Pub Date : 2024-11-01DOI: 10.1016/j.spl.2024.110291
Alfred Kume , Tomonari Sei , Andrew T.A. Wood
This article presents two new results relevant to a linear combination of random variables. The first result expresses the cumulative distribution function as a simple multiple of a certain tilted density. The second result shows that in important cases the inversion integral for the density may be expressed as a sum of relatively simple real integrals which, using suitable numerical methods, are straightforward to compute quickly and accurately.
{"title":"On the representation and computational aspects of the distribution of a linear combination of independent noncentral chi-squared random variables","authors":"Alfred Kume , Tomonari Sei , Andrew T.A. Wood","doi":"10.1016/j.spl.2024.110291","DOIUrl":"10.1016/j.spl.2024.110291","url":null,"abstract":"<div><div>This article presents two new results relevant to a linear combination of <span><math><msup><mrow><mi>χ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> random variables. The first result expresses the cumulative distribution function as a simple multiple of a certain tilted density. The second result shows that in important cases the inversion integral for the density may be expressed as a sum of relatively simple real integrals which, using suitable numerical methods, are straightforward to compute quickly and accurately.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"218 ","pages":"Article 110291"},"PeriodicalIF":0.9,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703929","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 : 2024-10-30DOI: 10.1016/j.spl.2024.110288
Tommy Wright
Klein, Wright, and Wieczorek (2020), hereafter KWW, constructs a simple novel measure of uncertainty for an estimated ranking using a joint confidence region for the true ranking of populations. In this current paper, our proposed framework permits some control over the amount of uncertainty and tightness in various portions of the estimated ranking with an optimal allocation of sample among the populations.
Klein、Wright 和 Wieczorek（2020 年）（以下简称 KWW）利用 K 种群真实排名的联合置信区域，为估计排名构建了一个简单的新型不确定性度量。在本文中，我们提出的框架允许通过在 K 个种群中优化样本分配，对估计排名各部分的不确定性和严密性进行一定程度的控制。
{"title":"Optimal tightening of the KWW joint confidence region for a ranking","authors":"Tommy Wright","doi":"10.1016/j.spl.2024.110288","DOIUrl":"10.1016/j.spl.2024.110288","url":null,"abstract":"<div><div>Klein, Wright, and Wieczorek (2020), hereafter KWW, constructs a simple novel measure of uncertainty for an estimated ranking using a joint confidence region for the true ranking of <span><math><mi>K</mi></math></span> populations. In this current paper, our proposed framework permits some control over the amount of uncertainty and tightness in various portions of the estimated ranking with an optimal allocation of sample among the <span><math><mi>K</mi></math></span> populations.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"217 ","pages":"Article 110288"},"PeriodicalIF":0.9,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142651260","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 : 2024-10-29DOI: 10.1016/j.spl.2024.110293
Minyuan Lu, Bu Zhou
This study focuses on the high-dimensional one-way analysis of variance problem, specifically, testing whether multiple population mean vectors are equal in the context of high-dimensional data. To solve the problem that classical multivariate analysis of variance (MANOVA) test statistics are undefined when the dimensionality surpasses the sample size, we propose a random permutation test using low-dimensional subspaces obtained by clustering of variables. The test statistics are derived from a one-way MANOVA decomposition for clustered variables and this approach utilizes the correlation information among variables to ensure high testing power. Simulation studies indicate that the proposed test performs well with high-dimensional data.
{"title":"A one-way MANOVA test for high-dimensional data using clustering subspaces","authors":"Minyuan Lu, Bu Zhou","doi":"10.1016/j.spl.2024.110293","DOIUrl":"10.1016/j.spl.2024.110293","url":null,"abstract":"<div><div>This study focuses on the high-dimensional one-way analysis of variance problem, specifically, testing whether multiple population mean vectors are equal in the context of high-dimensional data. To solve the problem that classical multivariate analysis of variance (MANOVA) test statistics are undefined when the dimensionality surpasses the sample size, we propose a random permutation test using low-dimensional subspaces obtained by clustering of variables. The test statistics are derived from a one-way MANOVA decomposition for clustered variables and this approach utilizes the correlation information among variables to ensure high testing power. Simulation studies indicate that the proposed test performs well with high-dimensional data.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"217 ","pages":"Article 110293"},"PeriodicalIF":0.9,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594045","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 : 2024-10-28DOI: 10.1016/j.spl.2024.110290
Chi Zhang, Danna Zhang
Probability and moment inequalities for quadratic forms are valuable tools in studying the properties of second-order statistics. There are extensive results regarding quadratic forms in random variables with finite exponential moments. However, the counterpart that allows for weaker moment conditions is inadequate. In this work, we present a new Nagaev-type tail probability inequality and a Rosenthal-type moment inequality for quadratic forms in random variables with fat tails.
{"title":"Probability and moment inequalities for quadratic forms in independent random variables with fat tails","authors":"Chi Zhang, Danna Zhang","doi":"10.1016/j.spl.2024.110290","DOIUrl":"10.1016/j.spl.2024.110290","url":null,"abstract":"<div><div>Probability and moment inequalities for quadratic forms are valuable tools in studying the properties of second-order statistics. There are extensive results regarding quadratic forms in random variables with finite exponential moments. However, the counterpart that allows for weaker moment conditions is inadequate. In this work, we present a new Nagaev-type tail probability inequality and a Rosenthal-type moment inequality for quadratic forms in random variables with fat tails.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"217 ","pages":"Article 110290"},"PeriodicalIF":0.9,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142572986","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 : 2024-10-26DOI: 10.1016/j.spl.2024.110289
Issey Sukeda , Tomonari Sei
In this work, we demonstrate that the Frank copula is the minimum information copula under fixed Kendall’s (MICK), both theoretically and numerically. First, we explain that both MICK and the Frank density follow the hyperbolic Liouville equation. Subsequently, we show that the copula density satisfying the Liouville equation is uniquely the Frank copula. Our result asserts that selecting the Frank copula as an appropriate copula model is equivalent to using Kendall’s as the sole available information about the true distribution, based on the entropy maximization principle.
在这项工作中，我们从理论和数值两方面证明了弗兰克协整是固定肯德尔τ（MICK）条件下的最小信息协整。首先，我们解释了 MICK 和 Frank 密度都遵循双曲 Liouville 方程。随后，我们证明满足 Liouville 方程的 copula 密度是唯一的 Frank copula。我们的结果证明，根据熵最大化原则，选择 Frank copula 作为合适的 copula 模型等同于使用 Kendall's τ 作为关于真实分布的唯一可用信息。
{"title":"Frank copula is minimum information copula under fixed Kendall’s τ","authors":"Issey Sukeda , Tomonari Sei","doi":"10.1016/j.spl.2024.110289","DOIUrl":"10.1016/j.spl.2024.110289","url":null,"abstract":"<div><div>In this work, we demonstrate that the Frank copula is the minimum information copula under fixed Kendall’s <span><math><mi>τ</mi></math></span> (MICK), both theoretically and numerically. First, we explain that both MICK and the Frank density follow the hyperbolic Liouville equation. Subsequently, we show that the copula density satisfying the Liouville equation is uniquely the Frank copula. Our result asserts that selecting the Frank copula as an appropriate copula model is equivalent to using Kendall’s <span><math><mi>τ</mi></math></span> as the sole available information about the true distribution, based on the entropy maximization principle.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"217 ","pages":"Article 110289"},"PeriodicalIF":0.9,"publicationDate":"2024-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142551915","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 : 2024-10-24DOI: 10.1016/j.spl.2024.110287
Ying Huang, Jun Peng
We consider a robust optimal investment and reinsurance problem with multiple dependent risks for an Ambiguity-Averse Insurer (AAI), who wishes to minimize the probability that the value of the wealth process reaches a low barrier before a high goal. We assume that the insurer can purchase per-loss reinsurance for every class of insurance business and invest its surplus in a risk-free asset and a risky asset. Using the technique of stochastic control theory and solving the associated Hamilton-Jacobi-Bellman (HJB) equation, we derive the robust optimal investment-reinsurance strategy and the associated value function. We conclude that the robust optimal investment-reinsurance strategy coincides with the one without model ambiguity, but the value function differs. We also illustrate our results by numerical examples.
{"title":"Minimizing the penalized goal-reaching probability with multiple dependent risks","authors":"Ying Huang, Jun Peng","doi":"10.1016/j.spl.2024.110287","DOIUrl":"10.1016/j.spl.2024.110287","url":null,"abstract":"<div><div>We consider a robust optimal investment and reinsurance problem with multiple dependent risks for an Ambiguity-Averse Insurer (AAI), who wishes to minimize the probability that the value of the wealth process reaches a low barrier before a high goal. We assume that the insurer can purchase per-loss reinsurance for every class of insurance business and invest its surplus in a risk-free asset and a risky asset. Using the technique of stochastic control theory and solving the associated Hamilton-Jacobi-Bellman (HJB) equation, we derive the robust optimal investment-reinsurance strategy and the associated value function. We conclude that the robust optimal investment-reinsurance strategy coincides with the one without model ambiguity, but the value function differs. We also illustrate our results by numerical examples.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"217 ","pages":"Article 110287"},"PeriodicalIF":0.9,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142572987","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 : 2024-10-17DOI: 10.1016/j.spl.2024.110283
Paek Il-Kwang , Kang Chol-Su , Kim Kyong-Hui
This paper deals with new explicit pricing formulae for Lookback option when underlying asset price processes are represented by stochastic delay differential equation (hereafter “SDDE”). We derive a lemma on the joint distribution of the minimum and itself of a Wiener process in the SDDE model. Using this lemma, we obtain the explicit pricing formulae for the Lookback option. Through some numerical comparison experiment, we assure the correctness of the obtained pricing formula.
{"title":"Pricing formula of Lookback option in stochastic delay differential equation model","authors":"Paek Il-Kwang , Kang Chol-Su , Kim Kyong-Hui","doi":"10.1016/j.spl.2024.110283","DOIUrl":"10.1016/j.spl.2024.110283","url":null,"abstract":"<div><div>This paper deals with new explicit pricing formulae for Lookback option when underlying asset price processes are represented by stochastic delay differential equation (hereafter “SDDE”). We derive a lemma on the joint distribution of the minimum and itself of a Wiener process in the SDDE model. Using this lemma, we obtain the explicit pricing formulae for the Lookback option. Through some numerical comparison experiment, we assure the correctness of the obtained pricing formula.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"216 ","pages":"Article 110283"},"PeriodicalIF":0.9,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527950","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 : 2024-10-15DOI: 10.1016/j.spl.2024.110285
Soham Gokhale
We consider the stochastic Landau–Lifshitz–Gilbert equation driven by pure jump noise. We assume non-zero contribution from the helicity term to the total energy. Using finite dimensional approximation followed by a generalization of the Jakubowski’s version of the Skorohod Theorem for non-metric spaces, we show that the considered problem admits a weak martingale solution. Restricting the problem to dimension 1, we show that the obtained solution is pathwise unique, thereby concluding the existence of a strong solution.
{"title":"Well-posedness for the stochastic Landau–Lifshitz–Gilbert equation with helicity driven by jump noise","authors":"Soham Gokhale","doi":"10.1016/j.spl.2024.110285","DOIUrl":"10.1016/j.spl.2024.110285","url":null,"abstract":"<div><div>We consider the stochastic Landau–Lifshitz–Gilbert equation driven by pure jump noise. We assume non-zero contribution from the helicity term to the total energy. Using finite dimensional approximation followed by a generalization of the Jakubowski’s version of the Skorohod Theorem for non-metric spaces, we show that the considered problem admits a weak martingale solution. Restricting the problem to dimension 1, we show that the obtained solution is pathwise unique, thereby concluding the existence of a strong solution.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"216 ","pages":"Article 110285"},"PeriodicalIF":0.9,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527949","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 : 2024-10-15DOI: 10.1016/j.spl.2024.110284
Ryan McFadden, Fraser Daly, Seva Shneer
We consider a susceptible-infected-susceptible (SIS) epidemic model on an undirected graph, with a homogeneous infection rate and heterogeneous curing rates. We set an overall network curing rate, , and study optimal allocation of curing rates to nodes, in terms of the expected time to the extinction of the epidemic. As other parameters are fixed, we study these allocations as the infection rate tends to 0 and in both regular and non-regular graphs. We further illustrate this optimisation with some numerical examples. Our findings demonstrate that, while the uniform split of is optimal in some situations, it is typically not optimal, even for regular graphs.
{"title":"Optimal curing rate allocation in the SIS epidemic model","authors":"Ryan McFadden, Fraser Daly, Seva Shneer","doi":"10.1016/j.spl.2024.110284","DOIUrl":"10.1016/j.spl.2024.110284","url":null,"abstract":"<div><div>We consider a susceptible-infected-susceptible (SIS) epidemic model on an undirected graph, with a homogeneous infection rate and heterogeneous curing rates. We set an overall network curing rate, <span><math><mi>Δ</mi></math></span>, and study optimal allocation of curing rates to nodes, in terms of the expected time to the extinction of the epidemic. As other parameters are fixed, we study these allocations as the infection rate tends to 0 and <span><math><mi>∞</mi></math></span> in both regular and non-regular graphs. We further illustrate this optimisation with some numerical examples. Our findings demonstrate that, while the uniform split of <span><math><mi>Δ</mi></math></span> is optimal in some situations, it is typically not optimal, even for regular graphs.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"216 ","pages":"Article 110284"},"PeriodicalIF":0.9,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527951","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 : 2024-10-15DOI: 10.1016/j.spl.2024.110286
Dingwen Zhang
Low-frequency observations are a common occurrence in real-world applications, making statistical inference for stochastic processes driven by stochastic differential equations (SDEs) based on such observations an important issue. In this paper, we investigate the statistical inference for the Ornstein–Uhlenbeck (OU) process using low-frequency observations. We propose modified least squares estimators (MLSEs) for the drift parameters and a modified quadratic variation estimator for the diffusion parameter based on the solution of the OU process. The MLSEs are derived heuristically using the nonlinear least squares method, despite the OU process satisfying a linear SDE. Unlike previous approaches, these modified estimators are asymptotically unbiased. Leveraging the ergodic properties of the OU process, we also propose ergodic estimators for the three parameters. The asymptotic behavior of these estimators is established using the ergodic properties and central limit theorem for the OU process, achieved through linear model techniques and multivariate Markov chain central limit theorem. Monte Carlo simulation results are presented to illustrate and support our theoretical findings.
低频观测是现实世界应用中的常见现象，因此基于低频观测对由随机微分方程（SDE）驱动的随机过程进行统计推断是一个重要问题。本文研究了利用低频观测数据对奥恩斯坦-乌伦贝克（OU）过程进行统计推断的问题。我们根据 OU 过程的解，提出了漂移参数的修正最小二乘估计器（MLSE）和扩散参数的修正二次变化估计器。尽管 OU 过程满足线性 SDE，但 MLSE 是通过非线性最小二乘法启发式得出的。与以前的方法不同，这些修正估计器在渐近上是无偏的。利用 OU 过程的遍历特性，我们还提出了三个参数的遍历估计值。通过线性模型技术和多变量马尔可夫链中心极限定理，我们利用 OU 过程的遍历特性和中心极限定理确定了这些估计器的渐近行为。蒙特卡罗模拟结果用于说明和支持我们的理论发现。
{"title":"Statistical inference for Ornstein–Uhlenbeck processes based on low-frequency observations","authors":"Dingwen Zhang","doi":"10.1016/j.spl.2024.110286","DOIUrl":"10.1016/j.spl.2024.110286","url":null,"abstract":"<div><div>Low-frequency observations are a common occurrence in real-world applications, making statistical inference for stochastic processes driven by stochastic differential equations (SDEs) based on such observations an important issue. In this paper, we investigate the statistical inference for the Ornstein–Uhlenbeck (OU) process using low-frequency observations. We propose modified least squares estimators (MLSEs) for the drift parameters and a modified quadratic variation estimator for the diffusion parameter based on the solution of the OU process. The MLSEs are derived heuristically using the nonlinear least squares method, despite the OU process satisfying a linear SDE. Unlike previous approaches, these modified estimators are asymptotically unbiased. Leveraging the ergodic properties of the OU process, we also propose ergodic estimators for the three parameters. The asymptotic behavior of these estimators is established using the ergodic properties and central limit theorem for the OU process, achieved through linear model techniques and multivariate Markov chain central limit theorem. Monte Carlo simulation results are presented to illustrate and support our theoretical findings.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"216 ","pages":"Article 110286"},"PeriodicalIF":0.9,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527948","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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