Asymptotic normality of a covariance estimator for nonsynchronously observed diffusion processes

We consider the problem of estimating the covariance of two diffusion-type processes when they are observed only at discrete times in a nonsynchronous manner. In our previous work in 2003, we proposed a new estimator which is free of any ‘synchronization’ processing of the original data and showed that it is consistent for the true covariance of the processes as the observation interval shrinks to zero; Hayashi and Yoshida (Bernoulli, 11, 359–379, 2005). This paper is its sequel. Specifically, it establishes asymptotic normality of the estimator in a general nonsynchronous sampling scheme.     

Interactive Statistical Mechanics and Nonlinear Filtering

This paper connects non-equilibrium statistical mechanics and optimal nonlinear filtering. The latter concerns the observation-conditional behaviour of Markov signal processes, and thus provides a tool for investigating statistical mechanics with partial observations. These allow entropy reduction, illustrating Landauer’s Principle in a quantitative way. The joint process comprising a signal and its nonlinear filter is irreversible in its invariant distribution, which therefore corresponds to a non-equilibrium stationary state of the associated joint system. Macroscopic entropy and energy flows are identified for this state. Since these are driven by observations internal to the system, they do not cause entropy increase, and so the joint system makes statistical mechanical sense in reverse time. Time reversal yields a dual system in which the signal and filter exchange roles. Despite the structural similarity of the original and dual systems, there is a substantial asymmetry in their complexities. This reveals the direction of time, despite the systems being in stationary states that do not produce entropy. This work was partially supported by Leverhulme Trust Research Fellowship 2003/0426, by MURI Grant F49620-02-1-0325 (Complex Adaptive Networks for Co-operative Control), by ARO-MURI Grant DAAD19-00-1-0466 (Data Fusion in Large Arrays of Microsensors (sensor web)) and by NSF-ITR Grant CCR-0325774 Collaborative Research: New Approaches to Experimental Design and Statistical Analysis of Genomic and Structural Biological Data from Multiple Sources.     

A remark on control of partially observed Markov chains

A new state variable is introduced for the problem of controlling a Markov chain under partial observations, which, under a suitably altered probability measure, has a simple evolution.     

On the computation of the optimal cost function for discrete time Markov models with partial observations

We consider several applications of two state, finite action, infinite horizon, discrete-time Markov decision processes with partial observations, for two special cases of observation quality, and show that in each of these cases the optimal cost function is piecewise linear. This in turn allows us to obtain either explicit formulas or simplified algorithms to compute the optimal cost function and the associated optimal control policy. Several examples are presented.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-86-0029, in part by the National Science Foundation under Grant ECS-8617860, in part by the Advanced Technology Program of the State of Texas, and in part by the DoD Joint Services Electronics Program through the Air Force Office of Scientific Research (AFSC) Contract F49620-86-C-0045.     

Quasi-likelihood analysis for nonsynchronously observed diffusion processes

We consider nonsynchronous sampling of parameterized stochastic regression models, which contain stochastic differential equations. Constructing a quasi-likelihood function, we prove that the quasi-maximum likelihood estimator and the Bayes type estimator are consistent and asymptotically mixed normal when the sampling frequency of the nonsynchronous data becomes large.     

On almost sure max-limit theorems of complete and incomplete samples from stationary sequences

Let M _n denote the partial maximum of a strictly stationary sequence (X _n ). Suppose that some of the random variables of (X _n ) can be observed and let [(M)tilde]_ntilde M_n stand for the maximum of observed random variables from the set {X ₁, ..., X _n }. In this paper, the almost sure limit theorems related to random vector ([(M)tilde]_ntilde M_n , M _n ) are considered in terms of i.i.d. case. The related results are also extended to weakly dependent stationary Gaussian sequence as its covariance function satisfies some regular conditions.     

Local Sensitivity Analysis in Estimation Problems

This article deals with the problem of local sensitivity analysis, that is, how sensitive are the results of a statistical analysis to changes in the data? A general methodology of sensitivity analysis is applied to some statistical problems. The proposed methods are applicable to any statistical problem that can be expressed as an optimization problem or that involves solving a system of equations. As some particular examples, the methodology is applied to the maximum likelihood method, the standard and constrained methods of moments and the standard and constrained probability weighted moments methods. Unlike the standard method of moments, the constrained method of moments ensures that the obtained estimates are always consistent with the data. Closed analytical formulas for the calculation of these local sensitivities are derived. The obtained sensitivities include: (a) the objective function sensitivities to data points and (b) the sensitivities of the estimated parameters to data points. The derived formulas for the sensitivities are based on recent results in the area of mathematical programming. Several examples of parameter estimation problems are used to illustrate the methodology.     

Parametric inference for discrete observations of diffusion processes with mixed effects

Stochastic differential equations with mixed effects provide means to model intra-individual and inter-individual variability in repeated experiments leading to longitudinal data. We consider N i.i.d. stochastic processes defined by a stochastic differential equation with linear mixed effects which are discretely observed. We study a parametric framework with distributions leading to explicit approximate likelihood functions and investigate the asymptotic behavior of estimators under the asymptotic framework : the number N of individuals (trajectories) and the number n of observations per individual tend to infinity within a fixed time interval. The estimation method is assessed on simulated data for various models.     

几种新的回归诊断图示模型及其医学应用

本文在变量选择问题的基础上，提出了一种新的图示模型──减变残差图。并给出它的两种推广形式：均值平移异常值检验图和部分影响诊断图。通过它们不但可以容易地考察一个变量在模型中的作用和检验异常值，而且可以诊断样本点对模型和变量的影响大小。     

Limit theorems for the pre-averaged Hayashi–Yoshida estimator with random sampling

We will focus on estimating the integrated covariance of two diffusion processes observed in a nonsynchronous manner. The observation data is contaminated by some noise, which possibly depends on the time and the latent diffusion processes, while the sampling times also possibly depend on the observed processes. In a high-frequency setting, we consider a modified version of the pre-averaged Hayashi–Yoshida estimator, and we show that such a kind of estimator has the consistency and the asymptotic mixed normality, and attains the optimal rate of convergence.