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Graphical posterior predictive classification: Bayesian model averaging with particle Gibbs 图形后验预测分类:粒子吉布斯贝叶斯平均模型
Q3 Mathematics Pub Date : 2023-10-03 DOI: 10.1090/tpms/1198
Tatjana Pavlenko, Felix Rios
In this study, we present a multi-class graphical Bayesian predictive classifier that incorporates the uncertainty in the model selection into the standard Bayesian formalism. For each class, the dependence structure underlying the observed features is represented by a set of decomposable Gaussian graphical models. Emphasis is then placed on the Bayesian model averaging which takes full account of the class-specific model uncertainty by averaging over the posterior graph model probabilities. An explicit evaluation of the model probabilities is well known to be infeasible. To address this issue, we consider the particle Gibbs strategy of J. Olsson, T. Pavlenko, and F. L. Rios [Electron. J. Statist. 13 (2019), no. 2, 2865–2897] for posterior sampling from decomposable graphical models which utilizes the so-called Christmas tree algorithm of J. Olsson, T. Pavlenko, and F. L. Rios [Stat. Comput. 32 (2022), no. 5, Paper No. 80, 18] as proposal kernel. We also derive a strong hyper Markov law which we call the hyper normal Wishart law that allows to perform the resultant Bayesian calculations locally. The proposed predictive graphical classifier reveals superior performance compared to the ordinary Bayesian predictive rule that does not account for the model uncertainty, as well as to a number of out-of-the-box classifiers.
在这项研究中,我们提出了一个多类图形贝叶斯预测分类器,将模型选择中的不确定性纳入标准贝叶斯形式。对于每一类,观察到的特征的依赖结构由一组可分解的高斯图形模型表示。然后将重点放在贝叶斯模型平均上,该模型通过平均后验图模型概率,充分考虑了特定类别的模型不确定性。众所周知,对模型概率的明确评估是不可行的。为了解决这个问题,我们考虑了J. Olsson, T. Pavlenko和F. L. Rios [Electron]的粒子吉布斯策略。统计学家。13 (2019),no。J. Olsson, T. Pavlenko和F. L. Rios的所谓圣诞树算法[Stat. Comput. 32 (2022), no. 1],用于可分解图形模型的后测抽样。5、论文第80,18号作为提案核心。我们还推导了一个强超马尔可夫定律,我们称之为超正态Wishart定律,它允许在局部执行所得到的贝叶斯计算。与不考虑模型不确定性的普通贝叶斯预测规则以及一些开箱即用的分类器相比,所提出的预测图形分类器显示出优越的性能。
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引用次数: 0
Test for mean matrix in GMANOVA model under heteroscedasticity and non-normality for high-dimensional data 高维数据异方差和非正态下GMANOVA模型均值矩阵的检验
Q3 Mathematics Pub Date : 2023-10-03 DOI: 10.1090/tpms/1200
Takayuki Yamada, Tetsuto Himeno, Annika Tillander, Tatjana Pavlenko
This paper is concerned with the testing bilateral linear hypothesis on the mean matrix in the context of the generalized multivariate analysis of variance (GMANOVA) model when the dimensions of the observed vector may exceed the sample size, the design may become unbalanced, the population may not be normal, or the true covariance matrices may be unequal. The suggested testing methodology can treat many problems such as the one- and two-way MANOVA tests, the test for parallelism in profile analysis, etc., as specific ones. We propose a bias-corrected estimator of the Frobenius norm for the mean matrix, which is a key component of the test statistic. The null and non-null distributions are derived under a general high-dimensional asymptotic framework that allows the dimensionality to arbitrarily exceed the sample size of a group, thereby establishing consistency for the testing criterion. The accuracy of the proposed test in a finite sample is investigated through simulations conducted for several high-dimensional scenarios and various underlying population distributions in combination with different within-group covariance structures. Finally, the proposed test is applied to a high-dimensional two-way MANOVA problem for DNA microarray data.
本文讨论了在广义多元方差分析(GMANOVA)模型中,当观测向量的维度可能超过样本量、设计可能变得不平衡、总体可能不正常或真协方差矩阵不相等时,在平均矩阵上检验双边线性假设的问题。所建议的测试方法可以将许多问题,如单、双向方差分析测试、配置文件分析中的并行性测试等,作为具体的问题来处理。我们提出了Frobenius范数的偏校正估计的平均矩阵,这是检验统计量的一个关键组成部分。零分布和非零分布是在一般的高维渐近框架下推导出来的,该框架允许维数任意超过一个组的样本量,从而建立了检验标准的一致性。通过对几个高维场景和各种潜在种群分布结合不同组内协方差结构的模拟,研究了在有限样本中提出的测试的准确性。最后,将提出的测试应用于DNA微阵列数据的高维双向方差分析问题。
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引用次数: 0
Matrix variate generalized asymmetric Laplace distributions 矩阵变量广义非对称拉普拉斯分布
Q3 Mathematics Pub Date : 2023-10-03 DOI: 10.1090/tpms/1197
Tomasz Kozubowski, Stepan Mazur, Krzysztof Podgórski
The generalized asymmetric Laplace (GAL) distributions, also known as the variance/mean-gamma models, constitute a popular flexible class of distributions that can account for peakedness, skewness, and heavier-than-normal tails, often observed in financial or other empirical data. We consider extensions of the GAL distribution to the matrix variate case, which arise as covariance mixtures of matrix variate normal distributions. Two different mixing mechanisms connected with the nature of the random scaling matrix are considered, leading to what we term matrix variate GAL distributions of Type I and II. While Type I matrix variate GAL distribution has been studied before, there is no comprehensive account of Type II in the literature, except for their rather brief treatment as a special case of matrix variate generalized hyperbolic distributions. With this work we fill this gap, and present an account for basic distributional properties of Type II matrix variate GAL distributions. In particular, we derive their probability density function and the characteristic function, as well as provide stochastic representations related to matrix variate gamma distribution. We also show that this distribution is closed under linear transformations, and study the relevant marginal distributions. In addition, we also briefly account for Type I and discuss the intriguing connections with Type II. We hope that this work will be useful in the areas where matrix variate distributions provide an appropriate probabilistic tool for three-way or, more generally, panel data sets, which can arise across different applications.
广义非对称拉普拉斯(GAL)分布,也称为方差/均值-伽马模型,构成了一类流行的灵活分布,可以解释峰态、偏态和比正态更重的尾部,通常在金融或其他经验数据中观察到。我们考虑将GAL分布扩展到矩阵变量情况,即矩阵变量正态分布的协方差混合。考虑了与随机标度矩阵的性质有关的两种不同的混合机制,导致我们称之为I型和II型矩阵变量GAL分布。虽然以前已经研究过I型矩阵变量GAL分布,但除了将II型作为矩阵变量广义双曲分布的特殊情况进行了相当简短的处理外,文献中没有对II型进行全面的描述。通过这项工作,我们填补了这一空白,并提出了II型矩阵变量GAL分布的基本分布性质的解释。特别是,我们推导了它们的概率密度函数和特征函数,并提供了与矩阵变量伽马分布相关的随机表示。我们还证明了该分布在线性变换下是封闭的,并研究了相关的边际分布。此外,我们还简要说明了I型,并讨论了与II型的有趣联系。我们希望这项工作将在矩阵变量分布为三方或更普遍的面板数据集提供适当概率工具的领域有用,这些数据集可能出现在不同的应用中。
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引用次数: 0
A note on the prediction error of principal component regression in high dimensions 关于高维主成分回归预测误差的注记
Q3 Mathematics Pub Date : 2023-10-03 DOI: 10.1090/tpms/1196
Laura Hucker, Martin Wahl
We analyze the prediction error of principal component regression (PCR) and prove high probability bounds for the corresponding squared risk conditional on the design. Our first main result shows that PCR performs comparably to the oracle method obtained by replacing empirical principal components by their population counterparts, provided that an effective rank condition holds. On the other hand, if the latter condition is violated, then empirical eigenvalues start to have a significant upward bias, resulting in a self-induced regularization of PCR. Our approach relies on the behavior of empirical eigenvalues, empirical eigenvectors and the excess risk of principal component analysis in high-dimensional regimes.
我们分析了主成分回归(PCR)的预测误差,并证明了在设计条件下相应风险平方的高概率界。我们的第一个主要结果表明,只要有效的秩条件成立,PCR的性能与用种群对应的主成分代替经验主成分得到的oracle方法相当。另一方面,如果违反后一个条件,则经验特征值开始具有显著的向上偏差,导致PCR的自诱导正则化。我们的方法依赖于经验特征值、经验特征向量的行为和高维状态下主成分分析的超额风险。
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引用次数: 1
Statistical inference for models driven by 𝑛-th order fractional Brownian motion 由驱动的模型的统计推断𝑛-阶分数布朗运动
IF 0.9 Q3 Mathematics Pub Date : 2023-05-02 DOI: 10.1090/tpms/1185
Hicham Chaouch, H. Maroufy, Mohamed Omari

We consider the following stochastic integral equation X ( t ) = μ t + σ 0 t φ ( s ) d B H n ( s ) X(t)=mu t + sigma int _0^t varphi (s) dB_H^n(s) , t 0 tgeq 0 , where φ varphi is a known function and B H n B^n_H is the n n -th order fractional Brownian motion. We provide explicit maximum likelihood estimators for both

我们考虑以下随机积分方程X(t)=μ,其中φvarphi是一个已知函数,并且BhnB^n_H是n阶分数布朗运动。我们给出了μμ和σ2sigma^2的显式最大似然估计量,然后我们用幂变分法显式地公式化了μμμ的最小二乘估计量和σ2sigma^2的估计量。当观测次数或时间范围足够大时,建立了这些估计量的一致性和渐近正态性。
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引用次数: 0
Approximations for success run probabilities in Bernoulli trials 伯努利试验成功运行概率的近似
IF 0.9 Q3 Mathematics Pub Date : 2023-05-02 DOI: 10.1090/tpms/1186
S. Kaczkowski
Concise and convenient bounds are obtained for the probability mass and cumulative distribution functions associated with the first success run of length k k in a sequence of n n Bernoulli trials. Results are compared to an approximation obtained by the Stein–Chen method as well as to bounds obtained from statistical reliability theory. These approximation formulas are used to obtain precise estimates of the expectation value associated with the occurrence of at least one success run of length k k within N N concurrent sequences of Bernoulli trials.
在一系列n个伯努利试验中,获得了与长度为k k的第一次成功游程相关的概率质量和累积分布函数的简明和方便的边界。将结果与Stein–Chen方法获得的近似值以及统计可靠性理论获得的边界进行比较。这些近似公式用于获得期望值的精确估计,该期望值与伯努利试验的N个并发序列中至少一个长度为k k的成功游程的发生有关。
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引用次数: 0
A comment on rates of convergence for density function in extreme value theory and Rényi entropy 关于极值理论中密度函数与Rényi熵的收敛速度
IF 0.9 Q3 Mathematics Pub Date : 2023-05-02 DOI: 10.1090/tpms/1191
Ali Saeb
De Haan and Resnick [Ann. Probab. 10 (1982), no. 2, 396–413] have shown that the Rényi entropy of order β beta ( β > 1 beta >1 ) of normalized sample maximum of independent and identically distributed (iid) random variables with continuous differentiable density converges to the Rényi entropy of order β beta of a max stable law. In this paper, we review the rate of convergence for density function in extreme value theory. Finally, we study the rate of convergence for Rényi entropy in the case of normalized sample maxima.
De Haan和Resnick[Ann.Probab.10(1982),no.2396–413]已经表明,具有连续可微密度的独立同分布(iid)随机变量的归一化样本极大值的ββ阶Rényi熵(β>1β>1)收敛于极大稳定律的β贝塔阶Rény熵。本文讨论了极值理论中密度函数的收敛速度。最后,我们研究了Rényi熵在归一化样本最大值情况下的收敛速度。
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引用次数: 1
Distribution of the product of a Wishart matrix and a normal vector 一个Wishart矩阵和一个法向量的乘积的分布
IF 0.9 Q3 Mathematics Pub Date : 2023-05-02 DOI: 10.1090/tpms/1193
Koshiro Yonenaga, A. Suzukawa
We consider the distribution of the product of a Wishart matrix and a normal vector with uncommon covariance matrices. We derive the stochastic representation which reduces the computational burden for the generation of realizations of the product. Using this representation, the density function and higher order moments of the product are derived. In a numerical illustration, we investigate some properties of the distribution of the product. We further suggest the Edgeworth type expansions for the product, and we observe that the suggested approximations provide a good performance for moderately large degrees of freedom of a Wishart matrix.
我们考虑了具有不常见协方差矩阵的Wishart矩阵与法向量乘积的分布。我们推导了随机表示,减少了生成产品实现的计算负担。利用这种表示,导出了乘积的密度函数和高阶矩。在一个数值说明中,我们研究了乘积分布的一些性质。我们进一步提出了该乘积的Edgeworth型展开式,并且我们观察到,所建议的近似对于Wishart矩阵的中等大自由度提供了良好的性能。
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引用次数: 0
On recurrence and transience of some Lévy-type processes in ℝ 若干l<s:1>型过程的递归性和暂态性
IF 0.9 Q3 Mathematics Pub Date : 2023-05-02 DOI: 10.1090/tpms/1187
V. Knopova

In this note we prove some sufficient conditions for transience and recurrence of a Lévy-type process in R mathbb {R} , whose generator defined on the test functions is of the form L f ( x ) = R ( f ( x + u ) f ( x ) f ( x ) u 1 | u | 1

在本文中,我们证明了R mathbb R{中l型过程的暂态和递归的几个充分条件。其生成函数定义在测试函数上的形式为L f (x) =∫R (f (x + u) - f (x) -∇f (x)·u 1 | u |≤1)ν (x, du), f∈C∞2 (R)。}begin{equation*} Lf(x) =int _{mathbb {R}} left ( f(x+u)-f(x)- nabla f(x)cdot u mathbb {1}_{|u|leq 1} right ) nu (x,du), quad fin C_infty ^2(mathbb {R}). end{equation*}这里的ν (x,du) nu (x,du)是一个l型核,它的尾部要么有规律地扩展变化,要么衰减得足够快。为了证明,使用了Foster-Lyapunov方法。
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引用次数: 0
Gaussian Volterra processes: Asymptotic growth and statistical estimation 高斯Volterra过程:渐近增长与统计估计
IF 0.9 Q3 Mathematics Pub Date : 2023-02-07 DOI: 10.1090/tpms/1190
Y. Mishura, K. Ralchenko, S. Shklyar
The paper is devoted to three-parametric self-similar Gaussian Volterra processes that generalize fractional Brownian motion. We study the asymptotic growth of such processes and the properties of long- and short-range dependence. Then we consider the problem of the drift parameter estimation for Ornstein–Uhlenbeck process driven by Gaussian Volterra process under consideration. We construct a strongly consistent estimator and investigate its asymptotic properties. Namely, we prove that it has the Cauchy asymptotic distribution.
本文研究了推广分数布朗运动的三个参数自相似高斯-Volterra过程。我们研究了这类过程的渐近增长和长短程依赖性的性质。然后,我们考虑高斯-Volterra过程驱动的Ornstein–Uhlenbeck过程的漂移参数估计问题。我们构造了一个强一致估计量,并研究了它的渐近性质。也就是说,我们证明了它具有柯西渐近分布。
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引用次数: 0
期刊
Theory of Probability and Mathematical Statistics
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