{"title":"具有线性漂移的泊松过程及相关函数序列","authors":"V. E. Mosyagin","doi":"10.1137/s0040585x97t99191x","DOIUrl":null,"url":null,"abstract":"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 281-293, August 2024. <br/> Consider the random process $Y(t)=at-\\nu_+(pt)+\\nu_-(-qt)$, $t\\in(-\\infty,\\infty)$, where $\\nu_{\\pm}(t)$ are independent standard Poisson processes for $t\\geqslant 0$ and $\\nu_{\\pm}(t)=0$ for $t<0$. The parameters $a$, $p$, and $q$ are such that $\\mathbf{E}Y(t)<0$, $t\\neq0$. We evaluate the sums $\\varphi_m(z,r)=\\sum_{k\\geq0}(re^{-r})^{k}(z+k)^{m+k-1}/k!$, $m=1,2,\\dots$, $z\\geq0$, of function series with parameter $ r\\in(0,1) $. These series are used for recursive evaluation of the moments $\\mathbf{E}(t^*)^m$, $m\\geq 1$, 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 $\\widehat\\theta_n$ is a maximum likelihood estimator for the true parameter $\\theta_0$, then the limit distribution as $n\\to\\infty$ for the normalized estimators $n(\\widehat\\theta_n-\\theta_0)$ 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":"28 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Poisson Process with Linear Drift and Related Function Series\",\"authors\":\"V. E. Mosyagin\",\"doi\":\"10.1137/s0040585x97t99191x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Theory of Probability &Its Applications, Volume 69, Issue 2, Page 281-293, August 2024. <br/> Consider the random process $Y(t)=at-\\\\nu_+(pt)+\\\\nu_-(-qt)$, $t\\\\in(-\\\\infty,\\\\infty)$, where $\\\\nu_{\\\\pm}(t)$ are independent standard Poisson processes for $t\\\\geqslant 0$ and $\\\\nu_{\\\\pm}(t)=0$ for $t<0$. The parameters $a$, $p$, and $q$ are such that $\\\\mathbf{E}Y(t)<0$, $t\\\\neq0$. We evaluate the sums $\\\\varphi_m(z,r)=\\\\sum_{k\\\\geq0}(re^{-r})^{k}(z+k)^{m+k-1}/k!$, $m=1,2,\\\\dots$, $z\\\\geq0$, of function series with parameter $ r\\\\in(0,1) $. These series are used for recursive evaluation of the moments $\\\\mathbf{E}(t^*)^m$, $m\\\\geq 1$, 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 $\\\\widehat\\\\theta_n$ is a maximum likelihood estimator for the true parameter $\\\\theta_0$, then the limit distribution as $n\\\\to\\\\infty$ for the normalized estimators $n(\\\\widehat\\\\theta_n-\\\\theta_0)$ 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\":\"28 1\",\"pages\":\"\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-08-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theory of Probability and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1137/s0040585x97t99191x\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theory of Probability and its Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/s0040585x97t99191x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Poisson Process with Linear Drift and Related Function Series
Theory of Probability &Its Applications, Volume 69, Issue 2, Page 281-293, August 2024. Consider the random process $Y(t)=at-\nu_+(pt)+\nu_-(-qt)$, $t\in(-\infty,\infty)$, where $\nu_{\pm}(t)$ are independent standard Poisson processes for $t\geqslant 0$ and $\nu_{\pm}(t)=0$ for $t<0$. The parameters $a$, $p$, and $q$ are such that $\mathbf{E}Y(t)<0$, $t\neq0$. We evaluate the sums $\varphi_m(z,r)=\sum_{k\geq0}(re^{-r})^{k}(z+k)^{m+k-1}/k!$, $m=1,2,\dots$, $z\geq0$, of function series with parameter $ r\in(0,1) $. These series are used for recursive evaluation of the moments $\mathbf{E}(t^*)^m$, $m\geq 1$, 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 $\widehat\theta_n$ is a maximum likelihood estimator for the true parameter $\theta_0$, then the limit distribution as $n\to\infty$ for the normalized estimators $n(\widehat\theta_n-\theta_0)$ 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.
期刊介绍:
Theory of Probability and Its Applications (TVP) accepts original articles and communications on the theory of probability, general problems of mathematical statistics, and applications of the theory of probability to natural science and technology. Articles of the latter type will be accepted only if the mathematical methods applied are essentially new.