Parametric estimation for a simple branching diffusion process

A simple branching diffusion process is given as an elementary model of spatial evolution. A parametric estimation theory is presented for this model. As side results, a spatial central limit theorem and spatial strong law of large numbers are also obtained.     

A limit theorem for branching one-dimensional periodic diffusion processes

We give almost sure convergence of appropriately normalized particle numbers in bounded domains of locally supercritical branching diffusion processes with one-dimensional periodic diffusions as their non-branching part processes. Some spectral properties of periodic diffusion operators including Hill's ones are also studied.     

On sequential estimation for branching processes with immigration

Consider a Galton–Watson process with immigration. The limiting distributions of the nonsequential estimators of the offspring mean have been proved to be drastically different for the critical case and subcritical and supercritical cases. A sequential estimator, proposed by Sriram et al. (Ann. Statist. 19 (1991) 2232), was shown to be asymptotically normal for both the subcritical and critical cases. Based on a certain stopping rule, we construct a class of two-stage estimators for the offspring mean. These estimators are shown to be asymptotically normal for all the three cases. This gives, without assuming any prior knowledge, a unified estimation and inference procedure for the offspring mean.     

Bivariate estimation with left-truncated data

1. Introductionffendomly truncated data frequently arise in medical studies; other application areas include economics, insurance and astronomy in a broad senses random truncation correspondsto biased sampling, where only partial or incomplete data are aVailable about the variableof interest. A typical realization. can occur as follows: Suppose that individuals/items experience tWO consecutive events in time, an initiating eveal at t and a terminating eveal ats. Usuajly, statistical niterest …     

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In this paper, linear regression models with contaminated data are considered. Estimation methods for the regression parameters based on least absolute deviations (LAD) are proposed, and properties of consistency and asymptotic normality of the proposed method are proved under some regular conditions. Simulations are done to assess the properties of the method when sample size is small, and simulation results show that the methods works well.     

On long time almost sure asymptotics of renormalized branching diffusion processes

We are concerned with the long time behavior of branching diffusion processes. We give a partial answer to the following question: given a smooth density g₀, a branching rate and a spatial motion, does there exist a (nonspatially homogeneous) binary offspring distribution such that the corresponding renormalized branching process tends a.s. to g₀(y) dy as time grows to infinity?     

Bayesian non-parametric estimation for age-dependent branching processes

The purpose of this paper is to obtain Bayes estimators for both the offspring and life-length distribution in the context of a Bellman-Harris age-dependent branching process. We take a non-parametric approach by letting the prior random distributions, for the offspring and life-length distributions, be independent Dirichlet processes. Our primary results concern the derivation of Bayes estimators, under weighted squared error loss for each distribution. We also indicate some of their asymptotic properties and briefly discuss the modifications that become necessary when the initial information is such that the prior random distribution cannot be taken to be independent.     

Parametric estimation for linear stochastic differential equations driven by mixed fractional Brownian motion

ABSTRACT

We investigate the asymptotic properties of the maximum likelihood estimator and Bayes estimator of the drift parameter for stochastic processes satisfying linear stochastic differential equations driven by a mixed fractional Brownian motion. We obtain a Bernstein–von Mises-type theorem also for such a class of processes.     

Wavelet estimation of the diffusion coefficient in time dependent diffusion models

The estimation problem for diffusion coefficients in diffusion processes has been studied in many papers,where the diffusion coefficient function is assumed to be a 1-dimensional bounded Lipschitzian function of the state or the time only.There is no previous work for the nonparametric estimation of time-dependent diffusion models where the diffusion coefficient depends on both the state and the time.This paper introduces and studies a wavelet estimation of the time-dependent diffusion coefficient under a more general assumption that the diffusion coefficient is a linear growth Lipschitz function.Using the properties of martingale,we translate the problems in diffusion into the nonparametric regression setting and give the L~r convergence rate.A strong consistency of the estimate is established.With this result one can estimate the time-dependent diffusion coefficient using the same structure of the wavelet estimators under any equivalent probability measure.For example, in finance,the wavelet estimator is strongly consistent under the market probability measure as well as the risk neutral probability measure.     

Maximum likelihood estimation for noncausal autoregressive processes

We discuss a maximum likelihood procedure for estimating parameters in possibly noncausal autoregressive processes driven by i.i.d. non-Gaussian noise. Under appropriate conditions, estimates of the parameters that are solutions to the likelihood equations exist and are asymptotically normal. The estimation procedure is illustrated with a simulation study for AR(2) processes.     

Inference for thinned point processes, with application to Cox processes

Given i.i.d. point processes N₁, N₂,…, let the observations be p-thinnings N′₁, N′₂,…, where p is a function from the underlying space E (a compact metric space) to [0, 1], whose interpretation is that a point of N_i at x is retained with probability p(x) and deleted with probability 1−p(x). Strongly consistent estimators of the thinning function p and the Laplace functional L_N(f) = E[e^−N(f)] of the N_i are constructed; associated “central limit” properties are given. Tests are presented, for the case when the N_i and N′_i are both observable, of the hypothesis that the N′_i are p-thinnings of the N_i. State estimation techniques are developed for the case where the N_i are Cox processes directed by unobservable random measures M_i; these techniques yield minimum mean-squared error estimators, based on observation of only the thinned processes N′_i of the N_i and the directing measures M_i. Limit theorems for empirical Laplace functionals of point processes are given.     

Reducing component estimation for varying coefficient models with longitudinal data

Varying-coefficient models with longitudinal observations are very useful in epidemiology and some other practical fields.In this paper,a reducing component procedure is proposed for es- timating the unknown functions and their derivatives in very general models,in which the unknown coefficient functions admit different or the same degrees of smoothness and the covariates can be time- dependent.The asymptotic properties of the estimators,such as consistency,rate of convergence and asymptotic distribution,are derived.The asymptotic results show that the asymptotic variance of the reducing component estimators is smaller than that of the existing estimators when the coefficient functions admit different degrees of smoothness.Finite sample properties of our procedures are studied through Monte Carlo simulations.     

Instrumental variable estimation for a linear stochastic differential equation driven by a mixed fractional Brownian motion

We investigate the asymptotic properties of instrumental variable estimators of the drift parameter for stochastic processes satisfying linear stochastic differential equations driven by mixed fractional Brownian motion.     

局部平稳扩散模型中时变参数的加权最小二乘估计

基于离散观测样本,利用局部线性拟合,得到了局部平稳扩散模型中时变漂移参数的加权最小二乘估计,并讨论了估计量的相合性,渐近正态性和一致收敛速度.同时,通过模拟研究说明了估计量的有效性.     

Brillinger type mixing conditions for a simple branching diffusion process

A simple branching diffusion process is described. Formulae for intensity functions and factorial cumulant density functions at several times are given. Mixing conditions in terms of integrals of these cumulants are defined and proven for this stochastic evolutionary point process. The mixing conditions then allow a spatial central limit theorem and a strong law of large numbers to be easily obtained.     

Local asymptotic normality of a sequential model for marked point processes and its applications

This paper deals with statistical inference problems for a special type of marked point processes based on the realization in random time intervals [0,_u]. Sufficient conditions to establish the local asymptotic normality (LAN) of the model are presented, and then, certain class of stopping times _u satisfying them is proposed. Using these stopping rules, one can treat the processes within the framework of LAN, which yields asymptotic optimalities of various inference procedures. Applications for compound Poisson processes and continuous time Markov branching processes (CMBP) are discussed. Especially, asymptotically uniformly most powerful tests for criticality of CMBP can be obtained. Such tests do not exist in the case of the non-sequential approach. Also, asymptotic normality of the sequential maximum likelihood estimators (MLE) of the Malthusian parameter of CMBP can be derived, although the non-sequential MLE is not asymptotically normal in the supercritical case.     

Minimum contrast estimator for fractional Ornstein-Uhlenbeck processes

This paper proposes a minimum contrast methodology to estimate the drift parameter for the Ornstein-Uhlenbeck process driven by fractional Brownian motion of Hurst index, which is greater than one half. Both the strong consistency and the asymptotic normality of this minimum contrast estimator are studied based on the Laplace transform. The numerical simulation results confirm the theoretical analysis and show that the minimum contrast technique is effective and efficient.     

局部平稳短期利率扩散模型的半参数估计

主要研究局部平稳扩散模型的半参数估计．首先,基于局部常数拟合,利用局部加权最小二乘法得到了漂移参数函数的估计量．同时,通过Kolmogorov向前方程,得到了扩散函数的估计量．然后,分别讨论了所得估计量的相合性和渐近正态性．最后,通过模拟研究说明了估计量的有效性．