We study one-dimensional stochastic differential equations of the form dXt=σ(Xt)dYtdX_t = sigma (X_t)dY_t, where YY is a suitable Hölder continuous driver such as the fractional Brownian motion BHB^H with H>12H>frac 12. The innovative aspect of the present paper lies in the assumptions on diffusion coefficients σsigma for which we assume very mild conditions. In particular, we allow σsigma to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.
我们研究了dX t = σ (X t)dY t dX＿t = sigma (X＿t)dY＿t的一维随机微分方程，其中Y Y是一个合适的Hölder连续驱动器，如分数阶布朗运动B H B^H with H ＆gt;12 H＆gt;frac本文的创新之处在于对扩散系数σ sigma的假设，我们假设了非常温和的条件。特别地，我们允许σ sigma具有不连续，因此我们的结果可以应用于研究具有不连续扩散的方程。
{"title":"Stochastic differential equations with discontinuous diffusion coefficients","authors":"Soledad Torres, Lauri Viitasaari","doi":"10.1090/tpms/1201","DOIUrl":"https://doi.org/10.1090/tpms/1201","url":null,"abstract":"We study one-dimensional stochastic differential equations of the form <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d upper X Subscript t Baseline equals sigma left-parenthesis upper X Subscript t Baseline right-parenthesis d upper Y Subscript t\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">dX_t = sigma (X_t)dY_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Y\"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=\"application/x-tex\">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a suitable Hölder continuous driver such as the fractional Brownian motion <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B Superscript upper H\"> <mml:semantics> <mml:msup> <mml:mi>B</mml:mi> <mml:mi>H</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">B^H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H greater-than one half\"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">H>frac 12</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The innovative aspect of the present paper lies in the assumptions on diffusion coefficients <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma\"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which we assume very mild conditions. In particular, we allow <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma\"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135695492","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}
Jonathan von Schroeder, Thorsten Dickhaus, Taras Bodnar
Stylized facts about financial data comprise skewed and heavy-tailed (log-)returns. Therefore, we revisit previous results on reverse stress testing under elliptical models, and we extend them to the broader class of skew-elliptical models. In the elliptical case, an explicit formula for the solution is provided. In the skew-elliptical case, we characterize the solution in terms of an easy-to-implement numerical optimization problem. As specific examples, we investigate the classes of skew-normal and skew-t models in detail. Since the solutions depend on population parameters, which are often unknown in practice, we also tackle the statistical task of estimating these parameters and provide confidence regions for the most likely scenarios.
{"title":"Reverse stress testing in skew-elliptical models","authors":"Jonathan von Schroeder, Thorsten Dickhaus, Taras Bodnar","doi":"10.1090/tpms/1199","DOIUrl":"https://doi.org/10.1090/tpms/1199","url":null,"abstract":"Stylized facts about financial data comprise skewed and heavy-tailed (log-)returns. Therefore, we revisit previous results on reverse stress testing under elliptical models, and we extend them to the broader class of skew-elliptical models. In the elliptical case, an explicit formula for the solution is provided. In the skew-elliptical case, we characterize the solution in terms of an easy-to-implement numerical optimization problem. As specific examples, we investigate the classes of skew-normal and skew-t models in detail. Since the solutions depend on population parameters, which are often unknown in practice, we also tackle the statistical task of estimating these parameters and provide confidence regions for the most likely scenarios.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135695495","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}
In MM-estimation we consider the sets of all minimizing points of convex empirical criterion functions. These sets are random closed sets. We derive distributional convergence in the hyperspace of all closed subsets of the real line endowed with the Fell-topology. As a special case single minimizing points converge in distribution in the classical sense. In contrast to the literature so far, unusual rates of convergence and non-normal limits emerge, which go far beyond the square-root asymptotic normality. Moreover, our theory can be applied to the sets of zero-estimators.
{"title":"Distributional hyperspace-convergence of Argmin-sets in convex 𝑀-estimation","authors":"Dietmar Ferger","doi":"10.1090/tpms/1195","DOIUrl":"https://doi.org/10.1090/tpms/1195","url":null,"abstract":"In <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-estimation we consider the sets of all minimizing points of convex empirical criterion functions. These sets are random closed sets. We derive distributional convergence in the hyperspace of all closed subsets of the real line endowed with the Fell-topology. As a special case single minimizing points converge in distribution in the classical sense. In contrast to the literature so far, unusual rates of convergence and non-normal limits emerge, which go far beyond the square-root asymptotic normality. Moreover, our theory can be applied to the sets of zero-estimators.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135695499","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}
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.
{"title":"Test for mean matrix in GMANOVA model under heteroscedasticity and non-normality for high-dimensional data","authors":"Takayuki Yamada, Tetsuto Himeno, Annika Tillander, Tatjana Pavlenko","doi":"10.1090/tpms/1200","DOIUrl":"https://doi.org/10.1090/tpms/1200","url":null,"abstract":"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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135648619","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}
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。
{"title":"Graphical posterior predictive classification: Bayesian model averaging with particle Gibbs","authors":"Tatjana Pavlenko, Felix Rios","doi":"10.1090/tpms/1198","DOIUrl":"https://doi.org/10.1090/tpms/1198","url":null,"abstract":"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 <italic>Bayesian model averaging</italic> 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 <italic>Christmas tree algorithm</italic> 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 <italic>hyper normal Wishart law</italic> 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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":"98 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135695501","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}
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.
{"title":"Matrix variate generalized asymmetric Laplace distributions","authors":"Tomasz Kozubowski, Stepan Mazur, Krzysztof Podgórski","doi":"10.1090/tpms/1197","DOIUrl":"https://doi.org/10.1090/tpms/1197","url":null,"abstract":"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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135696137","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 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.
{"title":"A note on the prediction error of principal component regression in high dimensions","authors":"Laura Hucker, Martin Wahl","doi":"10.1090/tpms/1196","DOIUrl":"https://doi.org/10.1090/tpms/1196","url":null,"abstract":"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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135695493","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}
{"title":"Statistical inference for models driven by 𝑛-th order fractional Brownian motion","authors":"Hicham Chaouch, H. Maroufy, Mohamed Omari","doi":"10.1090/tpms/1185","DOIUrl":"https://doi.org/10.1090/tpms/1185","url":null,"abstract":"<p>We consider the following stochastic integral equation <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X left-parenthesis t right-parenthesis equals mu t plus sigma integral Subscript 0 Superscript t Baseline phi left-parenthesis s right-parenthesis d upper B Subscript upper H Superscript n Baseline left-parenthesis s right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>μ<!-- μ --></mml:mi>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>σ<!-- σ --></mml:mi>\u0000 <mml:msubsup>\u0000 <mml:mo>∫<!-- ∫ --></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mi>t</mml:mi>\u0000 </mml:msubsup>\u0000 <mml:mi>φ<!-- φ --></mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>s</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:msubsup>\u0000 <mml:mi>B</mml:mi>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msubsup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>s</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">X(t)=mu t + sigma int _0^t varphi (s) dB_H^n(s)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t greater-than-or-equal-to 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">tgeq 0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, where <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"phi\">\u0000 <mml:semantics>\u0000 <mml:mi>φ<!-- φ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">varphi</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a known function and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B Subscript upper H Superscript n\">\u0000 <mml:semantics>\u0000 <mml:msubsup>\u0000 <mml:mi>B</mml:mi>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msubsup>\u0000 <mml:annotation encoding=\"application/x-tex\">B^n_H</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-th order fractional Brownian motion. We provide explicit maximum likelihood estimators for both <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http:","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41595813","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}
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.
{"title":"Approximations for success run probabilities in Bernoulli trials","authors":"S. Kaczkowski","doi":"10.1090/tpms/1186","DOIUrl":"https://doi.org/10.1090/tpms/1186","url":null,"abstract":"Concise and convenient bounds are obtained for the probability mass and cumulative distribution functions associated with the first success run of length \u0000\u0000 \u0000 k\u0000 k\u0000 \u0000\u0000 in a sequence of \u0000\u0000 \u0000 n\u0000 n\u0000 \u0000\u0000 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 \u0000\u0000 \u0000 k\u0000 k\u0000 \u0000\u0000 within \u0000\u0000 \u0000 N\u0000 N\u0000 \u0000\u0000 concurrent sequences of Bernoulli trials.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45708546","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}
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。
{"title":"A comment on rates of convergence for density function in extreme value theory and Rényi entropy","authors":"Ali Saeb","doi":"10.1090/tpms/1191","DOIUrl":"https://doi.org/10.1090/tpms/1191","url":null,"abstract":"De Haan and Resnick [Ann. Probab. 10 (1982), no. 2, 396–413] have shown that the Rényi entropy of order \u0000\u0000 \u0000 β\u0000 beta\u0000 \u0000\u0000 (\u0000\u0000 \u0000 \u0000 β\u0000 >\u0000 1\u0000 \u0000 beta >1\u0000 \u0000\u0000) of normalized sample maximum of independent and identically distributed (iid) random variables with continuous differentiable density converges to the Rényi entropy of order \u0000\u0000 \u0000 β\u0000 beta\u0000 \u0000\u0000 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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45040022","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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