Local Asymptotic Normality Property Research Articles

Locally optimal tests against periodic linear regression in short panels

This paper aims to detect the periodic coefficients of a Linear Regression model with Panel Data. Nonparametric locally and asymptotically optimal tests are proposed for testing the null hypothesis of a traditional linear regression model, against an alternative of a Periodic Coefficient Regression model. The Uniform Local Asymptotic Normality (ULAN) property is established for Periodic Linear Regression (PLR) model, to construct Locally asymptotically optimal tests. The rank-based version of the optimal parametric test is provided in particular cases. The performance of the proposed tests is valid with numerical simulations using the Monte Carlo study. Real data was studied for applying the periodic linear regression model which compared to both traditional regression and random coefficient regression models.

Tests in functional autoregressive processes via local asymptotic normality condition

Local asymptotic normality (LAN) and uniform local asymptotic normality (ULAN) conditions are obtained for a linear functional autoregressive process of order p, p≥1, using a martingale central limit theorem. As a consequence of the LAN property, asymptotically efficient tests for determining the order are constructed. An example is discussed as a special cases.

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.

Estimation in short-panel data models with bilinear errors

Many estimation methods have been proposed for the parameters of the regression models with serially correlated errors. In this work, we develop an asymptotic theory for estimation in the short panel data models with bilinear error. We propose a comparative study by simulation between several estimators (adaptive, ordinary and weighted least squares) for the coefficients of panel data models when the errors are bilinear serially correlated. As a consequence of the uniform local asymptotic normality property, we obtain adaptive estimates of the parameters. Finally, we illustrate the performance of the proposed estimators via Monte Carlo simulation study. We show that the adaptive estimates are more efficient than the weighted and ordinary least squares estimates.

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.

The LAN property for McKean–Vlasov models in a mean-field regime

We establish the local asymptotic normality (LAN) property for estimating a multidimensional parameter in the drift of a system of N interacting particles observed over a fixed time horizon in a mean-field regime N→∞. By implementing the classical theory of Ibragimov and Hasminski, we obtain in particular sharp results for the maximum likelihood estimator that go beyond its simple asymptotic normality thanks to Hájek’s convolution theorem and strong controls of the likelihood process that yield asymptotic minimax optimality (up to constants). Our structural results shed some light to the accompanying nonlinear McKean–Vlasov experiment, and enable us to derive simple and explicit criteria to obtain identifiability and non-degeneracy of the Fisher information matrix. These conditions are also of interest for other recent studies on the topic of parametric inference for interacting diffusions.

LOCAL ASYMPTOTIC NORMALITY OF GENERAL CONDITIONALLY HETEROSKEDASTIC AND SCORE-DRIVEN TIME-SERIES MODELS

The paper establishes the local asymptotic normality property for general conditionally heteroskedastic time series models of multiplicative form, $\epsilon _t=\sigma _t(\boldsymbol {\theta }_0)\eta _t$ , where the volatility $\sigma _t(\boldsymbol {\theta }_0)$ is a parametric function of $\{\epsilon _{s}, s< t\}$ , and $(\eta _t)$ is a sequence of i.i.d. random variables with common density $f_{\boldsymbol {\theta }_0}$ . In contrast with earlier results, the finite dimensional parameter $\boldsymbol {\theta }_0$ enters in both the volatility and the density specifications. To deal with nondifferentiable functions, we introduce a conditional notion of the familiar quadratic mean differentiability condition which takes into account parameter variation in both the volatility and the errors density. Our results are illustrated on two particular models: the APARCH with asymmetric Student-t distribution, and the Beta-t-GARCH model, and are extended to handle a conditional mean.

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.

Locally asymptotically efficient estimation for parametric PINAR(p) models

This article focuses on the efficient estimation problem of an arbitrary‐order periodic integer‐valued autoregressive (PINAR(p)) model. Both the local asymptotic normality (LAN) property and the local asymptotic linearity property satisfied by the central sequence of the underlying model are established. Using these results, we construct efficient estimators for the parameters in a parametric framework. The consistency property of these efficient estimations is evaluated via an intensive simulation study. Moreover, the performances of these efficient estimations, over the conditional maximum likelihood (CML) and the conditional least squares (CLS) estimations, are also illustrated via an intensive simulation study and an application on real data set.

Adaptive test for periodicity in restrictive EXPAR(p) models

Locally asymptotically most stringent test for testing the periodicity in the restricted exponential autoregressive model EXPAR(1) has been constructed by Merzougui et al. This test is generalized here to the restricted EXPAR of order p. On the other hand, we construct a semi-parametric optimal test of periodicity when the innovation density is unknown. The sufficient conditions of the Local Asymptotic Normality (LAN) property are adapted to the periodic model.

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.

Local asymptotic normality for a periodically time varying long memory parameter

A periodically time varying long memory parameter version of the fixed long memory parameter case local asymptotic normality (LAN) results are implemented here. The process is assumed to be invertible and causal for simplicity. We prove that the general condition of LAN is verified by our model. The simulation experiment is carried out for studying the distribution of the central sequence in LAN property. The most interesting in this note is to construct an adaptive estimate in future consideration of this type of model.

Optimal Detection of Bilinear Dependence in Short Panels of Regression Data

In this paper, we propose parametric and nonparametric locally andasymptotically optimal tests for regression models with superdiagonal bilinear time series errors in short panel data (large n, small T). We establish a local asymptotic normality property– with respect to intercept μ, regression coefficient β, the scale parameter σ of the error, and the parameter b of panel superdiagonal bilinear model (which is the parameter of interest)– for a given density f1 of the error terms. Rank-based versions of optimal parametric tests are provided. This result, which allows, by Hájek’s representation theorem, the construction of locally asymptotically optimal rank-based tests for the null hypothesis b = 0 (absence of panel superdiagonal bilinear model). These tests –at specified innovation densities f1– are optimal (most stringent), but remain valid under any actual underlying density. From contiguity, we obtain the limiting distribution of our test statistics under the null and local sequences of alternatives. The asymptotic relative efficiencies, with respect to the pseudo-Gaussian parametric tests, are derived. A Monte Carlo study confirms the good performance of the proposed tests.

The nonparametric LAN expansion for discretely observed diffusions

Consider a scalar reflected diffusion $(X_{t}:t\geq 0)$, where the unknown drift function $b$ is modelled nonparametrically. We show that in the low frequency sampling case, when the sample consists of $(X_{0},X_{\Delta },...,X_{n\Delta })$ for some fixed sampling distance $\Delta >0$, the model satisfies the local asymptotic normality (LAN) property, assuming that $b$ satisfies some mild regularity assumptions. This is established by using the connections of diffusion processes to elliptic and parabolic PDEs. The key tools used are regularity estimates for certain parabolic PDEs as well as a detailed analysis of the spectral properties of the elliptic differential operator related to $(X_{t}:t\geq 0)$.

Local asymptotic normality and efficient estimation for multivariate GINAR(p) models

AbstractWe derive the Local Asymptotic Normality (LAN) property for a multivariate generalized integer-valued autoregressive (MGINAR) process with order p. The generalized thinning operator in the ...

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)].

Local asymptotic normality property for fractional Gaussian noise under high-frequency observations

Local Asymptotic Normality (LAN) property for fractional Gaussian noise under high-frequency observations is proved with nondiagonal rate matrices depending on the parameter to be estimated. In contrast to the LAN families in the literature, nondiagonal rate matrices are inevitable. As consequences of the LAN property, a maximum likelihood sequence of estimators is shown to be asymptotically efficient and the likelihood ratio test on the Hurst parameter is shown to be an asymptotically uniformly most powerful unbiased test for two-sided hypotheses.

LAN property for stochastic differential equations with additive fractional noise and continuous time observation

We consider a stochastic differential equation with additive fractional noise with Hurst parameter H>1∕2, and a non-linear drift depending on an unknown parameter. We show the Local Asymptotic Normality property (LAN) of this parametric model with rate τ as τ→∞, when the solution is observed continuously on the time interval [0,τ]. The proof uses ergodic properties of the equation and a Girsanov-type transform. We analyze the particular case of the fractional Ornstein–Uhlenbeck process and show that the Maximum Likelihood Estimator is asymptotically efficient in the sense of the Minimax Theorem.

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.