This paper is about the modeling of cumulative hazard functions using martingale posterior distributions. The focus is on uncertainty quantification from a nonparametric perspective. The foundational Bayesian model in this case is the beta process and the classic estimator is the Nelson–Aalen. We use a sequence of estimators which form a martingale in order to obtain a random cumulative hazard function from the martingale posterior. The connection with the beta process is established and a number of illustrations is presented.
{"title":"Martingale posterior distributions for cumulative hazard functions","authors":"Stephen G. Walker","doi":"10.1111/sjos.12712","DOIUrl":"https://doi.org/10.1111/sjos.12712","url":null,"abstract":"This paper is about the modeling of cumulative hazard functions using martingale posterior distributions. The focus is on uncertainty quantification from a nonparametric perspective. The foundational Bayesian model in this case is the beta process and the classic estimator is the Nelson–Aalen. We use a sequence of estimators which form a martingale in order to obtain a random cumulative hazard function from the martingale posterior. The connection with the beta process is established and a number of illustrations is presented.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"8 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140591815","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}
This paper deals with a Skorokhod's integral‐based least squares‐ (LS) type estimator of the drift parameter computed from multiple (possibly dependent) copies of the solution of a stochastic differential equation (SDE) driven by a fractional Brownian motion of Hurst index . On the one hand, some convergence results are established on our LS estimator when . On the other hand, when , Skorokhod's integral‐based estimators cannot be computed from data, but in this paper some convergence results are established on a computable approximation of our LS estimator.
本文论述了一种基于斯科洛克霍德积分的最小二乘法（LS）型漂移参数估计器，该估计器由赫斯特指数为.的分数布朗运动驱动的随机微分方程（SDE）解的多个（可能依赖的）副本计算得出。一方面，当......时，我们的 LS 估计器建立了一些收敛结果。另一方面，当 , 时，Skorokhod 基于积分的估计器无法从数据中计算出来，但本文对我们的 LS 估计器的可计算近似值建立了一些收敛结果。
{"title":"On a computable Skorokhod's integral‐based estimator of the drift parameter in fractional SDE","authors":"Nicolas Marie","doi":"10.1111/sjos.12711","DOIUrl":"https://doi.org/10.1111/sjos.12711","url":null,"abstract":"This paper deals with a Skorokhod's integral‐based least squares‐ (LS) type estimator of the drift parameter computed from multiple (possibly dependent) copies of the solution of a stochastic differential equation (SDE) driven by a fractional Brownian motion of Hurst index . On the one hand, some convergence results are established on our LS estimator when . On the other hand, when , Skorokhod's integral‐based estimators cannot be computed from data, but in this paper some convergence results are established on a computable approximation of our LS estimator.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"124 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140197324","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}
This paper studies generative adversarial networks (GANs) from the perspective of statistical inference. A GAN is a popular machine learning method in which the parameters of two neural networks, a generator and a discriminator, are estimated to solve a particular minimax problem. This minimax problem typically has a multitude of solutions and the focus of this paper are the statistical properties of these solutions. We address two key statistical issues for the generator and discriminator network parameters, consistent estimation and confidence sets. We first show that the set of solutions to the sample GAN problem is a (Hausdorff) consistent estimator of the set of solutions to the corresponding population GAN problem. We then devise a computationally intensive procedure to form confidence sets and show that these sets contain the population GAN solutions with the desired coverage probability. Small numerical experiments and a Monte Carlo study illustrate our results and verify our theoretical findings. We also show that our results apply in general minimax problems that may be nonconvex, nonconcave, and have multiple solutions.
本文从统计推断的角度研究生成对抗网络（GAN）。生成式对抗网络（GAN）是一种流行的机器学习方法，通过估算生成器和判别器这两个神经网络的参数来解决一个特定的最小问题。这个最小问题通常有多种解决方案，本文的重点是这些解决方案的统计特性。我们探讨了生成器和判别器网络参数的两个关键统计问题：一致估计和置信集。我们首先证明，样本 GAN 问题的解集是相应群体 GAN 问题解集的（豪斯多夫）一致性估计。然后，我们设计了一种计算密集型程序来形成置信集，并证明这些置信集包含具有所需覆盖概率的群体 GAN 解。小型数值实验和蒙特卡罗研究说明了我们的结果，并验证了我们的理论发现。我们还证明，我们的结果适用于一般的最小问题，这些问题可能是非凸、非凹和多解的。
{"title":"Statistical inference for generative adversarial networks and other minimax problems","authors":"Mika Meitz","doi":"10.1111/sjos.12710","DOIUrl":"https://doi.org/10.1111/sjos.12710","url":null,"abstract":"This paper studies generative adversarial networks (GANs) from the perspective of statistical inference. A GAN is a popular machine learning method in which the parameters of two neural networks, a generator and a discriminator, are estimated to solve a particular minimax problem. This minimax problem typically has a multitude of solutions and the focus of this paper are the statistical properties of these solutions. We address two key statistical issues for the generator and discriminator network parameters, consistent estimation and confidence sets. We first show that the set of solutions to the sample GAN problem is a (Hausdorff) consistent estimator of the set of solutions to the corresponding population GAN problem. We then devise a computationally intensive procedure to form confidence sets and show that these sets contain the population GAN solutions with the desired coverage probability. Small numerical experiments and a Monte Carlo study illustrate our results and verify our theoretical findings. We also show that our results apply in general minimax problems that may be nonconvex, nonconcave, and have multiple solutions.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"15 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140197717","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 consistency and asymptotic normality results for quasi-maximum likelihood methods for drift parameters of ergodic stochastic processes observed in discrete time in an underlying continuous-time setting. The special feature of our analysis is that the stochastic integral part is unobserved and nonparametric. Additionally, the drift may depend on the (unknown and unobserved) stochastic integrand. Our results hold for ergodic semi-parametric diffusions and backward SDEs. Simulation studies confirm that the methods proposed yield good convergence results.
{"title":"Efficient drift parameter estimation for ergodic solutions of backward SDEs","authors":"Teppei Ogihara, Mitja Stadje","doi":"10.1111/sjos.12709","DOIUrl":"https://doi.org/10.1111/sjos.12709","url":null,"abstract":"We derive consistency and asymptotic normality results for quasi-maximum likelihood methods for drift parameters of ergodic stochastic processes observed in discrete time in an underlying continuous-time setting. The special feature of our analysis is that the stochastic integral part is unobserved and nonparametric. Additionally, the drift may depend on the (unknown and unobserved) stochastic integrand. Our results hold for ergodic semi-parametric diffusions and backward SDEs. Simulation studies confirm that the methods proposed yield good convergence results.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"170 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140001947","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}
This paper studies the autoregressive and moving average (ARMA) model with time-functional variance (TFV) noises, called the ARMA-TFV model. We first establish the consistency and asymptotic normality of its least squares estimator (LSE). The Wald tests and portmanteau tests are constructed based on the theory for variable selection and model checking. A simulation study is carried out to assess the performance of our approach in finite samples, and two real examples are given. It should be mentioned that the process generated from the ARMA-TFV model is not stationary, and the technique in this paper is nonstandard and may provide insights for future research in this area.
{"title":"Asymptotic inference of the ARMA model with time-functional variance noises","authors":"Bibi Cai, Enwen Zhu, Shiqing Ling","doi":"10.1111/sjos.12708","DOIUrl":"https://doi.org/10.1111/sjos.12708","url":null,"abstract":"This paper studies the autoregressive and moving average (ARMA) model with time-functional variance (TFV) noises, called the ARMA-TFV model. We first establish the consistency and asymptotic normality of its least squares estimator (LSE). The Wald tests and portmanteau tests are constructed based on the theory for variable selection and model checking. A simulation study is carried out to assess the performance of our approach in finite samples, and two real examples are given. It should be mentioned that the process generated from the ARMA-TFV model is not stationary, and the technique in this paper is nonstandard and may provide insights for future research in this area.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"22 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139759892","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}
In a placebo-controlled clinical study one may calculate the average treatment effect to convey the effect of the active treatment on some outcome. However, if it is speculated that the treatment only has an effect if the patient responds to the treatment defined by a certain biomarker response, then it is arguably more relevant to estimate the treatment effect among such responders. We present such a causal parameter that is based on principal stratification and is identified under the exclusion of a treatment effect among the non-responders. We focus on time-;to-event outcomes allowing for right censoring, and construct a doubly robust and efficient estimator based on the associated efficient influence function. The properties of the estimator are showcased in a simulation study and the methodology is applied to the Leader trial investigating the effect of liraglutide on the occurrence of cardiovascular events.
{"title":"Estimation of treatment effect among treatment responders with a time-to-event endpoint","authors":"Andreas Nordland, Torben Martinussen","doi":"10.1111/sjos.12706","DOIUrl":"https://doi.org/10.1111/sjos.12706","url":null,"abstract":"In a placebo-controlled clinical study one may calculate the average treatment effect to convey the effect of the active treatment on some outcome. However, if it is speculated that the treatment only has an effect if the patient responds to the treatment defined by a certain biomarker response, then it is arguably more relevant to estimate the treatment effect among such responders. We present such a causal parameter that is based on principal stratification and is identified under the exclusion of a treatment effect among the non-responders. We focus on time-;to-event outcomes allowing for right censoring, and construct a doubly robust and efficient estimator based on the associated efficient influence function. The properties of the estimator are showcased in a simulation study and the methodology is applied to the Leader trial investigating the effect of liraglutide on the occurrence of cardiovascular events.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"56 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139501684","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}
Christophe Denis, Charlotte Dion-Blanc, Eddy Ella-Mintsa, Viet Chi Tran
We study the multiclass classification problem where the features come from a mixture of time-homogeneous diffusion.Specifically, the classes are discriminated by their drift functions while the diffusion coefficient is common to all classes and unknown.In this framework, we build a plug-in classifier which relies on nonparamateric estimators of the drift and diffusion functions.We first establish the consistency of our classification procedure under mild assumptions and then provide rates of convergence under different setof assumptions. Finally, a numerical study supports our theoretical findings.
{"title":"Nonparametric plug-in classifier for multiclass classification of S.D.E. paths","authors":"Christophe Denis, Charlotte Dion-Blanc, Eddy Ella-Mintsa, Viet Chi Tran","doi":"10.1111/sjos.12702","DOIUrl":"https://doi.org/10.1111/sjos.12702","url":null,"abstract":"We study the multiclass classification problem where the features come from a mixture of time-homogeneous diffusion.Specifically, the classes are discriminated by their drift functions while the diffusion coefficient is common to all classes and unknown.In this framework, we build a plug-in classifier which relies on nonparamateric estimators of the drift and diffusion functions.We first establish the consistency of our classification procedure under mild assumptions and then provide rates of convergence under different setof assumptions. Finally, a numerical study supports our theoretical findings.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"33 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139475214","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 present a detailed discussion of the theoretical properties of quadratic inference function estimators of the parameters in marginal linear regression models. We consider the effect of the choice of working correlation on fundamental questions including the existence of quadratic inference function estimators, their relationship with generalized estimating equations estimators, and the robustness and asymptotic relative efficiency of quadratic inference function and generalized estimating equations estimators. We show that the quadratic inference function estimators do not always exist and propose a way to handle this. We then show that they have unbounded influence functions and can be more or less asymptotically efficient than generalized estimating equations estimators. We also present empirical evidence to demonstrate these results. We conclude that the choice of working correlation can have surprisingly large effects.
{"title":"The effect of the working correlation on fitting models to longitudinal data","authors":"Samuel Muller, Suojin Wang, A. H. Welsh","doi":"10.1111/sjos.12704","DOIUrl":"https://doi.org/10.1111/sjos.12704","url":null,"abstract":"We present a detailed discussion of the theoretical properties of quadratic inference function estimators of the parameters in marginal linear regression models. We consider the effect of the choice of working correlation on fundamental questions including the existence of quadratic inference function estimators, their relationship with generalized estimating equations estimators, and the robustness and asymptotic relative efficiency of quadratic inference function and generalized estimating equations estimators. We show that the quadratic inference function estimators do not always exist and propose a way to handle this. We then show that they have unbounded influence functions and can be more or less asymptotically efficient than generalized estimating equations estimators. We also present empirical evidence to demonstrate these results. We conclude that the choice of working correlation can have surprisingly large effects.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"46 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077716","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}
A metric tensor for Riemann manifold Monte Carlo particularly suited for non-linear Bayesian hierarchical models is proposed. The metric tensor is built from symmetric positive semidefinite log-density gradient covariance (LGC) matrices, which are also proposed and further explored here. The LGCs generalize the Fisher information matrix by measuring the joint information content and dependence structure of both a random variable and the parameters of said variable. Consequently, positive definite Fisher/LGC-based metric tensors may be constructed not only from the observation likelihoods as is current practice, but also from arbitrarily complicated non-linear prior/latent variable structures, provided the LGC may be derived for each conditional distribution used to construct said structures. The proposed methodology is highly automatic and allows for exploitation of any sparsity associated with the model in question. When implemented in conjunction with a Riemann manifold variant of the recently proposed numerical generalized randomized Hamiltonian Monte Carlo processes, the proposed methodology is highly competitive, in particular for the more challenging target distributions associated with Bayesian hierarchical models.
{"title":"Log-density gradient covariance and automatic metric tensors for Riemann manifold Monte Carlo methods†","authors":"Tore Selland Kleppe","doi":"10.1111/sjos.12705","DOIUrl":"https://doi.org/10.1111/sjos.12705","url":null,"abstract":"A metric tensor for Riemann manifold Monte Carlo particularly suited for non-linear Bayesian hierarchical models is proposed. The metric tensor is built from symmetric positive semidefinite log-density gradient covariance (LGC) matrices, which are also proposed and further explored here. The LGCs generalize the Fisher information matrix by measuring the joint information content and dependence structure of both a random variable and the parameters of said variable. Consequently, positive definite Fisher/LGC-based metric tensors may be constructed not only from the observation likelihoods as is current practice, but also from arbitrarily complicated non-linear prior/latent variable structures, provided the LGC may be derived for each conditional distribution used to construct said structures. The proposed methodology is highly automatic and allows for exploitation of any sparsity associated with the model in question. When implemented in conjunction with a Riemann manifold variant of the recently proposed numerical generalized randomized Hamiltonian Monte Carlo processes, the proposed methodology is highly competitive, in particular for the more challenging target distributions associated with Bayesian hierarchical models.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"41 11 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139053743","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}
In this paper we study two important representations for extreme value distributions and their max-domains of attraction (MDA), namely von Mises representation (vMR) and variation representation (VR), which are convenient ways to gain limit results. Both VR and vMR are defined via so-called auxiliary functions ψ. Up to now, however, the set of valid auxiliary functions for vMR has neither been characterized completely nor separated from those for VR. We contribute to the current literature by introducing “universal” auxiliary functions which are valid for both VR and vMR representations for the entire MDA distribution families. Then we identify exactly the sets of valid auxiliary functions for both VR and vMR. Moreover, we propose a method for finding appropriate auxiliary functions with analytically simple structure and provide them for several important distributions.
{"title":"Characterization of valid auxiliary functions for representations of extreme value distributions and their max-domains of attraction","authors":"Miriam Isabel Seifert","doi":"10.1111/sjos.12701","DOIUrl":"https://doi.org/10.1111/sjos.12701","url":null,"abstract":"In this paper we study two important representations for extreme value distributions and their max-domains of attraction (MDA), namely von Mises representation (vMR) and variation representation (VR), which are convenient ways to gain limit results. Both VR and vMR are defined via so-called auxiliary functions <i>ψ</i>. Up to now, however, the set of valid auxiliary functions for vMR has neither been characterized completely nor separated from those for VR. We contribute to the current literature by introducing “universal” auxiliary functions which are valid for both VR and vMR representations for the entire MDA distribution families. Then we identify exactly the sets of valid auxiliary functions for both VR and vMR. Moreover, we propose a method for finding appropriate auxiliary functions with analytically simple structure and provide them for several important distributions.","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"11 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139053691","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}
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