doi":"arxiv-2311.15528","DOIUrl":"https://doi.org/arxiv-2311.15528","url":null,"abstract":"Graphical and sparse (inverse) covariance models have found widespread use in\u0000modern sample-starved high dimensional applications. A part of their wide\u0000appeal stems from the significantly low sample sizes required for the existence\u0000of estimators, especially in comparison with the classical full covariance\u0000model. For undirected Gaussian graphical models, the minimum sample size\u0000required for the existence of maximum likelihood estimators had been an open\u0000question for almost half a century, and has been recently settled. The very\u0000same question for pseudo-likelihood estimators has remained unsolved ever since\u0000their introduction in the '70s. Pseudo-likelihood estimators have recently\u0000received renewed attention as they impose fewer restrictive assumptions and\u0000have better computational tractability, improved statistical performance, and\u0000appropriateness in modern high dimensional applications, thus renewing interest\u0000in this longstanding problem. In this paper, we undertake a comprehensive study\u0000of this open problem within the context of the two classes of pseudo-likelihood\u0000methods proposed in the literature. We provide a precise answer to this\u0000question for both pseudo-likelihood approaches and relate the corresponding\u0000solutions to their Gaussian counterpart.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"34 9","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526166","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}
引用次数: 0
Low-Degree Hardness of Detection for Correlated Erdős-Rényi Graphs 关联Erdős-Rényi图的低度检测硬度
arXiv - MATH - Statistics Theory Pub Date : 2023-11-27 DOI: arxiv-2311.15931
Jian Ding, Hang Du, Zhangsong Li
{"title":"Low-Degree Hardness of Detection for Correlated Erdős-Rényi Graphs","authors":"Jian Ding, Hang Du, Zhangsong Li","doi":"arxiv-2311.15931","DOIUrl":"https://doi.org/arxiv-2311.15931","url":null,"abstract":"Given two ErdH{o}s-R'enyi graphs with $n$ vertices whose edges are\u0000correlated through a latent vertex correspondence, we study complexity lower\u0000bounds for the associated correlation detection problem for the class of\u0000low-degree polynomial algorithms. We provide evidence that any\u0000degree-$O(rho^{-1})$ polynomial algorithm fails for detection, where $rho$ is\u0000the edge correlation. Furthermore, in the sparse regime where the edge density\u0000$q=n^{-1+o(1)}$, we provide evidence that any degree-$d$ polynomial algorithm\u0000fails for detection, as long as $log d=obig( frac{log n}{log nq} wedge\u0000sqrt{log n} big)$ and the correlation $rho<sqrt{alpha}$ where\u0000$alphaapprox 0.338$ is the Otter's constant. Our result suggests that several\u0000state-of-the-art algorithms on correlation detection and exact matching\u0000recovery may be essentially the best possible.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526168","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}
引用次数: 0
The Local Landscape of Phase Retrieval Under Limited Samples 有限样本条件下相位检索的局部图景
arXiv - MATH - Statistics Theory Pub Date : 2023-11-26 DOI: arxiv-2311.15221
Kaizhao Liu, Zihao Wang, Lei Wu
{"title":"The Local Landscape of Phase Retrieval Under Limited Samples","authors":"Kaizhao Liu, Zihao Wang, Lei Wu","doi":"arxiv-2311.15221","DOIUrl":"https://doi.org/arxiv-2311.15221","url":null,"abstract":"In this paper, we provide a fine-grained analysis of the local landscape of\u0000phase retrieval under the regime with limited samples. Our aim is to ascertain\u0000the minimal sample size necessary to guarantee a benign local landscape\u0000surrounding global minima in high dimensions. Let $n$ and $d$ denote the sample\u0000size and input dimension, respectively. We first explore the local convexity\u0000and establish that when $n=o(dlog d)$, for almost every fixed point in the\u0000local ball, the Hessian matrix must have negative eigenvalues as long as $d$ is\u0000sufficiently large. Consequently, the local landscape is highly non-convex. We\u0000next consider the one-point strong convexity and show that as long as\u0000$n=omega(d)$, with high probability, the landscape is one-point strongly\u0000convex in the local annulus: ${winmathbb{R}^d: o_d(1)leqslant\u0000|w-w^*|leqslant c}$, where $w^*$ is the ground truth and $c$ is an absolute\u0000constant. This implies that gradient descent initialized from any point in this\u0000domain can converge to an $o_d(1)$-loss solution exponentially fast.\u0000Furthermore, we show that when $n=o(dlog d)$, there is a radius of\u0000$widetildeThetaleft(sqrt{1/d}right)$ such that one-point convexity breaks\u0000in the corresponding smaller local ball. This indicates an impossibility to\u0000establish a convergence to exact $w^*$ for gradient descent under limited\u0000samples by relying solely on one-point convexity.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526137","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}
引用次数: 0
Characterization of valid auxiliary functions for representations of extreme value distributions and their max-domains of attraction 表示极值分布及其最大吸引域的有效辅助函数的表征
arXiv - MATH - Statistics Theory Pub Date : 2023-11-26 DOI: arxiv-2311.15355
Miriam Isabel Seifert
{"title":"Characterization of valid auxiliary functions for representations of extreme value distributions and their max-domains of attraction","authors":"Miriam Isabel Seifert","doi":"arxiv-2311.15355","DOIUrl":"https://doi.org/arxiv-2311.15355","url":null,"abstract":"In this paper we study two important representations for extreme value\u0000distributions and their max-domains of attraction (MDA), namely von Mises\u0000representation (vMR) and variation representation (VR), which are convenient\u0000ways to gain limit results. Both VR and vMR are defined via so-called auxiliary\u0000functions psi. Up to now, however, the set of valid auxiliary functions for vMR\u0000has neither been characterized completely nor separated from those for VR. We\u0000contribute to the current literature by introducing ''universal'' auxiliary\u0000functions which are valid for both VR and vMR representations for the entire\u0000MDA distribution families. Then we identify exactly the sets of valid auxiliary\u0000functions for both VR and vMR. Moreover, we propose a method for finding\u0000appropriate auxiliary functions with analytically simple structure and provide\u0000them for several important distributions.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526154","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}
引用次数: 0
Quickest Change Detection with Post-Change Density Estimation 最快的变化检测与后变化密度估计
arXiv - MATH - Statistics Theory Pub Date : 2023-11-25 DOI: arxiv-2311.15128
Yuchen Liang, Venugopal V. Veeravalli
{"title":"Quickest Change Detection with Post-Change Density Estimation","authors":"Yuchen Liang, Venugopal V. Veeravalli","doi":"arxiv-2311.15128","DOIUrl":"https://doi.org/arxiv-2311.15128","url":null,"abstract":"The problem of quickest change detection in a sequence of independent\u0000observations is considered. The pre-change distribution is assumed to be known,\u0000while the post-change distribution is unknown. Two tests based on post-change\u0000density estimation are developed for this problem, the window-limited\u0000non-parametric generalized likelihood ratio (NGLR) CuSum test and the\u0000non-parametric window-limited adaptive (NWLA) CuSum test. Both tests do not\u0000assume any knowledge of the post-change distribution, except that the\u0000post-change density satisfies certain smoothness conditions that allows for\u0000efficient non-parametric estimation. Also, they do not require any\u0000pre-collected post-change training samples. Under certain convergence\u0000conditions on the density estimator, it is shown that both tests are\u0000first-order asymptotically optimal, as the false alarm rate goes to zero. The\u0000analysis is validated through numerical results, where both tests are compared\u0000with baseline tests that have distributional knowledge.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"83 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138521329","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}
引用次数: 0
Moment-Type Estimators for the Dirichlet and the Multivariate Gamma Distributions Dirichlet分布和多元伽玛分布的矩型估计
arXiv - MATH - Statistics Theory Pub Date : 2023-11-25 DOI: arxiv-2311.15025
Ioannis Oikonomidis, Samis Trevezas
{"title":"Moment-Type Estimators for the Dirichlet and the Multivariate Gamma Distributions","authors":"Ioannis Oikonomidis, Samis Trevezas","doi":"arxiv-2311.15025","DOIUrl":"https://doi.org/arxiv-2311.15025","url":null,"abstract":"This study presents new closed-form estimators for the Dirichlet and the\u0000Multivariate Gamma distribution families, whose maximum likelihood estimator\u0000cannot be explicitly derived. The methodology builds upon the score-adjusted\u0000estimators for the Beta and Gamma distributions, extending their applicability\u0000to the Dirichlet and Multivariate Gamma distributions. Expressions for the\u0000asymptotic variance-covariance matrices are provided, demonstrating the\u0000superior performance of score-adjusted estimators over the traditional moment\u0000ones. Leveraging well-established connections between Dirichlet and\u0000Multivariate Gamma distributions, a novel class of estimators for the latter is\u0000introduced, referred to as \"Dirichlet-based moment-type estimators\". The\u0000general asymptotic variance-covariance matrix form for this estimator class is\u0000derived. To facilitate the application of these innovative estimators, an R\u0000package called estimators is developed and made publicly available.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526184","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}
引用次数: 0
Kernel-based measures of association between inputs and outputs based on ANOVA 基于方差分析的输入和输出之间关联的基于核的度量
arXiv - MATH - Statistics Theory Pub Date : 2023-11-25 DOI: arxiv-2311.14894
Matieyendou Lamboni
{"title":"Kernel-based measures of association between inputs and outputs based on ANOVA","authors":"Matieyendou Lamboni","doi":"arxiv-2311.14894","DOIUrl":"https://doi.org/arxiv-2311.14894","url":null,"abstract":"ANOVA decomposition of function with random input variables provides ANOVA\u0000functionals (AFs), which contain information about the contributions of the\u0000input variables on the output variable(s). By embedding AFs into an appropriate\u0000reproducing kernel Hilbert space regarding their distributions, we propose an\u0000efficient statistical test of independence between the input variables and\u0000output variable(s). The resulting test statistic leads to new dependent\u0000measures of association between inputs and outputs that allow for i) dealing\u0000with any distribution of AFs, including the Cauchy distribution, ii) accounting\u0000for the necessary or desirable moments of AFs and the interactions among the\u0000input variables. In uncertainty quantification for mathematical models, a\u0000number of existing measures are special cases of this framework. We then\u0000provide unified and general global sensitivity indices and their consistent\u0000estimators, including asymptotic distributions. For Gaussian-distributed AFs,\u0000we obtain Sobol' indices and dependent generalized sensitivity indices using\u0000quadratic kernels.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526142","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}
引用次数: 0
An Identification and Dimensionality Robust Test for Instrumental Variables Models 工具变量模型的识别和量纲稳健性检验
arXiv - MATH - Statistics Theory Pub Date : 2023-11-25 DOI: arxiv-2311.14892
Manu Navjeevan
{"title":"An Identification and Dimensionality Robust Test for Instrumental Variables Models","authors":"Manu Navjeevan","doi":"arxiv-2311.14892","DOIUrl":"https://doi.org/arxiv-2311.14892","url":null,"abstract":"I propose a new identification-robust test for the structural parameter in a\u0000heteroskedastic linear instrumental variables model. The proposed test\u0000statistic is similar in spirit to a jackknife version of the K-statistic and\u0000the resulting test has exact asymptotic size so long as an auxiliary parameter\u0000can be consistently estimated. This is possible under approximate sparsity even\u0000when the number of instruments is much larger than the sample size. As the\u0000number of instruments is allowed, but not required, to be large, the limiting\u0000behavior of the test statistic is difficult to examine via existing central\u0000limit theorems. Instead, I derive the asymptotic chi-squared distribution of\u0000the test statistic using a direct Gaussian approximation technique. To improve\u0000power against certain alternatives, I propose a simple combination with the\u0000sup-score statistic of Belloni et al. (2012) based on a thresholding rule. I\u0000demonstrate favorable size control and power properties in a simulation study\u0000and apply the new methods to revisit the effect of social spillovers in movie\u0000consumption.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"EM-13 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526172","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}
引用次数: 0
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