Fisher Information for Fractional Brownian Motion Under High-Frequency Discrete Sampling

We investigate the issue of the validation of the local asymptotic normality property of three characterizing parameters of the fractional Brownian motion under high-frequency discrete sampling. We prove that the local asymptotic normality property holds true for the likelihood only when at least one of the volatility parameter and the Hurst exponent is known. We provide optimal rates of convergence of the three parameters and Fisher information matrix in closed form.     

On Minimum Distance Estimation in Recurrent Markov Step Processes. I

Consider a Markov step process X=(X_t)_t≥0 whose generator depends on an unknown d -dimensional parameter ϑ. We look at certain empirical measures for recurrent Markov step processes and their a.s. convergence; based on this, we introduce a class of minimum distance estimators. For broad families of sequential observation schemes (at stage n, the trajectory of X is observed up to time S_n, (S_n)_n a sequence of stopping times increasing to ∞), we formulate a stochastic expansion of the suitably rescaled estimation error; for a particular scheme, asymptotic normality is obtained as n →∞. A minimax property under misspecification of the model (in the sense that the true probability law is contiguous to the parametric model but not contained in it) is given.     

Parameter identification for the discretely observed geometric fractional Brownian motion

This paper deals with the problem of estimating all the unknown parameters of geometric fractional Brownian processes from discrete observations. The estimation procedure is built upon the marriage of the quadratic variation and the maximum likelihood approach. The asymptotic properties of the estimators are provided. Moveover, we compare our derived method with the approach proposed by Misiran et al. [Fractional Black-Scholes models: complete MLE with application to fractional option pricing. In International conference on optimization and control; Guiyang, China; 2010. p. 573–586.], namely the complete maximum likelihood estimation. Simulation studies confirm theoretical findings and illustrate that our methodology is efficient and reliable. To show how to apply our approach in realistic contexts, an empirical study of Chinese financial market is also presented.     

Variance Estimation for Fractional Brownian Motions with Fixed Hurst Parameters

Some real-world phenomena in geo-science, micro-economy, and turbulence, to name a few, can be effectively modeled by a fractional Brownian motion indexed by a Hurst parameter, a regularity level, and a scaling parameter σ², an energy level. This article discusses estimation of a scaling parameter σ² when a Hurst parameter is known. To estimate σ², we propose three approaches based on maximum likelihood estimation, moment-matching, and concentration inequalities, respectively, and discuss the theoretical characteristics of the estimators and optimal-filtering guidelines. We also justify the improvement of the estimation of σ² when a Hurst parameter is known. Using the three approaches and a parametric bootstrap methodology in a simulation study, we compare the confidence intervals of σ² in terms of their lengths, coverage rates, and computational complexity and discuss empirical attributes of the tested approaches. We found that the approach based on maximum likelihood estimation was optimal in terms of efficiency and accuracy, but computationally expensive. The moment-matching approach was found to be not only comparably efficient and accurate but also computationally fast and robust to deviations from the fractional Brownian motion model.     

Estimation for Discretely Observed Small Diffusions Based on Approximate Martingale Estimating Functions

Abstract.  We consider an asymptotically efficient estimator of the drift parameter for a multi-dimensional diffusion process with small dispersion parameter ɛ . In the situation where the sample path is observed at equidistant times k / n , k  = 0, 1, …,  n , we study asymptotic properties of an M -estimator derived from an approximate martingale estimating function as ɛ tends to 0 and n tends to ∞ simultaneously.     

Asymptotic Inference for Jump Diffusions with State‐Dependent Intensity

We establish the local asymptotic normality property for a class of ergodic parametric jump‐diffusion processes with state‐dependent intensity and known volatility function sampled at high frequency. We prove that the inference problem about the drift and jump parameters is adaptive with respect to parameters in the volatility function that can be consistently estimated.     

Estimation for Continuous Branching Processes

The maximum-likelihood estimator for the curved exponential family given by continuous branching processes with immigration is investigated. These processes originated from population biology but also model the dynamics of interest rates and development of the state of technology in economics. It is proved that in contrast to branching processes with discrete space and/or time the MLE gives a unified approach to the inference. In order to include singular subdomains of the parameter space we modify the MLE slightly. Consistency and asymptotic normality for the MLE are considered. Concerning the asymptotic theory of the experiments, all three properties LAQ, LAN, and LAMN occur for different submodels     

Parameter Estimation for a Discretely Observed Integrated Diffusion Process

Abstract.  We consider the estimation of unknown parameters in the drift and diffusion coefficients of a one-dimensional ergodic diffusion X when the observation is a discrete sampling of the integral of X at times i Δ , i  =  1 ,…, n . Assuming that the sampling interval tends to 0 while the total length time interval tends to infinity, we first prove limit theorems for functionals associated with our observations. We apply these results to obtain a contrast function. The associated minimum contrast estimators are shown to be consistent and asymptotically Gaussian with different rates for drift and diffusion coefficient parameters.     

Estimation of an Ergodic Diffusion from Discrete Observations

We consider a one-dimensional diffusion process X , with ergodic property, with drift b ( x , θ) and diffusion coefficient a ( x , σ) depending on unknown parameters θ and σ. We are interested in the joint estimation of (θ, σ). For that purpose, we dispose of a discretized trajectory, observed at n equidistant times tⁿ_i = ih_n , 1 ≤ i ≤ n . We assume that h_n ← 0 and nh_n ←∞. Under the condition nh_n^p ← 0 for an arbitrary integer p , we exhibit a contrast dependent on p which provides us with an asymptotically normal and efficient estimator of (θ, σ).     

An Orthogonality‐Based Estimation of Moments for Linear Mixed Models

Abstract. Estimating higher‐order moments, particularly fourth‐order moments in linear mixed models is an important, but difficult issue. In this article, an orthogonality‐based estimation of moments is proposed. Under only moment conditions, this method can easily be used to estimate the model parameters and moments, particularly those of higher order than the second order, and in the estimators the random effects and errors do not affect each other. The asymptotic normality of all the estimators is provided. Moreover, the method is readily extended to handle non‐linear, semiparametric and non‐linear models. A simulation study is carried out to examine the performance of the new method.     

On Minimum Distance Estimation Using Kolmogorov-Lévy Type Metrics

Let X ₁, . . ., X_n be independent identically distributed random variables with a common continuous (cumulative) distribution function (d.f.) F , and F^_n the empirical d.f. (e.d.f.) based on X ₁, . . ., X_n . Let G be a smooth d.f. and Gθ = G (·–θ) its translation through θ∈ R . Using a Kolmogorov-Lévy type metric ρ_α defined on the space of d.f.s. on R , the paper derives both null and non-null limiting distributions of √ n [ ρ_α ( F_n , Gθ_n ) – ρ_α ( F, G_θ )], √ n (θ _n –θ) and √ nρ_α ( Gθ , Gθ ), where θ _n and θ are the minimum ρ_α -distance parameters for F_n and F from G , respectively. These distributions are known explicitly in important particular cases; with some complementary Monte Carlo simulations, they help us clarify our understanding of estimation using minimum distance methods and supremum type metrics. We advocate use of the minimum distance method with supremum type metrics in cases of non-null models. The resulting functionals are Hadamard differentiable and efficient. For small scale parameters the minimum distance functionals are close to medians of the parent distributions. The optimal small scale models result in minimum distance estimators having asymptotic variances very competitive and comparable with best known robust estimators.     

Local asymptotic properties for Cox-Ingersoll-Ross process with discrete observations

In this paper, we consider a one-dimensional Cox-Ingersoll-Ross (CIR) process whose drift coefficient depends on unknown parameters. Considering the process discretely observed at high frequency, we prove the local asymptotic normality property in the subcritical case, the local asymptotic quadraticity in the critical case, and the local asymptotic mixed normality property in the supercritical case. To obtain these results, we use the Malliavin calculus techniques developed recently for CIR process together with the estimation for positive and negative polynomial moments of the CIR process. In this study, we require the same conditions of high frequency and infinite horizon as in the case of ergodic diffusions with globally Lipschitz coefficients studied earlier in the literature. However, in the non-ergodic cases, additional assumptions on the decreasing rate are required due to the fact that the square root diffusion coefficient of the CIR process is not regular enough.     

End-Point Estimation for Decreasing Densities: Asymptotic Behaviour of the Penalized Likelihood Ratio

Abstract.  We consider the problem of estimating the modal value of a decreasing density on the positive real line. This has application in several interesting phenomena arising, for example, in renewal theory, and in biased and distance samplings. We use a penalized likelihood ratio-based approach for inference and derive the scale-free universal large sample null distribution of the log-likelihood ratio, using a suitably chosen penalty parameter. We present simulation results and a real data analysis to corroborate our findings, and compare the performance of the confidence sets with the existing results.     

LAN property for an ergodic diffusion with jumps

In this paper, we consider a multidimensional ergodic diffusion with jumps driven by a Brownian motion and a Poisson random measure associated with a compound Poisson process, whose drift coefficient depends on an unknown parameter. Considering the process discretely observed at high frequency, we derive the local asymptotic normality (LAN) property.     

Asymptotic Likelihood Based Inference for Co-integrated Homogenous Gaussian Diffusions

In this paper we consider inference for a multivariate Gaussian homogenous diffusion which is co-integrated, i.e. admits a representation in terms of stable relations (ergodic diffusions) plus Brownian motions. We show that inference on co-integration rank (the number of stable relations) in continuous time can be based on existing asymptotic distributions from discrete time co-integration analysis. Likewise the asymptotic distributions of the co-integration parameters are shown to be mixed Gaussian. Special attention is given to the parametrization of the drift terms. It is shown that the asymptotic distribution of the co-integration rank test statistic does not depend on the level of the process as a result of the chosen parametrization.     

Estimation and testing stationarity for double-autoregressive models

Summary.  The paper considers the double-autoregressive model y _t  =  φ y _{t −1}+ ɛ _t with ɛ _t  =     . Consistency and asymptotic normality of the estimated parameters are proved under the condition E  ln | φ  +√ α η _t |<0, which includes the cases with | φ |=1 or | φ |>1 as well as     . It is well known that all kinds of estimators of φ in these cases are not normal when ɛ _t are independent and identically distributed. Our result is novel and surprising. Two tests are proposed for testing stationarity of the model and their asymptotic distributions are shown to be a function of bivariate Brownian motions. Critical values of the tests are tabulated and some simulation results are reported. An application to the US 90-day treasury bill rate series is given.     

Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects

Abstract. We consider N independent stochastic processes (X _i (t), t ∈ [0,T _i ]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φ _i . The distribution of the random effect φ _i depends on unknown parameters which are to be estimated from the continuous observation of the processes X_i. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ _i and φ _i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T _i =T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.     

On the likelihood function of small time variance Gamma Lévy processes

We investigate the likelihood function of small generalized Laplace laws and variance gamma Lévy processes in the short time framework. We prove the local asymptotic normality property in statistical inference for the variance gamma Lévy process under high-frequency sampling with its associated optimal convergence rate and Fisher information matrix. The location parameter is required to be given in advance for this purpose, while the remaining three parameters are jointly well behaved with an invertible Fisher information matrix. The results are discussed with relation to equivalent formulations of the variance gamma Lévy process, that is, as a time-changed Brownian motion and as a difference of two independent gamma processes.