Local Asymptotic Mixed Normality Research Articles

Malliavin calculus techniques for local asymptotic mixed normality and their application to hypoelliptic diffusions

Malliavin calculus techniques for local asymptotic mixed normality and their application to hypoelliptic diffusions

Local asymptotic normality for ergodic jump-diffusion processes via transition density approximation

We show local asymptotic normality (LAN) for a statistical model of discretely observed ergodic jump-diffusion processes, where the drift coefficient, diffusion coefficient, and jump structure are parametrized. Under the LAN property, we can discuss the asymptotic efficiency of regular estimators, and the quasi-maximum-likelihood and Bayes-type estimators proposed in Shimizu and Yoshida (Stat. Inference Stoch. Process. 9 (2006) 227–277) and Ogihara and Yoshida (Stat. Inference Stoch. Process. 14 (2011) 189–229) are shown to be asymptotically efficient in this model. Moreover, we can construct asymptotically uniformly most powerful tests for the parameters. Unlike with a model for diffusion processes, Aronson-type estimates of the transition density functions do not hold, which makes it difficult to prove LAN. Therefore, instead of Aronson-type estimates, we employ the idea of Theorem 1 in Jeganathan (Sankhyā Ser. A 44 (1982) 173–212) and use the L2 regularity condition. Moreover, we show that local asymptotic mixed normality of a statistical model is implied from that for a model generated by approximated transition density functions under suitable conditions. Together with density approximation by means of thresholding techniques, the LAN property for the jump-diffusion processes is proved.

LAMN property for jump diffusion processes with discrete observations on a fixed time interval

We consider a one-dimensional stochastic differential equation with jumps driven by a Brownian motion and an independent Lévy process with finite Lévy measure, whose drift and diffusion coefficients depend on an unknown parameter. Under smoothness and non-degeneracy assumptions on the drift and diffusion coefficients and integrability assumption of jump size distribution, we prove the Local Asymptotic Mixed Normality property when the solution process is observed discretely at high frequency on a fixed time interval. The proof is essentially based on Malliavin calculus techniques and an analysis of the jump structure of the Lévy process.

How to estimate the memory of the elephant random walk

We introduce an original way to estimate the memory parameter of the elephant random walk, a fascinating discrete time random walk on integers having a complete memory of its entire history. Our estimator is nothing more than a quasi-maximum likelihood estimator, based on a second order Taylor approximation of the log-likelihood function. We show the almost sure convergence of our estimate in the diffusive, critical and superdiffusive regimes. The local asymptotic normality of our statistical procedure is established in the diffusive regime, while the local asymptotic mixed normality is proven in the superdiffusive regime. Asymptotic and exact confidence intervals as well as statistical tests are also provided. All our analysis relies on asymptotic results for martingales and the quadratic variations associated.

Maximum likelihood estimation and the local asymptotic mixed normality in a second‐order branching process with continuous state space

A second‐order branching process with continuous state space (SOBPCS) is considered. Maximum likelihood (ML) estimation in SOBPCS is studied under the assumption that the underlying subordinators are gamma processes. This estimation is discussed under two sampling schemes. In the first scheme, we assume that all generation sizes as well as the contributions of lag 1 and lag 2 in each generation are observable. In the second scheme, we assume that only generation sizes are observable. The derived estimators under each scheme are shown to be consistent and asymptotically normal (CAN). Also, it is shown that the SOBPCS belongs to the local asymptotic mixed normal (LAMN) family of models. A simulation study is carried out to verify the results. We apply our results to the optical density data emerging through growth of Bacilllus subtilis bacteria.

LAMN property for multivariate inhomogeneous diffusions with discrete observations

We consider a class of multidimensional inhomogeneous diffusions whose drift coefficient depends on a multidimensional unknown parameter. Under some appropriate assumptions, we prove the local asymptotic mixed normality property for the drift parameter from high-frequency observations when the length of the observation window tends to infinity. To obtain the result, we use the Malliavin calculus techniques and the change of measures. Our approach is applicable for both ergodic and non-ergodic diffusions.

A note on the local asymptotic mixed normality of a controlled branching process with a random control function

This note discusses the local asymptotic mixed normality property of a controlled branching process with a random control function. We establish this property under the assumption that the process is supercritical and on a non-extinction path.

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

AbstractIn 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.

LAMN property for the drift and volatility parameters of a sde driven by a stable Lévy process

This work focuses on the local asymptotic mixed normality (LAMN) property from high frequency observations, of a continuous time process solution of a stochastic differential equation driven by a truncated α-stable process with index α ∈ (0, 2). The process is observed on the fixed time interval [0,1] and the parameters appear in both the drift coefficient and scale coefficient. This extends the results of Clément and Gloter [Stoch. Process. Appl. 125 (2015) 2316–2352] where the index α ∈ (1, 2) and the parameter appears only in the drift coefficient. We compute the asymptotic Fisher information and find that the rate in the LAMN property depends on the behavior of the Lévy measure near zero. The proof relies on the small time asymptotic behavior of the transition density of the process obtained in Clément et al. [Preprint HAL-01410989v2 (2017)].

LAMN in a class of parametric models for null recurrent diffusions

We study statistical models for one-dimensional diffusions which are null recurrent. A first parameter in the drift is the principal one, and determines regular varying rates of convergence for the score and the information process. A finite number of other parameters, of secondary importance, introduces additional flexibility for the modelization of the drift, and does not perturb the null recurrent behaviour. Under time-continuous observation we obtain local asymptotic mixed normality, state a local asymptotic minimax bound, and specify asymptotically optimal estimators.

Asymptotics in small time for the density of a stochastic differential equation driven by a stable Lévy process

This work focuses on the asymptotic behavior of the density in small time of a stochastic differential equation driven by a truncatedα-stable process with indexα∈ (0, 2). We assume that the process depends on a parameterβ= (θ,σ)Tand we study the sensitivity of the density with respect to this parameter. This extends the results of [E. Clément and A. Gloter, Local asymptotic mixed normality property for discretely observed stochastic dierential equations driven by stable Lévy processes.Stochastic Process. Appl.125 (2015) 2316–2352.] which was restricted to the indexα∈ (1, 2) and considered only the sensitivity with respect to the drift coefficient. By using Malliavin calculus, we obtain the representation of the density and its derivative as an expectation and a conditional expectation. This permits to analyze the asymptotic behavior in small time of the density, using the time rescaling property of the stable process.

Quantum non demolition measurements: Parameter estimation for mixtures of multinomials

In Quantum Non Demolition measurements, the sequence of observations is distributed as a mixture of multinomial random variables. Parameters of the dynamics are naturally encoded into this family of distributions. We show the local asymptotic mixed normality of the underlying statistical model and the consistency of the maximum likelihood estimator. Furthermore, we prove the asymptotic optimality of this estimator as it saturates the usual Cram\'er Rao bound.

Local asymptotic quadraticity of statistical experiments connected with a Heston model

We study local asymptotic properties of likelihood ratios of certain Heston models. We distinguish three cases: subcritical, critical and supercritical models. For the drift parameters, local asymptotic normality is proved in the subcritical case, only local asymptotic quadraticity is shown in the critical case, while in the supercritical case not even local asymptotic quadraticity holds. For certain submodels, local asymptotic normality is proved in the critical case, and local asymptotic mixed normality is shown in the supercritical case. As a consequence, asymptotically optimal (randomized) tests are constructed in cases of local asymptotic normality. Moreover, local asymptotic minimax bound, and hence, asymptotic efficiency in the convolution theorem sense are concluded for the maximum likelihood estimators in cases of local asymptotic mixed normality.

One-parameter statistical model for linear stochastic differential equation with time delay

ABSTRACTAssume that we observe a stochastic process , which satisfies the linear stochastic delay differential equation where a is a finite signed measure on . The local asymptotic properties of the likelihood function are studied. Local asymptotic normality is proved in case of , local asymptotic quadraticity is shown if , and, under some additional conditions, local asymptotic mixed normality or periodic local asymptotic mixed normality is valid if , where is an appropriately defined quantity. As an application, the asymptotic behaviour of the maximum likelihood estimator of based on can be derived as.

Local asymptotic mixed normality property for nonsynchronously observed diffusion processes

We prove the local asymptotic mixed normality (LAMN) property for a family of probability measures defined by parametrized diffusion processes with nonsynchronous observations. We assume that observation times of processes are independent of processes and we will study asymptotics when the maximum length of observation intervals goes to zero in probability. We also prove that the quasi-maximum likelihood estimator and the Bayes-type estimator proposed in Ogihara and Yoshida (Stochastic Process. Appl. 124 (2014) 2954-3008) are asymptotically efficient.

Asymptotic inference for a stochastic differential equation with uniformly distributed time delay

For the affine stochastic delay differential equation dX(t)=a∫−10X(t+u)dudt+dW(t),t⩾0, the local asymptotic properties of the likelihood function are studied. Local asymptotic normality is proved in case of a∈(−π22,0), local asymptotic mixed normality is shown if a∈(0,∞), periodic local asymptotic mixed normality is valid if a∈(−∞,−π22), and only local asymptotic quadraticity holds at the points −π22 and 0. Applications to the asymptotic behaviour of the maximum likelihood estimator âT of a based on (X(t))t∈[0,T] are given as T→∞.

Local Asymptotic Mixed Normality property for discretely observed stochastic differential equations driven by stable Lévy processes

We prove the Local Asymptotic Mixed Normality property from high frequency observations, of a continuous time process solution of a stochastic differential equation driven by a pure jump Lévy process. The process is observed on the fixed time interval [0,1] and the parameter appears in the drift coefficient only. We compute the asymptotic Fisher information and find that the rate in the LAMN property depends on the behavior of the Lévy measure near zero. The proof of this result contains a sharp study of the asymptotic behavior, in small time, of the transition probability density of the process and of its logarithm derivative.

Local asymptotic mixed normality for discretely observed non-recurrent Ornstein–Uhlenbeck processes

Consider non-recurrent Ornstein–Uhlenbeck processes with unknown drift and diffusion parameters. Our purpose is to estimate the parameters jointly from discrete observations with a certain asymptotics. We show that the likelihood ratio of the discrete samples has the uniform LAMN property, and that some kind of approximated MLE is asymptotically optimal in a sense of asymptotic maximum concentration probability. The estimator is also asymptotically efficient in ergodic cases.

Bayesian prediction and model selection for locally asymptotically mixed normal models

An information criterion for models with local asymptotic mixed normality (LAMN) is proposed. Since the widely known Akaike's Information Criterion (AIC) is derived on the basis of local asymptotic normality (LAN), it cannot be directly used to model selection of LAMN models, and so a criterion for these models is required. The proposed criterion for LAMN models is an asymptotically unbiased estimator of the Kullback–Leibler risk of Bayesian prediction. We present the results of simulation studies for a mixed normal model, a discretely observed diffusion model and a partially explosive Gaussian AR model.

Local asymptotic mixed normality of transformed Gaussian models for random fields

Local asymptotic mixed normality (LAMN) of a class of transformed Gaussian models for discretely observed random fields is proved. The original Gaussian random field is assumed to be the product of a deterministic process and a process with independent increments. The transformed process is observed only on discrete lattice points in the unit cube and fixed domain asymptotics is investigated. This model is useful for modeling random fields with non-Gaussian marginal distributions.