{"title":"Le Cam–Stratonovich–Itôdiffusions的布尔理论","authors":"J. Bishwal","doi":"10.1515/rose-2023-2004","DOIUrl":null,"url":null,"abstract":"Abstract We connect the theory of local asymptotic normality (LAN) of Le Cam to Boole’s approximation of the Stratonovich stochastic integral by estimating the parameter in the nonlinear drift coefficient of an ergodic diffusion process satisfying a homogeneous Itô stochastic differential equation based on discretely spaced dense observations of the process. The asymptotic normality and local asymptotic minimaxity (in the Hajek–Le Cam sense) of approximate maximum likelihood estimators, approximate maximum probability estimators and approximate Bayes estimators based on Itô and Boole’s approximations of the continuous likelihood are obtained under an almost slowly increasing experimental design (ASIED) condition ( T n 6 / 7 → 0 {\\frac{T}{n^{6/7}}\\to 0} as T → ∞ {T\\to\\infty} and n → ∞ {n\\to\\infty} , where T is the length of the observation time and n is the number of observations) through the weak convergence of the approximate likelihood ratio random fields. Among other things, the Bernstein–von Mises type theorems concerning the convergence of suitably normalized and centered approximate posterior distributions to normal distribution under the same design condition are proved. Asymptotic normality and asymptotic efficiency of the conditional least squares estimator under the same design condition are obtained as a by-product. The log-likelihood derivatives based on Itô approximations are martingales, but the log-likelihood derivatives based on Boole’s approximations are not martingales but weighted averages of forward and backward martingales. These new approximations have faster rate of convergence than the martingale approximations. The methods would have advantages over Euler and Milstein approximations for Monte Carlo simulations.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Le Cam–Stratonovich–Boole theory for Itô diffusions\",\"authors\":\"J. 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The asymptotic normality and local asymptotic minimaxity (in the Hajek–Le Cam sense) of approximate maximum likelihood estimators, approximate maximum probability estimators and approximate Bayes estimators based on Itô and Boole’s approximations of the continuous likelihood are obtained under an almost slowly increasing experimental design (ASIED) condition ( T n 6 / 7 → 0 {\\\\frac{T}{n^{6/7}}\\\\to 0} as T → ∞ {T\\\\to\\\\infty} and n → ∞ {n\\\\to\\\\infty} , where T is the length of the observation time and n is the number of observations) through the weak convergence of the approximate likelihood ratio random fields. Among other things, the Bernstein–von Mises type theorems concerning the convergence of suitably normalized and centered approximate posterior distributions to normal distribution under the same design condition are proved. Asymptotic normality and asymptotic efficiency of the conditional least squares estimator under the same design condition are obtained as a by-product. The log-likelihood derivatives based on Itô approximations are martingales, but the log-likelihood derivatives based on Boole’s approximations are not martingales but weighted averages of forward and backward martingales. These new approximations have faster rate of convergence than the martingale approximations. 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引用次数: 0
摘要
摘要我们将Le Cam的局部渐近正态性(LAN)理论与Stratonovich随机积分的Boole近似联系起来,通过估计遍历扩散过程的非线性漂移系数中的参数,该过程满足齐次Itô随机微分方程,该过程基于离散间隔的稠密观测。在几乎缓慢增长的实验设计(ASIED)条件下(T n 6/7),获得了基于连续似然的Itô和Boole近似的近似最大似然估计量、近似最大概率估计量和近似贝叶斯估计量的渐近正态性和局部渐近极小性(在Hajek–Le Cam意义上)→ 0{\frac{T}{n^{6/7}}\到0}作为T→ ∞ {T \ to \ infty}和n→ ∞ {n\to\infty},其中T是观测时间的长度,n是观测次数)。证明了Bernstein–von Mises型定理,证明了在相同设计条件下,适当归一化和中心的近似后验分布收敛于正态分布。作为副产品,得到了在相同设计条件下条件最小二乘估计量的渐近正态性和渐近有效性。基于Itô近似的对数似然导数是鞅,但基于Boole近似的对数可能性导数不是鞅,而是前向和后向鞅的加权平均。这些新的近似比鞅近似具有更快的收敛速度。在蒙特卡洛模拟中,该方法将比欧拉和米尔斯坦近似具有优势。
Le Cam–Stratonovich–Boole theory for Itô diffusions
Abstract We connect the theory of local asymptotic normality (LAN) of Le Cam to Boole’s approximation of the Stratonovich stochastic integral by estimating the parameter in the nonlinear drift coefficient of an ergodic diffusion process satisfying a homogeneous Itô stochastic differential equation based on discretely spaced dense observations of the process. The asymptotic normality and local asymptotic minimaxity (in the Hajek–Le Cam sense) of approximate maximum likelihood estimators, approximate maximum probability estimators and approximate Bayes estimators based on Itô and Boole’s approximations of the continuous likelihood are obtained under an almost slowly increasing experimental design (ASIED) condition ( T n 6 / 7 → 0 {\frac{T}{n^{6/7}}\to 0} as T → ∞ {T\to\infty} and n → ∞ {n\to\infty} , where T is the length of the observation time and n is the number of observations) through the weak convergence of the approximate likelihood ratio random fields. Among other things, the Bernstein–von Mises type theorems concerning the convergence of suitably normalized and centered approximate posterior distributions to normal distribution under the same design condition are proved. Asymptotic normality and asymptotic efficiency of the conditional least squares estimator under the same design condition are obtained as a by-product. The log-likelihood derivatives based on Itô approximations are martingales, but the log-likelihood derivatives based on Boole’s approximations are not martingales but weighted averages of forward and backward martingales. These new approximations have faster rate of convergence than the martingale approximations. The methods would have advantages over Euler and Milstein approximations for Monte Carlo simulations.