Pub Date : 2024-06-13DOI: 10.1016/j.spl.2024.110182
Alexis Derumigny , Johannes Schmidt-Hieber
In nonparametric statistics, rate-optimal estimators typically balance bias and stochastic error. The recent work on overparametrization raises the question whether rate-optimal estimators exist that do not obey this trade-off. In this work we consider pointwise estimation in the Gaussian white noise model with regression function in a class of -Hölder smooth functions. Let ’worst-case’ refer to the supremum over all functions in the Hölder class. It is shown that any estimator with worst-case bias must necessarily also have a worst-case mean absolute deviation that is lower bounded by To derive the result, we establish abstract inequalities relating the change of expectation for two probability measures to the mean absolute deviation.
在非参数统计中,最优估计率通常会兼顾偏差和随机误差。最近关于超参数化的研究提出了一个问题:是否存在不服从这种权衡的最优估计率?在本研究中,我们将考虑在高斯白噪声模型中,用一类 β-Hölder 平滑函数中的回归函数 f 进行点估计。让 "最坏情况 "指的是 Hölder 类中所有函数 f 的上集。结果表明,任何具有最坏情况偏差≲n-β/(2β+1)≕ψn 的估计器,其最坏情况平均绝对偏差的下限必然也是≳ψn。为了推导出这一结果,我们建立了有关两个概率度量的期望变化与平均绝对偏差的抽象不等式。
{"title":"Lower bounds for the trade-off between bias and mean absolute deviation","authors":"Alexis Derumigny , Johannes Schmidt-Hieber","doi":"10.1016/j.spl.2024.110182","DOIUrl":"https://doi.org/10.1016/j.spl.2024.110182","url":null,"abstract":"<div><p>In nonparametric statistics, rate-optimal estimators typically balance bias and stochastic error. The recent work on overparametrization raises the question whether rate-optimal estimators exist that do not obey this trade-off. In this work we consider pointwise estimation in the Gaussian white noise model with regression function <span><math><mi>f</mi></math></span> in a class of <span><math><mi>β</mi></math></span>-Hölder smooth functions. Let ’worst-case’ refer to the supremum over all functions <span><math><mi>f</mi></math></span> in the Hölder class. It is shown that any estimator with worst-case bias <span><math><mrow><mo>≲</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>β</mi><mo>/</mo><mrow><mo>(</mo><mn>2</mn><mi>β</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup><mo>≕</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> must necessarily also have a worst-case mean absolute deviation that is lower bounded by <span><math><mrow><mo>≳</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>.</mo></mrow></math></span> To derive the result, we establish abstract inequalities relating the change of expectation for two probability measures to the mean absolute deviation.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"213 ","pages":"Article 110182"},"PeriodicalIF":0.8,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167715224001512/pdfft?md5=51662c001093a4b55185099a04b71ff2&pid=1-s2.0-S0167715224001512-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141424191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-07DOI: 10.1016/j.spl.2024.110171
Koya Endo, Yumiharu Nakano
We consider the weak convergence of the Euler–Maruyama approximation for Schrödinger–Föllmer diffusions, which are solutions of Schrödinger bridge problems and can be used for sampling from given distributions. We show that the distribution of the terminal random variable of the time-discretized process weakly converges to the target one under mild regularity conditions.
{"title":"Weak approximation of Schrödinger–Föllmer diffusion","authors":"Koya Endo, Yumiharu Nakano","doi":"10.1016/j.spl.2024.110171","DOIUrl":"https://doi.org/10.1016/j.spl.2024.110171","url":null,"abstract":"<div><p>We consider the weak convergence of the Euler–Maruyama approximation for Schrödinger–Föllmer diffusions, which are solutions of Schrödinger bridge problems and can be used for sampling from given distributions. We show that the distribution of the terminal random variable of the time-discretized process weakly converges to the target one under mild regularity conditions.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"213 ","pages":"Article 110171"},"PeriodicalIF":0.8,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167715224001408/pdfft?md5=b5afb3ddb3bfc9440dc9d67d44f31603&pid=1-s2.0-S0167715224001408-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141314597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-06DOI: 10.1016/j.spl.2024.110170
Paul Kabaila
Yu and Hoff constructed novel confidence intervals for the treatment means in a one-way layout. They assessed the expected lengths of these intervals using a semi-Bayesian analysis. We provide a revealing assessment of these expected lengths using a fully frequentist analysis.
{"title":"On Yu and Hoff’s confidence intervals for treatment means","authors":"Paul Kabaila","doi":"10.1016/j.spl.2024.110170","DOIUrl":"https://doi.org/10.1016/j.spl.2024.110170","url":null,"abstract":"<div><p>Yu and Hoff constructed novel confidence intervals for the treatment means in a one-way layout. They assessed the expected lengths of these intervals using a semi-Bayesian analysis. We provide a revealing assessment of these expected lengths using a fully frequentist analysis.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"213 ","pages":"Article 110170"},"PeriodicalIF":0.8,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0167715224001391/pdfft?md5=2d100e660e5414aefc69743d01f0c8c4&pid=1-s2.0-S0167715224001391-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141302933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-05DOI: 10.1016/j.spl.2024.110169
Oleksii Galganov , Andrii Ilienko
We study the asymptotic behavior of short cycles of random permutations with cycle weights. More specifically, on a specially constructed metric space whose elements encode all possible cycles, we consider a point process containing all information on cycles of a given random permutation on . The main result of the paper is the distributional convergence with respect to the vague topology of the above processes towards a Poisson point process as for a wide range of cycle weights. As an application, we give several limit theorems for various statistics of cycles.
{"title":"Short cycles of random permutations with cycle weights: Point processes approach","authors":"Oleksii Galganov , Andrii Ilienko","doi":"10.1016/j.spl.2024.110169","DOIUrl":"https://doi.org/10.1016/j.spl.2024.110169","url":null,"abstract":"<div><p>We study the asymptotic behavior of short cycles of random permutations with cycle weights. More specifically, on a specially constructed metric space whose elements encode all possible cycles, we consider a point process containing all information on cycles of a given random permutation on <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></mrow></math></span>. The main result of the paper is the distributional convergence with respect to the vague topology of the above processes towards a Poisson point process as <span><math><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math></span> for a wide range of cycle weights. As an application, we give several limit theorems for various statistics of cycles.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"213 ","pages":"Article 110169"},"PeriodicalIF":0.8,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141302851","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}
Pub Date : 2024-06-04DOI: 10.1016/j.spl.2024.110174
Xiangyu Huang
Arboreal Gas is a type of (unrooted) random forest on a graph, where the probability is determined by a parameter per edge. This model can be considered as the limit of the -states random cluster model with as . A natural question arises regarding the existence and performance of the weak limit of Arboreal Gas as the graph size goes to infinity. The answer to this question relies on the negative correlation of Arboreal Gas, which is still an open problem. This paper primarily focuses on the negative correlation of Arboreal Gas and provides some results for specific parameters.
Arboreal Gas 是一种图上的(无根)随机森林,其概率由每条边的参数 β>0 决定。该模型可视为 q→0 时 p=βq 的 q 态随机簇模型的极限。当图的大小达到无穷大时,自然会产生一个关于 Arboreal Gas 弱极限的存在和性能的问题。这个问题的答案取决于 Arboreal Gas 的负相关,而这仍是一个未决问题。本文主要关注 Arboreal Gas 的负相关性,并提供了一些特定参数的结果。
{"title":"On negative correlation of Arboreal Gas for specific parameters","authors":"Xiangyu Huang","doi":"10.1016/j.spl.2024.110174","DOIUrl":"https://doi.org/10.1016/j.spl.2024.110174","url":null,"abstract":"<div><p>Arboreal Gas is a type of (unrooted) random forest on a graph, where the probability is determined by a parameter <span><math><mrow><mi>β</mi><mo>></mo><mn>0</mn></mrow></math></span> per edge. This model can be considered as the limit of the <span><math><mi>q</mi></math></span>-states random cluster model with <span><math><mrow><mi>p</mi><mo>=</mo><mi>β</mi><mi>q</mi></mrow></math></span> as <span><math><mrow><mi>q</mi><mo>→</mo><mn>0</mn></mrow></math></span>. A natural question arises regarding the existence and performance of the weak limit of Arboreal Gas as the graph size goes to infinity. The answer to this question relies on the negative correlation of Arboreal Gas, which is still an open problem. This paper primarily focuses on the negative correlation of Arboreal Gas and provides some results for specific parameters.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"213 ","pages":"Article 110174"},"PeriodicalIF":0.8,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141290682","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}
Pub Date : 2024-06-04DOI: 10.1016/j.spl.2024.110173
Lijian Yang
Under nearly minimal conditions, continuity of extreme distribution function is established for both continuous Gaussian processes and finite Gaussian sequences, which entails existence of exact quantiles at any level. Also proved under simple conditions is strict monotonicity of extreme distribution functions that ensures uniqueness of exact quantiles at any level. These results provide convenient tools for developing statistical theory about global inference on functions.
{"title":"Exact quantiles of Gaussian process extremes","authors":"Lijian Yang","doi":"10.1016/j.spl.2024.110173","DOIUrl":"https://doi.org/10.1016/j.spl.2024.110173","url":null,"abstract":"<div><p>Under nearly minimal conditions, continuity of extreme distribution function is established for both continuous Gaussian processes and finite Gaussian sequences, which entails existence of exact quantiles at any level. Also proved under simple conditions is strict monotonicity of extreme distribution functions that ensures uniqueness of exact quantiles at any level. These results provide convenient tools for developing statistical theory about global inference on functions.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"213 ","pages":"Article 110173"},"PeriodicalIF":0.8,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141289146","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}
Pub Date : 2024-06-04DOI: 10.1016/j.spl.2024.110168
Ping Chen , Tusheng Zhang
In this paper, we consider reflected stochastic differential equations (SDEs) with interaction in a multidimensional general domain. The well-posedness is established under a monotone condition, and the exponential ergodicity is derived in the Wasserstein distance.
{"title":"Exponential ergodicity for reflected SDEs with interaction in a multidimensional general domain","authors":"Ping Chen , Tusheng Zhang","doi":"10.1016/j.spl.2024.110168","DOIUrl":"https://doi.org/10.1016/j.spl.2024.110168","url":null,"abstract":"<div><p>In this paper, we consider reflected stochastic differential equations (SDEs) with interaction in a multidimensional general domain. The well-posedness is established under a monotone condition, and the exponential ergodicity is derived in the Wasserstein distance.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"213 ","pages":"Article 110168"},"PeriodicalIF":0.8,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141289444","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}
Pub Date : 2024-06-03DOI: 10.1016/j.spl.2024.110172
Tianshi Lu
In this paper we characterized isotropic random tangential vector fields on -spheres for by the cross-covariance, and derived their Karhunen–Loève expansion. The tangential vector field can be decomposed into a curl-free part and a divergence-free part by the Helmholtz–Hodge decomposition. We proved that the two parts can be correlated on a 2-sphere, while they must be uncorrelated on a -sphere for . On a 3-sphere, the divergence-free part can be further decomposed into two isotropic flows.
在本文中,我们通过交叉协方差描述了 d≥1 时 d 球上各向同性随机切向矢量场的特征,并推导出了它们的卡尔胡宁-洛夫展开(Karhunen-Loève expansion)。切向矢量场可以通过亥姆霍兹-霍奇分解分解为无卷曲部分和无发散部分。我们证明了这两部分在 2 球面上可以相关,而在 d≥3 的 d 球面上必须不相关。在 3 球面上,无发散部分可以进一步分解为两个各向同性流。
{"title":"Isotropic random tangential vector fields on spheres","authors":"Tianshi Lu","doi":"10.1016/j.spl.2024.110172","DOIUrl":"10.1016/j.spl.2024.110172","url":null,"abstract":"<div><p>In this paper we characterized isotropic random tangential vector fields on <span><math><mi>d</mi></math></span>-spheres for <span><math><mrow><mi>d</mi><mo>≥</mo><mn>1</mn></mrow></math></span> by the cross-covariance, and derived their Karhunen–Loève expansion. The tangential vector field can be decomposed into a curl-free part and a divergence-free part by the Helmholtz–Hodge decomposition. We proved that the two parts can be correlated on a 2-sphere, while they must be uncorrelated on a <span><math><mi>d</mi></math></span>-sphere for <span><math><mrow><mi>d</mi><mo>≥</mo><mn>3</mn></mrow></math></span>. On a 3-sphere, the divergence-free part can be further decomposed into two isotropic flows.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"213 ","pages":"Article 110172"},"PeriodicalIF":0.8,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141280781","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}
Pub Date : 2024-06-01DOI: 10.1016/j.spl.2024.110167
Yuttana Ratibenyakool , Kritsana Neammanee
The Black–Scholes formula which was introduced by three economists, Black et al. (1973) has been widely used to calculate the theoretical price of the European call option. In 1979, Cox, Ross and Rubinstein (Cox et al., 1979) gave the binomial formula which is a tool to find the price of European option and showed that this formula converges to the Black–Scholes formula as the number of periods converges to infinity. In 1988, Boyle investigated another formula that is used to find the price of European option, that is the trinomial formula. In 2015, Puspita et al. gave examples to show that the trinomial formula is closed to the Black–Scholes formula. After that, Ratibenyakool and Neammanee (2020) gave the rigorous proof of this convergence. In this paper, we show that the rate of convergence is of order .
{"title":"Rate of convergence of trinomial formula to Black–Scholes formula","authors":"Yuttana Ratibenyakool , Kritsana Neammanee","doi":"10.1016/j.spl.2024.110167","DOIUrl":"10.1016/j.spl.2024.110167","url":null,"abstract":"<div><p>The Black–Scholes formula which was introduced by three economists, Black et al. (1973) has been widely used to calculate the theoretical price of the European call option. In 1979, Cox, Ross and Rubinstein (<span>Cox et al., 1979</span>) gave the binomial formula which is a tool to find the price of European option and showed that this formula converges to the Black–Scholes formula as the number of periods <span><math><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math></span> converges to infinity. In 1988, Boyle investigated another formula that is used to find the price of European option, that is the trinomial formula. In 2015, Puspita et al. gave examples to show that the trinomial formula is closed to the Black–Scholes formula. After that, Ratibenyakool and Neammanee (2020) gave the rigorous proof of this convergence. In this paper, we show that the rate of convergence is of order <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></math></span>.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"213 ","pages":"Article 110167"},"PeriodicalIF":0.8,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141281124","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}
Pub Date : 2024-05-28DOI: 10.1016/j.spl.2024.110156
Pingbo Hu , Xiuyuan Peng , Xinglin Hu
This paper presents some asymptotic results for series estimation of a nonparametric regression model under clustered dependence. A mean square rate of convergence for the series regression estimator is established. Moreover, asymptotic pointwise normality is shown for the series estimator.
{"title":"Some new asymptotic results for series estimation under clustered dependence","authors":"Pingbo Hu , Xiuyuan Peng , Xinglin Hu","doi":"10.1016/j.spl.2024.110156","DOIUrl":"https://doi.org/10.1016/j.spl.2024.110156","url":null,"abstract":"<div><p>This paper presents some asymptotic results for series estimation of a nonparametric regression model under clustered dependence. A mean square rate of convergence for the series regression estimator is established. Moreover, asymptotic pointwise normality is shown for the series estimator.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"213 ","pages":"Article 110156"},"PeriodicalIF":0.8,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141289443","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}