We develop a theory of high-arity PAC learning, which is statistical learning in the presence of "structured correlation". In this theory, hypotheses are either graphs, hypergraphs or, more generally, structures in finite relational languages, and i.i.d. sampling is replaced by sampling an induced substructure, producing an exchangeable distribution. We prove a high-arity version of the fundamental theorem of statistical learning by characterizing high-arity (agnostic) PAC learnability in terms of finiteness of a purely combinatorial dimension and in terms of an appropriate version of uniform convergence.
{"title":"High-arity PAC learning via exchangeability","authors":"Leonardo N. Coregliano, Maryanthe Malliaris","doi":"arxiv-2402.14294","DOIUrl":"https://doi.org/arxiv-2402.14294","url":null,"abstract":"We develop a theory of high-arity PAC learning, which is statistical learning\u0000in the presence of \"structured correlation\". In this theory, hypotheses are\u0000either graphs, hypergraphs or, more generally, structures in finite relational\u0000languages, and i.i.d. sampling is replaced by sampling an induced substructure,\u0000producing an exchangeable distribution. We prove a high-arity version of the\u0000fundamental theorem of statistical learning by characterizing high-arity\u0000(agnostic) PAC learnability in terms of finiteness of a purely combinatorial\u0000dimension and in terms of an appropriate version of uniform convergence.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"145 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139954524","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}
Omar Fawzi, Richard Kueng, Damian Markham, Aadil Oufkir
We develop a framework for learning properties of quantum states beyond the assumption of independent and identically distributed (i.i.d.) input states. We prove that, given any learning problem (under reasonable assumptions), an algorithm designed for i.i.d. input states can be adapted to handle input states of any nature, albeit at the expense of a polynomial increase in copy complexity. Furthermore, we establish that algorithms which perform non-adaptive incoherent measurements can be extended to encompass non-i.i.d. input states while maintaining comparable error probabilities. This allows us, among others applications, to generalize the classical shadows of Huang, Kueng, and Preskill to the non-i.i.d. setting at the cost of a small loss in efficiency. Additionally, we can efficiently verify any pure state using Clifford measurements, in a way that is independent of the ideal state. Our main techniques are based on de Finetti-style theorems supported by tools from information theory. In particular, we prove a new randomized local de Finetti theorem that can be of independent interest.
{"title":"Learning Properties of Quantum States Without the I.I.D. Assumption","authors":"Omar Fawzi, Richard Kueng, Damian Markham, Aadil Oufkir","doi":"arxiv-2401.16922","DOIUrl":"https://doi.org/arxiv-2401.16922","url":null,"abstract":"We develop a framework for learning properties of quantum states beyond the\u0000assumption of independent and identically distributed (i.i.d.) input states. We\u0000prove that, given any learning problem (under reasonable assumptions), an\u0000algorithm designed for i.i.d. input states can be adapted to handle input\u0000states of any nature, albeit at the expense of a polynomial increase in copy\u0000complexity. Furthermore, we establish that algorithms which perform\u0000non-adaptive incoherent measurements can be extended to encompass non-i.i.d.\u0000input states while maintaining comparable error probabilities. This allows us,\u0000among others applications, to generalize the classical shadows of Huang, Kueng,\u0000and Preskill to the non-i.i.d. setting at the cost of a small loss in\u0000efficiency. Additionally, we can efficiently verify any pure state using\u0000Clifford measurements, in a way that is independent of the ideal state. Our\u0000main techniques are based on de Finetti-style theorems supported by tools from\u0000information theory. In particular, we prove a new randomized local de Finetti\u0000theorem that can be of independent interest.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139646771","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}
We study the multi-agent multi-armed bandit (MAMAB) problem, where $m$ agents are factored into $rho$ overlapping groups. Each group represents a hyperedge, forming a hypergraph over the agents. At each round of interaction, the learner pulls a joint arm (composed of individual arms for each agent) and receives a reward according to the hypergraph structure. Specifically, we assume there is a local reward for each hyperedge, and the reward of the joint arm is the sum of these local rewards. Previous work introduced the multi-agent Thompson sampling (MATS) algorithm citep{verstraeten2020multiagent} and derived a Bayesian regret bound. However, it remains an open problem how to derive a frequentist regret bound for Thompson sampling in this multi-agent setting. To address these issues, we propose an efficient variant of MATS, the $epsilon$-exploring Multi-Agent Thompson Sampling ($epsilon$-MATS) algorithm, which performs MATS exploration with probability $epsilon$ while adopts a greedy policy otherwise. We prove that $epsilon$-MATS achieves a worst-case frequentist regret bound that is sublinear in both the time horizon and the local arm size. We also derive a lower bound for this setting, which implies our frequentist regret upper bound is optimal up to constant and logarithm terms, when the hypergraph is sufficiently sparse. Thorough experiments on standard MAMAB problems demonstrate the superior performance and the improved computational efficiency of $epsilon$-MATS compared with existing algorithms in the same setting.
{"title":"Finite-Time Frequentist Regret Bounds of Multi-Agent Thompson Sampling on Sparse Hypergraphs","authors":"Tianyuan Jin, Hao-Lun Hsu, William Chang, Pan Xu","doi":"arxiv-2312.15549","DOIUrl":"https://doi.org/arxiv-2312.15549","url":null,"abstract":"We study the multi-agent multi-armed bandit (MAMAB) problem, where $m$ agents\u0000are factored into $rho$ overlapping groups. Each group represents a hyperedge,\u0000forming a hypergraph over the agents. At each round of interaction, the learner\u0000pulls a joint arm (composed of individual arms for each agent) and receives a\u0000reward according to the hypergraph structure. Specifically, we assume there is\u0000a local reward for each hyperedge, and the reward of the joint arm is the sum\u0000of these local rewards. Previous work introduced the multi-agent Thompson\u0000sampling (MATS) algorithm citep{verstraeten2020multiagent} and derived a\u0000Bayesian regret bound. However, it remains an open problem how to derive a\u0000frequentist regret bound for Thompson sampling in this multi-agent setting. To\u0000address these issues, we propose an efficient variant of MATS, the\u0000$epsilon$-exploring Multi-Agent Thompson Sampling ($epsilon$-MATS) algorithm,\u0000which performs MATS exploration with probability $epsilon$ while adopts a\u0000greedy policy otherwise. We prove that $epsilon$-MATS achieves a worst-case\u0000frequentist regret bound that is sublinear in both the time horizon and the\u0000local arm size. We also derive a lower bound for this setting, which implies\u0000our frequentist regret upper bound is optimal up to constant and logarithm\u0000terms, when the hypergraph is sufficiently sparse. Thorough experiments on\u0000standard MAMAB problems demonstrate the superior performance and the improved\u0000computational efficiency of $epsilon$-MATS compared with existing algorithms\u0000in the same setting.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139057142","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}
Lachlan C. Astfalck, Adam M. Sykulski, Edward J. Cripps
Welch's method provides an estimator of the power spectral density that is statistically consistent. This is achieved by averaging over periodograms calculated from overlapping segments of a time series. For a finite length time series, while the variance of the estimator decreases as the number of segments increase, the magnitude of the estimator's bias increases: a bias-variance trade-off ensues when setting the segment number. We address this issue by providing a a novel method for debiasing Welch's method which maintains the computational complexity and asymptotic consistency, and leads to improved finite-sample performance. Theoretical results are given for fourth-order stationary processes with finite fourth-order moments and absolutely continuous fourth-order cumulant spectrum. The significant bias reduction is demonstrated with numerical simulation and an application to real-world data, where several empirical metrics indicate our debiased estimator compares favourably to Welch's. Our estimator also permits irregular spacing over frequency and we demonstrate how this may be employed for signal compression and further variance reduction. Code accompanying this work is available in the R and python languages.
韦尔奇方法提供了一种在统计上一致的功率谱密度估算器。这是通过对时间序列重叠段计算的周期图求取平均值来实现的。对于有限长度的时间序列，虽然估计器的方差会随着分段数的增加而减小，但估计器的偏差幅度却会增大：在设定分段数时，会出现偏差-方差权衡。为了解决这个问题，我们提供了一种新的韦尔奇去偏方法，这种方法保持了计算复杂性和渐进一致性，并提高了有限样本性能。该方法给出了具有有限四阶矩和绝对连续四阶累积谱的四阶平稳过程的理论结果。我们通过数值模拟和实际数据应用证明了偏差的显著减少，多个经验指标表明我们的去偏估计器优于韦尔奇估计器。我们的估计器还允许频率上的不规则间隔，并演示了如何将其用于信号压缩和进一步减小方差。本研究的相关代码使用 R 和python 语言编写。
{"title":"Debiasing Welch's Method for Spectral Density Estimation","authors":"Lachlan C. Astfalck, Adam M. Sykulski, Edward J. Cripps","doi":"arxiv-2312.13643","DOIUrl":"https://doi.org/arxiv-2312.13643","url":null,"abstract":"Welch's method provides an estimator of the power spectral density that is\u0000statistically consistent. This is achieved by averaging over periodograms\u0000calculated from overlapping segments of a time series. For a finite length time\u0000series, while the variance of the estimator decreases as the number of segments\u0000increase, the magnitude of the estimator's bias increases: a bias-variance\u0000trade-off ensues when setting the segment number. We address this issue by\u0000providing a a novel method for debiasing Welch's method which maintains the\u0000computational complexity and asymptotic consistency, and leads to improved\u0000finite-sample performance. Theoretical results are given for fourth-order\u0000stationary processes with finite fourth-order moments and absolutely continuous\u0000fourth-order cumulant spectrum. The significant bias reduction is demonstrated\u0000with numerical simulation and an application to real-world data, where several\u0000empirical metrics indicate our debiased estimator compares favourably to\u0000Welch's. Our estimator also permits irregular spacing over frequency and we\u0000demonstrate how this may be employed for signal compression and further\u0000variance reduction. Code accompanying this work is available in the R and\u0000python languages.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139028939","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}
Bayesian statistics has two common measures of central tendency of a posterior distribution: posterior means and Maximum A Posteriori (MAP) estimates. In this paper, we discuss a connection between MAP estimates and posterior means. We derive an asymptotic condition for a pair of prior densities under which the posterior mean based on one prior coincides with the MAP estimate based on the other prior. A sufficient condition for the existence of this prior pair relates to $alpha$-flatness of the statistical model in information geometry. We also construct a matching prior pair using $alpha$-parallel priors. Our result elucidates an interesting connection between regularization in generalized linear regression models and posterior expectation.
{"title":"Matching prior pairs connecting Maximum A Posteriori estimation and posterior expectation","authors":"Michiko Okudo, Keisuke Yano","doi":"arxiv-2312.09586","DOIUrl":"https://doi.org/arxiv-2312.09586","url":null,"abstract":"Bayesian statistics has two common measures of central tendency of a\u0000posterior distribution: posterior means and Maximum A Posteriori (MAP)\u0000estimates. In this paper, we discuss a connection between MAP estimates and\u0000posterior means. We derive an asymptotic condition for a pair of prior\u0000densities under which the posterior mean based on one prior coincides with the\u0000MAP estimate based on the other prior. A sufficient condition for the existence\u0000of this prior pair relates to $alpha$-flatness of the statistical model in\u0000information geometry. We also construct a matching prior pair using\u0000$alpha$-parallel priors. Our result elucidates an interesting connection\u0000between regularization in generalized linear regression models and posterior\u0000expectation.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138716531","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}
Recently defined expectile regions capture the idea of centrality with respect to a multivariate distribution, but fail to describe the tail behavior while it is not at all clear what should be understood by a tail of a multivariate distribution. Therefore, cone expectile sets are introduced which take into account a vector preorder for the multi-dimensional data points. This provides a way of describing and clustering a multivariate distribution/data cloud with respect to an order relation. Fundamental properties of cone expectiles including dual representations of both expectile regions and cone expectile sets are established. It is shown that set-valued sublinear risk measures can be constructed from cone expectile sets in the same way as in the univariate case. Inverse functions of cone expectiles are defined which should be considered as rank functions rather than depth functions. Finally, expectile orders for random vectors are introduced and characterized via expectile rank functions.
{"title":"Set-valued expectiles for ordered data analysis","authors":"Ha Thi Khanh Linh, Andreas H Hamel","doi":"arxiv-2312.09930","DOIUrl":"https://doi.org/arxiv-2312.09930","url":null,"abstract":"Recently defined expectile regions capture the idea of centrality with\u0000respect to a multivariate distribution, but fail to describe the tail behavior\u0000while it is not at all clear what should be understood by a tail of a\u0000multivariate distribution. Therefore, cone expectile sets are introduced which\u0000take into account a vector preorder for the multi-dimensional data points. This\u0000provides a way of describing and clustering a multivariate distribution/data\u0000cloud with respect to an order relation. Fundamental properties of cone\u0000expectiles including dual representations of both expectile regions and cone\u0000expectile sets are established. It is shown that set-valued sublinear risk\u0000measures can be constructed from cone expectile sets in the same way as in the\u0000univariate case. Inverse functions of cone expectiles are defined which should\u0000be considered as rank functions rather than depth functions. Finally, expectile\u0000orders for random vectors are introduced and characterized via expectile rank\u0000functions.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138716369","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}
Oliver Clarke, Serkan Hoşten, Nataliia Kushnerchuk, Janike Oldekop
We study the maximum likelihood (ML) degree of discrete exponential independence models and models defined by the second hypersimplex. For models with two independent variables, we show that the ML degree is an invariant of a matroid associated to the model. We use this description to explore ML degrees via hyperplane arrangements. For independence models with more variables, we investigate the connection between the vanishing of factors of its principal $A$-determinant and its ML degree. Similarly, for models defined by the second hypersimplex, we determine its principal $A$-determinant and give computational evidence towards a conjectured lower bound of its ML degree.
我们研究了离散指数独立模型和第二超复数定义模型的最大似然度（ML）。对于有两个自变量的模型，我们证明最大似然度是与模型相关的矢量的不变量。我们利用这一描述，通过超平面排列来探索 ML 度。对于有更多变量的独立模型，我们研究了其主$A$决定因素的消失与其 ML 度之间的联系。同样，对于由第二超复数定义的模型，我们确定了它的主$A$-决定因素，并给出了其 ML 度的一个猜想下限的计算证据。
{"title":"Matroid Stratification of ML Degrees of Independence Models","authors":"Oliver Clarke, Serkan Hoşten, Nataliia Kushnerchuk, Janike Oldekop","doi":"arxiv-2312.10010","DOIUrl":"https://doi.org/arxiv-2312.10010","url":null,"abstract":"We study the maximum likelihood (ML) degree of discrete exponential\u0000independence models and models defined by the second hypersimplex. For models\u0000with two independent variables, we show that the ML degree is an invariant of a\u0000matroid associated to the model. We use this description to explore ML degrees\u0000via hyperplane arrangements. For independence models with more variables, we\u0000investigate the connection between the vanishing of factors of its principal\u0000$A$-determinant and its ML degree. Similarly, for models defined by the second\u0000hypersimplex, we determine its principal $A$-determinant and give computational\u0000evidence towards a conjectured lower bound of its ML degree.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"116 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138716165","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}
We use Stein characterisations to derive new moment-type estimators for the parameters of several multivariate distributions in the i.i.d. case; we also derive the asymptotic properties of these estimators. Our examples include the multivariate truncated normal distribution and several spherical distributions. The estimators are explicit and therefore provide an interesting alternative to the maximum-likelihood estimator. The quality of these estimators is assessed through competitive simulation studies in which we compare their behaviour to the performance of other estimators available in the literature.
{"title":"Stein estimation in a multivariate setting","authors":"Adrian Fischer, Robert E. Gaunt, Yvik Swan","doi":"arxiv-2312.09344","DOIUrl":"https://doi.org/arxiv-2312.09344","url":null,"abstract":"We use Stein characterisations to derive new moment-type estimators for the\u0000parameters of several multivariate distributions in the i.i.d. case; we also\u0000derive the asymptotic properties of these estimators. Our examples include the\u0000multivariate truncated normal distribution and several spherical distributions.\u0000The estimators are explicit and therefore provide an interesting alternative to\u0000the maximum-likelihood estimator. The quality of these estimators is assessed\u0000through competitive simulation studies in which we compare their behaviour to\u0000the performance of other estimators available in the literature.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138716461","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}
Kurtosis minus squared skewness is bounded from below by 1, but for unimodal distributions this parameter is bounded by 189/125. In some applications it is natural to compare distributions by comparing their kurtosis-minus-squared-skewness parameters. The asymptotic behavior of the empirical version of this parameter is studied here for i.i.d. random variables. The result may be used to test the hypothesis of unimodality against the alternative that the kurtosis-minus-squared-skewness parameter is less than 189/125. However, such a test has to be applied with care, since this parameter can take arbitrarily large values, also for multimodal distributions. Numerical results are presented and for three classes of distributions the skewness-kurtosis sets are described in detail.
{"title":"Inference via the Skewness-Kurtosis Set","authors":"Chris A. J. Klaassen, Bert van Es","doi":"arxiv-2312.06212","DOIUrl":"https://doi.org/arxiv-2312.06212","url":null,"abstract":"Kurtosis minus squared skewness is bounded from below by 1, but for unimodal\u0000distributions this parameter is bounded by 189/125. In some applications it is\u0000natural to compare distributions by comparing their\u0000kurtosis-minus-squared-skewness parameters. The asymptotic behavior of the\u0000empirical version of this parameter is studied here for i.i.d. random\u0000variables. The result may be used to test the hypothesis of unimodality against\u0000the alternative that the kurtosis-minus-squared-skewness parameter is less than\u0000189/125. However, such a test has to be applied with care, since this parameter\u0000can take arbitrarily large values, also for multimodal distributions. Numerical\u0000results are presented and for three classes of distributions the\u0000skewness-kurtosis sets are described in detail.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138576968","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}
We address the problem of parameter estimation for degenerate diffusion processes defined via the solution of Stochastic Differential Equations (SDEs) with diffusion matrix that is not full-rank. For this class of hypo-elliptic diffusions recent works have proposed contrast estimators that are asymptotically normal, provided that the step-size in-between observations $Delta=Delta_n$ and their total number $n$ satisfy $n to infty$, $n Delta_n to infty$, $Delta_n to 0$, and additionally $Delta_n = o (n^{-1/2})$. This latter restriction places a requirement for a so-called `rapidly increasing experimental design'. In this paper, we overcome this limitation and develop a general contrast estimator satisfying asymptotic normality under the weaker design condition $Delta_n = o(n^{-1/p})$ for general $p ge 2$. Such a result has been obtained for elliptic SDEs in the literature, but its derivation in a hypo-elliptic setting is highly non-trivial. We provide numerical results to illustrate the advantages of the developed theory.
我们要解决的问题是，通过求解扩散矩阵非全秩的随机微分方程（SDE）定义的退化扩散过程的参数估计问题。对于这类下椭圆扩散，最近的研究提出了渐近正态分布的对比度估计器，条件是观测值之间的步长$Delta=Delta_n$及其总数$n$满足$n to infty$，$nDelta_n to infty$，$Delta_n to 0$，另外$Delta_n = o(n^{-1/2})$。后一种限制对所谓的 "快速增长实验设计 "提出了要求。在本文中，我们克服了这一限制，开发出了一种在较弱的设计条件 $Delta_n = o(n^{-1/p})$ 宽度一般为 $pge 2$ 下满足渐近正态性的一般对比度估计器。这样的结果在文献中已针对椭圆 SDE 得到，但在次椭圆环境中的推导却非常不容易。我们提供了数值结果来说明所发展理论的优势。
{"title":"Parameter Inference for Hypo-Elliptic Diffusions under a Weak Design Condition","authors":"Yuga Iguchi, Alexandros Beskos","doi":"arxiv-2312.04444","DOIUrl":"https://doi.org/arxiv-2312.04444","url":null,"abstract":"We address the problem of parameter estimation for degenerate diffusion\u0000processes defined via the solution of Stochastic Differential Equations (SDEs)\u0000with diffusion matrix that is not full-rank. For this class of hypo-elliptic\u0000diffusions recent works have proposed contrast estimators that are\u0000asymptotically normal, provided that the step-size in-between observations\u0000$Delta=Delta_n$ and their total number $n$ satisfy $n to infty$, $n\u0000Delta_n to infty$, $Delta_n to 0$, and additionally $Delta_n = o\u0000(n^{-1/2})$. This latter restriction places a requirement for a so-called\u0000`rapidly increasing experimental design'. In this paper, we overcome this\u0000limitation and develop a general contrast estimator satisfying asymptotic\u0000normality under the weaker design condition $Delta_n = o(n^{-1/p})$ for\u0000general $p ge 2$. Such a result has been obtained for elliptic SDEs in the\u0000literature, but its derivation in a hypo-elliptic setting is highly\u0000non-trivial. We provide numerical results to illustrate the advantages of the\u0000developed theory.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"103 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138553412","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}
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