We study the distribution of the unobserved states of two measure-valued diffusions of Fleming-Viot and Dawson-Watanabe type, conditional on observations from the underlying populations collected at past, present and future times. If seen as nonparametric hidden Markov models, this amounts to finding the smoothing distributions of these processes, which we show can be explicitly described in recursive form as finite mixtures of laws of Dirichlet and gamma random measures respectively. We characterize the time-dependent weights of these mixtures, accounting for potentially different time intervals between data collection times, and fully describe the implications of assuming a discrete or a nonatomic distribution for the underlying process that drives mutations. In particular, we show that with a nonatomic mutation offspring distribution, the inference automatically upweights mixture components that carry, as atoms, observed types shared at different collection times. The predictive distributions for further samples from the population conditional on the data are also identified and shown to be mixtures of generalized Polya urns, conditionally on a latent variable in the Dawson-Watanabe case.
{"title":"Smoothing distributions for conditional Fleming–Viot and Dawson–Watanabe diffusions","authors":"Filippo Ascolani, A. Lijoi, M. Ruggiero","doi":"10.3150/22-bej1504","DOIUrl":"https://doi.org/10.3150/22-bej1504","url":null,"abstract":"We study the distribution of the unobserved states of two measure-valued diffusions of Fleming-Viot and Dawson-Watanabe type, conditional on observations from the underlying populations collected at past, present and future times. If seen as nonparametric hidden Markov models, this amounts to finding the smoothing distributions of these processes, which we show can be explicitly described in recursive form as finite mixtures of laws of Dirichlet and gamma random measures respectively. We characterize the time-dependent weights of these mixtures, accounting for potentially different time intervals between data collection times, and fully describe the implications of assuming a discrete or a nonatomic distribution for the underlying process that drives mutations. In particular, we show that with a nonatomic mutation offspring distribution, the inference automatically upweights mixture components that carry, as atoms, observed types shared at different collection times. The predictive distributions for further samples from the population conditional on the data are also identified and shown to be mixtures of generalized Polya urns, conditionally on a latent variable in the Dawson-Watanabe case.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43165181","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}
The mixed-membership stochastic block model (MMSBM) is a common model for social networks. Given an $n$-node symmetric network generated from a $K$-community MMSBM, we would like to test $K=1$ versus $K>1$. We first study the degree-based $chi^2$ test and the orthodox Signed Quadrilateral (oSQ) test. These two statistics estimate an order-2 polynomial and an order-4 polynomial of a"signal"matrix, respectively. We derive the asymptotic null distribution and power for both tests. However, for each test, there exists a parameter regime where its power is unsatisfactory. It motivates us to propose a power enhancement (PE) test to combine the strengths of both tests. We show that the PE test has a tractable null distribution and improves the power of both tests. To assess the optimality of PE, we consider a randomized setting, where the $n$ membership vectors are independently drawn from a distribution on the standard simplex. We show that the success of global testing is governed by a quantity $beta_n(K,P,h)$, which depends on the community structure matrix $P$ and the mean vector $h$ of memberships. For each given $(K, P, h)$, a test is called $textit{ optimal}$ if it distinguishes two hypotheses when $beta_n(K, P,h)toinfty$. A test is called $textit{optimally adaptive}$ if it is optimal for all $(K, P, h)$. We show that the PE test is optimally adaptive, while many existing tests are only optimal for some particular $(K, P, h)$, hence, not optimally adaptive.
{"title":"Power enhancement and phase transitions for global testing of the mixed membership stochastic block model","authors":"Louis V. Cammarata, Z. Ke","doi":"10.3150/22-bej1519","DOIUrl":"https://doi.org/10.3150/22-bej1519","url":null,"abstract":"The mixed-membership stochastic block model (MMSBM) is a common model for social networks. Given an $n$-node symmetric network generated from a $K$-community MMSBM, we would like to test $K=1$ versus $K>1$. We first study the degree-based $chi^2$ test and the orthodox Signed Quadrilateral (oSQ) test. These two statistics estimate an order-2 polynomial and an order-4 polynomial of a\"signal\"matrix, respectively. We derive the asymptotic null distribution and power for both tests. However, for each test, there exists a parameter regime where its power is unsatisfactory. It motivates us to propose a power enhancement (PE) test to combine the strengths of both tests. We show that the PE test has a tractable null distribution and improves the power of both tests. To assess the optimality of PE, we consider a randomized setting, where the $n$ membership vectors are independently drawn from a distribution on the standard simplex. We show that the success of global testing is governed by a quantity $beta_n(K,P,h)$, which depends on the community structure matrix $P$ and the mean vector $h$ of memberships. For each given $(K, P, h)$, a test is called $textit{ optimal}$ if it distinguishes two hypotheses when $beta_n(K, P,h)toinfty$. A test is called $textit{optimally adaptive}$ if it is optimal for all $(K, P, h)$. We show that the PE test is optimally adaptive, while many existing tests are only optimal for some particular $(K, P, h)$, hence, not optimally adaptive.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45669660","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}
Characteristic functions that are radially symmetric have a dual interpretation, as they can be used as the isotropic correlation functions of spatial random fields. Extensions of isotropic correlation functions from balls into $d$-dimensional Euclidean spaces, $R^{d}$, have been understood after Rudin. Yet, extension theorems on product spaces are elusive, and a counterexample provided by Rudin on rectangles suggest that the problem is quite challenging. This paper provides extension theorem for multiradial characteristic functions that are defined in balls embedded in $R^d$ cross, either $R^{dd}$ or the unit sphere $S^{dd}$ embedded in $R^{dd+1}$, for any two positive integers $d$ and $dd$. We then examine Turning Bands operators that provide bijections between the class of multiradial correlation functions in given product spaces, and multiradial correlations in product spaces having different dimensions. The combination of extension theorems with Turning Bands provides a connection with random fields that are defined in balls cross linear or circular time.
{"title":"Rudin extension theorems on product spaces, turning bands, and random fields on balls cross time","authors":"E. Porcu, Samuel F. Feng, X. Emery, A. Peron","doi":"10.3150/22-bej1506","DOIUrl":"https://doi.org/10.3150/22-bej1506","url":null,"abstract":"Characteristic functions that are radially symmetric have a dual interpretation, as they can be used as the isotropic correlation functions of spatial random fields. Extensions of isotropic correlation functions from balls into $d$-dimensional Euclidean spaces, $R^{d}$, have been understood after Rudin. Yet, extension theorems on product spaces are elusive, and a counterexample provided by Rudin on rectangles suggest that the problem is quite challenging. This paper provides extension theorem for multiradial characteristic functions that are defined in balls embedded in $R^d$ cross, either $R^{dd}$ or the unit sphere $S^{dd}$ embedded in $R^{dd+1}$, for any two positive integers $d$ and $dd$. We then examine Turning Bands operators that provide bijections between the class of multiradial correlation functions in given product spaces, and multiradial correlations in product spaces having different dimensions. The combination of extension theorems with Turning Bands provides a connection with random fields that are defined in balls cross linear or circular time.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48616110","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}
In this paper, we investigate the exponential ergodicity in a Wasserstein-type distance for a damping Hamiltonian dynamics with state-dependent and non-local collisions, which indeed is a special case of piecewise deterministic Markov processes while is very popular in numerous modelling situations including stochastic algorithms. The approach adopted in this work is based on a combination of the refined basic coupling and the refined reflection coupling for non-local operators. In a certain sense, the main result developed in the present paper is a continuation of the counterpart in cite{BW2022} on exponential ergodicity of stochastic Hamiltonian systems with L'evy noises and a complement of cite{BA} upon exponential ergodicity for Andersen dynamics with constant jump rate functions.
{"title":"Exponential ergodicity for damping Hamiltonian dynamics with state-dependent and non-local collisions","authors":"J. Bao, Jian Wang","doi":"10.3150/22-bej1548","DOIUrl":"https://doi.org/10.3150/22-bej1548","url":null,"abstract":"In this paper, we investigate the exponential ergodicity in a Wasserstein-type distance for a damping Hamiltonian dynamics with state-dependent and non-local collisions, which indeed is a special case of piecewise deterministic Markov processes while is very popular in numerous modelling situations including stochastic algorithms. The approach adopted in this work is based on a combination of the refined basic coupling and the refined reflection coupling for non-local operators. In a certain sense, the main result developed in the present paper is a continuation of the counterpart in cite{BW2022} on exponential ergodicity of stochastic Hamiltonian systems with L'evy noises and a complement of cite{BA} upon exponential ergodicity for Andersen dynamics with constant jump rate functions.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47140403","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 consider the estimation of a non-linear reaction term in the stochastic heat or more generally in a semi-linear stochastic partial differential equation (SPDE). Consistent inference is achieved by studying a small diffusivity level, which is realistic in applications. Our main result is a central limit theorem for the estimation error of a parametric estimator, from which confidence intervals can be constructed. Statistical efficiency is demonstrated by establishing local asymptotic normality. The estimation method is extended to local observations in time and space, which allows for non-parametric estimation of a reaction intensity varying in time and space. Furthermore, discrete observations in time and space can be handled. The statistical analysis requires advanced tools from stochastic analysis like Malliavin calculus for SPDEs, the infinite-dimensional Gaussian Poincar'e inequality and regularity results for SPDEs in $L^p$-interpolation spaces.
{"title":"Estimation for the reaction term in semi-linear SPDEs under small diffusivity","authors":"Sascha Gaudlitz, M. Reiß","doi":"10.3150/22-bej1573","DOIUrl":"https://doi.org/10.3150/22-bej1573","url":null,"abstract":"We consider the estimation of a non-linear reaction term in the stochastic heat or more generally in a semi-linear stochastic partial differential equation (SPDE). Consistent inference is achieved by studying a small diffusivity level, which is realistic in applications. Our main result is a central limit theorem for the estimation error of a parametric estimator, from which confidence intervals can be constructed. Statistical efficiency is demonstrated by establishing local asymptotic normality. The estimation method is extended to local observations in time and space, which allows for non-parametric estimation of a reaction intensity varying in time and space. Furthermore, discrete observations in time and space can be handled. The statistical analysis requires advanced tools from stochastic analysis like Malliavin calculus for SPDEs, the infinite-dimensional Gaussian Poincar'e inequality and regularity results for SPDEs in $L^p$-interpolation spaces.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45090905","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}
The main result of this article is that we obtain an elementwise error bound for the Fused Lasso estimator for any general convex loss function $rho$. We then focus on the special cases when either $rho$ is the square loss function (for mean regression) or is the quantile loss function (for quantile regression) for which we derive new pointwise error bounds. Even though error bounds for the usual Fused Lasso estimator and its quantile version have been studied before; our bound appears to be new. This is because all previous works bound a global loss function like the sum of squared error, or a sum of Huber losses in the case of quantile regression in Padilla and Chatterjee (2021). Clearly, element wise bounds are stronger than global loss error bounds as it reveals how the loss behaves locally at each point. Our element wise error bound also has a clean and explicit dependence on the tuning parameter $lambda$ which informs the user of a good choice of $lambda$. In addition, our bound is nonasymptotic with explicit constants and is able to recover almost all the known results for Fused Lasso (both mean and quantile regression) with additional improvements in some cases.
{"title":"Element-wise estimation error of generalized Fused Lasso","authors":"Teng Zhang, S. Chatterjee","doi":"10.3150/22-bej1557","DOIUrl":"https://doi.org/10.3150/22-bej1557","url":null,"abstract":"The main result of this article is that we obtain an elementwise error bound for the Fused Lasso estimator for any general convex loss function $rho$. We then focus on the special cases when either $rho$ is the square loss function (for mean regression) or is the quantile loss function (for quantile regression) for which we derive new pointwise error bounds. Even though error bounds for the usual Fused Lasso estimator and its quantile version have been studied before; our bound appears to be new. This is because all previous works bound a global loss function like the sum of squared error, or a sum of Huber losses in the case of quantile regression in Padilla and Chatterjee (2021). Clearly, element wise bounds are stronger than global loss error bounds as it reveals how the loss behaves locally at each point. Our element wise error bound also has a clean and explicit dependence on the tuning parameter $lambda$ which informs the user of a good choice of $lambda$. In addition, our bound is nonasymptotic with explicit constants and is able to recover almost all the known results for Fused Lasso (both mean and quantile regression) with additional improvements in some cases.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42579811","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}
Gaussian couplings of partial sum processes are derived for the high-dimensional regime $d=o(n^{1/3})$. The coupling is derived for sums of independent random vectors and subsequently extended to nonstationary time series. Our inequalities depend explicitly on the dimension and on a measure of nonstationarity, and are thus also applicable to arrays of random vectors. To enable high-dimensional statistical inference, a feasible Gaussian approximation scheme is proposed. Applications to sequential testing and change-point detection are described.
{"title":"Sequential Gaussian approximation for nonstationary time series in high dimensions","authors":"Fabian Mies, A. Steland","doi":"10.3150/22-bej1577","DOIUrl":"https://doi.org/10.3150/22-bej1577","url":null,"abstract":"Gaussian couplings of partial sum processes are derived for the high-dimensional regime $d=o(n^{1/3})$. The coupling is derived for sums of independent random vectors and subsequently extended to nonstationary time series. Our inequalities depend explicitly on the dimension and on a measure of nonstationarity, and are thus also applicable to arrays of random vectors. To enable high-dimensional statistical inference, a feasible Gaussian approximation scheme is proposed. Applications to sequential testing and change-point detection are described.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46140493","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":"Erratum for Prediction and estimation consistency of sparse multi-class penalized optimal scoring","authors":"I. Gaynanova","doi":"10.3150/21-bej1359","DOIUrl":"https://doi.org/10.3150/21-bej1359","url":null,"abstract":"","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41690610","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 prove that the random simple cubic planar graph $mathsf{C}_n$ with an even number $n$ of vertices admits a novel uniform infinite cubic planar graph (UICPG) as quenched local limit. We describe how the limit may be constructed by a series of random blow-up operations applied to the dual map of the type~III Uniform Infinite Planar Triangulation established by Angel and Schramm (Comm. Math. Phys., 2003). Our main technical lemma is a contiguity relation between $mathsf{C}_n$ and a model where the networks inserted at the links of the largest $3$-connected component of $mathsf{C}_n$ are replaced by independent copies of a specific Boltzmann network. We prove that the number of vertices of the largest $3$-connected component concentrates at $kappa n$ for $kappa approx 0.85085$, with Airy-type fluctuations of order $n^{2/3}$. The second-largest component is shown to have significantly smaller size $O_p(n^{2/3})$.
{"title":"The uniform infinite cubic planar graph","authors":"Benedikt Stufler","doi":"10.3150/22-bej1568","DOIUrl":"https://doi.org/10.3150/22-bej1568","url":null,"abstract":"We prove that the random simple cubic planar graph $mathsf{C}_n$ with an even number $n$ of vertices admits a novel uniform infinite cubic planar graph (UICPG) as quenched local limit. We describe how the limit may be constructed by a series of random blow-up operations applied to the dual map of the type~III Uniform Infinite Planar Triangulation established by Angel and Schramm (Comm. Math. Phys., 2003). Our main technical lemma is a contiguity relation between $mathsf{C}_n$ and a model where the networks inserted at the links of the largest $3$-connected component of $mathsf{C}_n$ are replaced by independent copies of a specific Boltzmann network. We prove that the number of vertices of the largest $3$-connected component concentrates at $kappa n$ for $kappa approx 0.85085$, with Airy-type fluctuations of order $n^{2/3}$. The second-largest component is shown to have significantly smaller size $O_p(n^{2/3})$.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48636520","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}
This paper addresses the problem of deriving the asymptotic distribution of the empirical distribution function F n of the residuals in a general class of time series models, including conditional mean and conditional heteroscedaticity, whose independent and identically distributed errors have unknown distribution F. We show that, for a large class of time series models (including the standard ARMA-GARCH), the asymptotic distribution of √ n{ F n (·) − F (·)} is impacted by the estimation but does not depend on the model parameters. It is thus neither asymptotically estimation free, as is the case for purely linear models, nor asymptotically model dependent, as is the case for some nonlinear models. The asymptotic stochastic equicontinuity is also established. We consider an application to the estimation of the conditional Value-at-Risk.
{"title":"Adaptiveness of the empirical distribution of residuals in semi-parametric conditional location scale models","authors":"C. Francq, J. Zakoian","doi":"10.3150/21-bej1357","DOIUrl":"https://doi.org/10.3150/21-bej1357","url":null,"abstract":"This paper addresses the problem of deriving the asymptotic distribution of the empirical distribution function F n of the residuals in a general class of time series models, including conditional mean and conditional heteroscedaticity, whose independent and identically distributed errors have unknown distribution F. We show that, for a large class of time series models (including the standard ARMA-GARCH), the asymptotic distribution of √ n{ F n (·) − F (·)} is impacted by the estimation but does not depend on the model parameters. It is thus neither asymptotically estimation free, as is the case for purely linear models, nor asymptotically model dependent, as is the case for some nonlinear models. The asymptotic stochastic equicontinuity is also established. We consider an application to the estimation of the conditional Value-at-Risk.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49295738","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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