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Two-sided Estimates for the Sum of Probabilities of Errors in the Multiple Hypothesis Testing Problem with Finite Number of Hypotheses on a Nonhomogeneous Sample 非均质样本上有限假设数的多重假设检验问题中错误概率之和的双侧估计值
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-08-14 DOI: 10.1137/s0040585x97t991945
M. P. Savelov
{"title":"Two-sided Estimates for the Sum of Probabilities of Errors in the Multiple Hypothesis Testing Problem with Finite Number of Hypotheses on a Nonhomogeneous Sample","authors":"M. P. Savelov","doi":"10.1137/s0040585x97t991945","DOIUrl":"https://doi.org/10.1137/s0040585x97t991945","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 2, Page 322-330, August 2024. <br/> We obtain two-sided estimates for the weighted sum of probabilities of errors in the multiple hypothesis testing problem with finite number of hypotheses on a nonhomogeneous sample of size $n$. The obtained upper and lower estimates are shown to converge to zero exponentially fast with increasing $n$ in a wide class of cases. The results obtained can be used for deriving two-sided estimates for the size of a sample required for multiple hypothesis testing.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198604","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}
引用次数: 0
On One Nonparametric Estimation of the Poisson Regression Function 关于泊松回归函数的一次非参数估计
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-08-14 DOI: 10.1137/s0040585x97t991842
P. K. Babilua, E. A. Nadaraya
{"title":"On One Nonparametric Estimation of the Poisson Regression Function","authors":"P. K. Babilua, E. A. Nadaraya","doi":"10.1137/s0040585x97t991842","DOIUrl":"https://doi.org/10.1137/s0040585x97t991842","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 2, Page 173-185, August 2024. <br/> The limiting distribution of the integral square deviation of a nonparametric kernel-type estimator for the Poisson regression function is established. A criterion for testing the hypothesis on the Poisson regression function is constructed. The power asymptotic of the constructed criterion is studied for certain types of close alternatives.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198636","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}
引用次数: 0
Poisson Process with Linear Drift and Related Function Series 具有线性漂移的泊松过程及相关函数序列
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-08-14 DOI: 10.1137/s0040585x97t99191x
V. E. Mosyagin
{"title":"Poisson Process with Linear Drift and Related Function Series","authors":"V. E. Mosyagin","doi":"10.1137/s0040585x97t99191x","DOIUrl":"https://doi.org/10.1137/s0040585x97t99191x","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 2, Page 281-293, August 2024. <br/> Consider the random process $Y(t)=at-nu_+(pt)+nu_-(-qt)$, $tin(-infty,infty)$, where $nu_{pm}(t)$ are independent standard Poisson processes for $tgeqslant 0$ and $nu_{pm}(t)=0$ for $t&lt;0$. The parameters $a$, $p$, and $q$ are such that $mathbf{E}Y(t)&lt;0$, $tneq0$. We evaluate the sums $varphi_m(z,r)=sum_{kgeq0}(re^{-r})^{k}(z+k)^{m+k-1}/k!$, $m=1,2,dots$, $zgeq0$, of function series with parameter $ rin(0,1) $. These series are used for recursive evaluation of the moments $mathbf{E}(t^*)^m$, $mgeq 1$, for the time $t^*$ when the trajectory of the process $Y(t)$ attains its maximum value. The results obtained are applied to the problem of estimating the parameter $theta$ from $n$ observations with density $f(x,theta)$, which has a jump at the point $x=x(theta)$, $x'(theta)neq 0$. If $widehattheta_n$ is a maximum likelihood estimator for the true parameter $theta_0$, then the limit distribution as $ntoinfty$ for the normalized estimators $n(widehattheta_n-theta_0)$ is the distribution of the argument of the maximum $t^*_{theta_0}$ of the trajectory of the process $Y(t)$ with parameters $a$, $p$, and $q$, which depend on both the one-sided limits of the density at the point $x(theta_0)$ and the derivative $x'(theta_0)$. In this case, by evaluating the moments $mathbf{E}(t^*_{theta_0})^m$, $m=1, 2$, one can estimate both the asymptotic bias for the maximum likelihood estimator and its efficiency.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198505","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}
引用次数: 0
Conditional Functional Limit Theorem for a Random Recurrence Sequence Conditioned on a Large Deviation Event 以大偏差事件为条件的随机递推序列的条件函数极限定理
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-05-02 DOI: 10.1137/s0040585x97t991775
A. V. Shklyaev
{"title":"Conditional Functional Limit Theorem for a Random Recurrence Sequence Conditioned on a Large Deviation Event","authors":"A. V. Shklyaev","doi":"10.1137/s0040585x97t991775","DOIUrl":"https://doi.org/10.1137/s0040585x97t991775","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 1, Page 99-116, May 2024. <br/> Let ${Z_n,, nge 0}$ be a branching process in an independent and identically distributed (i.i.d.) random environment and ${S_n,, n,{ge}, 1}$ be the associated random walk with steps $xi_i$. Under the Cramér condition on $xi_1$ and moment assumptions on a number of descendants of one particle, we know the asymptotics of the large deviation probabilities $mathbf{P}(ln Z_n &gt; x)$, where $x/n &gt; mu^*$. Here, $mu^*$ is a parameter depending on the process type. We study the asymptotic behavior of the process trajectory under the condition of a large deviation event. In particular, we obtain a conditional functional limit theorem for the trajectory of $(Z_{[nt]},, tin [0,1])$ given $ln Z_n&gt;x$. This result is obtained in a more general model of linear recurrence sequence $Y_{n+1}=A_n Y_n + B_n$, $nge 0$, where ${A_i}$ is a sequence of i.i.d. random variables, $Y_0$, $B_i$, $ige 0$, are possibly dependent and have different distributions, and we need only some moment assumptions on them.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140839361","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}
引用次数: 0
Energy Saving Approximation of Wiener Process under Unilateral Constraints 单边约束条件下的维纳过程节能近似法
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-05-02 DOI: 10.1137/s0040585x97t99174x
M. A. Lifshits, S. E. Nikitin
{"title":"Energy Saving Approximation of Wiener Process under Unilateral Constraints","authors":"M. A. Lifshits, S. E. Nikitin","doi":"10.1137/s0040585x97t99174x","DOIUrl":"https://doi.org/10.1137/s0040585x97t99174x","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 1, Page 59-70, May 2024. <br/> We consider an energy saving approximation of a Wiener process under unilateral constraints. We show that, almost surely, on large time intervals the minimal energy necessary for the approximation logarithmically depends on the interval length. We also construct an adaptive approximation strategy optimal in a class of diffusion strategies and providing the logarithmic order of energy consumption.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140839357","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}
引用次数: 0
Utility Maximization of the Exponential Lévy Switching Models 指数莱维转换模型的效用最大化
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-05-02 DOI: 10.1137/s0040585x97t991799
Y. Dong, L. Vostrikova
{"title":"Utility Maximization of the Exponential Lévy Switching Models","authors":"Y. Dong, L. Vostrikova","doi":"10.1137/s0040585x97t991799","DOIUrl":"https://doi.org/10.1137/s0040585x97t991799","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 1, Page 127-149, May 2024. <br/> This article is devoted to maximization of HARA (hyperbolic absolute risk aversion) utilities of the exponential Lévy switching processes on a finite time interval via the dual method. The description of all $f$-divergence minimal martingale measures and the expression of their Radon--Nikodým densities involving the Hellinger and Kulback--Leibler processes are given. The optimal strategies in progressively enlarged filtration for the maximization of HARA utilities as well as the values of the corresponding maximal expected utilities are derived. As an example, the Brownian switching model is presented with financial interpretations of the results via the value process.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140839352","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}
引用次数: 0
News of Scientific Life - Information on the General Seminar of the Department of Probability, Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, Moscow, Russia, Spring Term 2023 科学生活新闻 - 莫斯科国立罗蒙诺索夫大学数学与力学学院概率系2023年春季学期综合研讨会信息
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-05-02 DOI: 10.1137/s0040585x97t991817
A.N. Shiryaev, E.B. Yarkovaya, V.A. Kutsenko
{"title":"News of Scientific Life - Information on the General Seminar of the Department of Probability, Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, Moscow, Russia, Spring Term 2023","authors":"A.N. Shiryaev, E.B. Yarkovaya, V.A. Kutsenko","doi":"10.1137/s0040585x97t991817","DOIUrl":"https://doi.org/10.1137/s0040585x97t991817","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 1, Page 161-165, May 2024. <br/> This paper presents summaries of talks given during the 2023 spring semester of the General Seminar of the Department of Probability, Moscow State University. The seminar was held under the direction of A. N. Kolmogorov and B. V. Gnedenko. Current information about the seminar is available at http://new.math.msu.su/department/probab/seminar.html.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140839547","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}
引用次数: 0
Joint Distributions of Generalized Integrable Increasing Processes and Their Generalized Compensators 广义积分递增过程的联合分布及其广义补偿器
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-05-02 DOI: 10.1137/s0040585x97t991714
D. A. Borzykh
{"title":"Joint Distributions of Generalized Integrable Increasing Processes and Their Generalized Compensators","authors":"D. A. Borzykh","doi":"10.1137/s0040585x97t991714","DOIUrl":"https://doi.org/10.1137/s0040585x97t991714","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 1, Page 1-24, May 2024. <br/> Let $Lambda$ be the set of all boundary joint laws $operatorname{Law} ([X_a, A_a], [X_b, A_b])$ at times $t=a$ and $t=b$ of integrable increasing processes $(X_t)_{t in [a, b]}$ and their compensators $(A_t)_{t in [a, b]}$, which start at the initial time from an arbitrary integrable initial condition $[X_a, A_a]$. We show that $Lambda$ is convex and closed relative to the $psi$-weak topology with linearly growing gauge function $psi$. We obtain necessary and sufficient conditions for a probability measure $lambda$ on $mathcal{B}(mathbf{R}^2 times mathbf{R}^2)$ to lie in the class of measures $Lambda$. The main result of the paper provides, for two measures $mu_a$ and $mu_b$ on $mathcal{B}(mathbf{R}^2)$, necessary and sufficient conditions for the set $Lambda$ to contain a measure $lambda$ for which $mu_a$ and $mu_b$ are marginal distributions.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140839312","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}
引用次数: 0
Limit Behavior of Order Statistics on Cycle Lengths of Random $A$-Permutations 随机 $A$-Permutations 循环长度的阶次统计极限行为
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-05-02 DOI: 10.1137/s0040585x97t991787
A. L. Yakymiv
{"title":"Limit Behavior of Order Statistics on Cycle Lengths of Random $A$-Permutations","authors":"A. L. Yakymiv","doi":"10.1137/s0040585x97t991787","DOIUrl":"https://doi.org/10.1137/s0040585x97t991787","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 1, Page 117-126, May 2024. <br/> We consider a random permutation $tau_n$ uniformly distributed on the set of all permutations of degree $n$ whose cycle lengths lie in a fixed set $A$ (the so-called $A$-permutations). Let $zeta_n$ be the total number of cycles, and let $eta_n(1)leqeta_n(2)leqdotsleqeta_n(zeta_n)$ be the ordered sample of cycle lengths of the permutation $tau_n$. We consider a class of sets $A$ with positive density in the set of natural numbers. We study the asymptotic behavior of $eta_n(m)$ with numbers $m$ in the left-hand and middle parts of this series for a class of sets of positive asymptotic density. A limit theorem for the rightmost terms of this series was proved by the author of this note earlier. The study of limit properties of the sequence $eta_n(m)$ dates back to the paper by Shepp and Lloyd [Trans. Amer. Math. Soc., 121 (1966), pp. 340--357] who considered the case $A=mathbf N$.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140839363","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}
引用次数: 0
Universal Nonparametric Kernel-Type Estimators for the Mean and Covariance Functions of a Stochastic Process 随机过程均值和协方差函数的通用非参数核型估计器
IF 0.6 4区 数学
Theory of Probability and its Applications Pub Date : 2024-05-02 DOI: 10.1137/s0040585x97t991738
Yu. Yu. Linke, I. S. Borisov
{"title":"Universal Nonparametric Kernel-Type Estimators for the Mean and Covariance Functions of a Stochastic Process","authors":"Yu. Yu. Linke, I. S. Borisov","doi":"10.1137/s0040585x97t991738","DOIUrl":"https://doi.org/10.1137/s0040585x97t991738","url":null,"abstract":"Theory of Probability &amp;Its Applications, Volume 69, Issue 1, Page 35-58, May 2024. <br/> Let $f_1(t), dots, f_n(t)$ be independent copies of some a.s. continuous stochastic process $f(t)$, $tin[0,1]$, which are observed with noise. We consider the problem of nonparametric estimation of the mean function $mu(t) = mathbf{E}f(t)$ and of the covariance function $psi(t,s)=operatorname{Cov}{f(t),f(s)}$ if the noise values of each of the copies $f_i(t)$, $i=1,dots,n$, are observed in some collection of generally random (in general) time points (regressors). Under wide assumptions on the time points, we construct uniformly consistent kernel estimators for the mean and covariance functions both in the case of sparse data (where the number of observations for each copy of the stochastic process is uniformly bounded) and in the case of dense data (where the number of observations at each of $n$ series is increasing as $ntoinfty$). In contrast to the previous studies, our kernel estimators are universal with respect to the structure of time points, which can be either fixed rather than necessarily regular, or random rather than necessarily formed of independent or weakly dependent random variables.","PeriodicalId":51193,"journal":{"name":"Theory of Probability and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140839317","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}
引用次数: 0
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