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Abstract: Abstract In this paper, we propose an estimator for the Ornstein–Uhlenbeck parameters based on observations of its supremum. We derive an analytic expression for the supremum density. Making use of the pseudo-likelihood method based on the supremum density, our estimator is constructed as the maximal argument of this function. Using weak-dependency results, we prove some statistical properties on the estimator such as consistency and asymptotic normality. Finally, we apply our estimator to simulated and real data. PubDate: 2024-02-26

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Abstract: Abstract This paper deals with a Skorokhod’s integral based projection type estimator \({\widehat{b}}_m\) of the drift function \(b_0\) computed from \(N\in \mathbb N^*\) independent copies \(X^1,\dots ,X^N\) of the solution X of \(dX_t = b_0(X_t)dt +\sigma dB_t\) , where B is a fractional Brownian motion of Hurst index \(H\in (1/2,1)\) . Skorokhod’s integral based estimators cannot be calculated directly from \(X^1,\dots ,X^N\) , but in this paper an \(\mathbb L^2\) -error bound is established on a calculable approximation of \({\widehat{b}}_m\) . PubDate: 2024-02-21

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Abstract: Abstract The problem of estimating the change point in a sequence of independent observations is considered. Hinkley (1970) demonstrated that the maximum likelihood estimate of the change point is associated with a two-sided random walk in which the ascending and descending epochs and heights are the key elements for its evaluation. The aim here is to expand the information generated from the random walks and from fluctuation theory and applied to the change point formulation. This permits us to obtain computable expressions for the asymptotic distribution of the change point with respect to convolutions and Laplace transforms of the likelihood ratios. Further, if moment expressions of the likelihood ratios are known, explicit representations of the asymptotic distribution of the change point become accessible up to the second order with respect to the moments. In addition, the rate of convergence between the finite and infinite distribution of the change point distribution is established and it is shown to be of polynomial order. PubDate: 2023-12-19

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Abstract: Abstract In recent years, the direction has turned to non-stationary time series. Here the situation is more complicated: it is often unclear how to set down a meaningful asymptotic for non-stationary processes. For this reason, the theory of locally stationary processes arose, and it is based on infill asymptotics created from non-parametric statistics. The present paper aims to develop a framework for inference of locally stationary functional time series based on the so-called conditional U-statistics introduced by Stute (Ann Probab 19:812–825, 1991), and may be viewed as a generalization of the Nadaraya-Watson regression function estimates. In this paper, we introduce an estimator of the conditional U-statistics operator that takes into account the nonstationary behavior of the data-generating process. We are mainly interested in establishing weak convergence of conditional U-processes in the locally stationary functional mixing data framework. More precisely, we investigate the weak convergence of conditional U-processes when the explicative variable is functional. We treat the weak convergence when the class of functions is bounded or unbounded, satisfying some moment conditions. These results are established under fairly general structural conditions on the classes of functions and the underlying models. The theoretical results established in this paper are (or will be) critical tools for further functional data analysis developments. PubDate: 2023-12-18

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Abstract: Abstract Estimating parameters of drift and diffusion coefficients for multidimensional stochastic delay equations with small noise are considered. The delay structure is written as an integral form with respect to a delay measure. Our contrast function is based on a local-Gauss approximation to the transition probability density of the process. We show consistency and asymptotic normality of the minimum-contrast estimator when a small dispersion coefficient \(\varepsilon \rightarrow 0\) and sample size \(n\rightarrow \infty \) simultaneously. PubDate: 2023-12-07 DOI: 10.1007/s11203-023-09299-7

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Abstract: Abstract We propose a nonparametric estimation of random effects from the following fractional diffusions \(dX^{j}(t) = \psi _{j}X^{j}(t)d t+X^{j}(t)d W^{H,j}(t), \) \(~X^j(0)=x^j_0,~t\ge 0, \) \( j=1,\ldots ,n,\) where \(\psi _j\) are random variables and \( W^{j,H}\) are fractional Brownian motions with a common known Hurst index \(H\in (0,1)\) . We are concerned with the study of Hermite projection and kernel density estimators for the \(\psi _j\) ’s common density, when the horizon time of observation is fixed or sufficiently large. We corroborate these theoretical results through simulations. An empirical application is made to the real Asian financial data. PubDate: 2023-12-02 DOI: 10.1007/s11203-023-09302-1

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Abstract: Abstract Motivated by the water pollution detection, this paper studies a nonlinear filtering problem over an infinite time interval. The signal to be estimated, which indicates the concentration of undesired chemical in a river, is driven by a stochastic partial differential equation involves unknown parameters. Based on discrete observation, strongly consistent estimators of unknown parameters are derived at first. With the optimal filter given by Bayes formula, the uniqueness of invariant measure for the signal-filter pair has been verified. The paper then establishes approximation to the optimal filter with estimators, showing that the pathwise average distance, per unit time, of the computed approximating filter from the optimal filter converges to zero in probability. Simulation results are presented at last. PubDate: 2023-12-02 DOI: 10.1007/s11203-023-09303-0

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Abstract: Abstract We study parameter estimation for a linear parabolic second-order stochastic partial differential equation (SPDE) in two space dimensions with a small dispersion parameter using high frequency data with respect to time and space. We set two types of Q-Wiener processes as a driving noise. We provide minimum contrast estimators of the coefficient parameters of the SPDE appearing in the eigenfunctions of the differential operator of the SPDE based on the thinned data in space, and approximate the coordinate process based on the thinned data in time. Moreover, we propose an estimator of the drift parameter using the fact that the coordinate process is the Ornstein-Uhlenbeck process and statistical inference for diffusion processes with a small noise. We also give an example and simulation results for the proposed estimators. PubDate: 2023-11-21 DOI: 10.1007/s11203-023-09301-2

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Abstract: Abstract We consider the problem of asymptotically efficient estimation of drift parameters of the ergodic fractional Ornstein-Uhlenbeck process under continuous observations when the Hurst parameter \(H<1/2\) and the mean of its stationary distribution is not equal to zero. In this paper, we derive asymptotically efficient rates and variances of estimators of drift parameters and prove an asymptotic efficiency of a maximum likelihood estimator of drift parameters. PubDate: 2023-11-21 DOI: 10.1007/s11203-023-09300-3

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Abstract: Abstract Asymptotic expansion is presented for an estimator of the Hurst coefficient of a fractional Brownian motion. We first derive the expansion formula of the principal term of the error of the estimator using a recently developed theory of asymptotic expansion of the distribution of Wiener functionals, and utilize the perturbation method on the obtained formula in order to calculate the expansion of the estimator. We also discuss some second-order modifications of the estimator. Numerical results show that the asymptotic expansion attains higher accuracy than the normal approximation. PubDate: 2023-09-25 DOI: 10.1007/s11203-023-09298-8

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Abstract: Abstract This paper studies the second-order asymptotics of maximum likelihood estimator (MLE) and Whittle estimator under \(\varepsilon \) -contaminated model for Gaussian stationary processes. We evaluate the robustness of MLE and Whittle estimator based on the second-order Edgeworth expansion with an \( \varepsilon \) -disturbance spectral density. The measures of second-order robustness of MLE and Whittle estimator are investigated for concrete models with numerical study. The findings show that the MLE of Gaussian autoregressive process is robust in second-order term to a disturbance in spectral density under the middle level of spectral frequency, while it is more sensitive to a contamination under a too low frequency spectral mass. The Whittle estimator is robust to a moving average contamination when the Gaussian autoregressive process is not near unit root case, while it is sensitive to the disturbance under a nonregular situation in the case of near unit root. PubDate: 2023-09-23 DOI: 10.1007/s11203-023-09296-w

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Abstract: Abstract The problem of localization on the plane of two radioactive sources by K detectors is considered. Each detector records a realization of an inhomogeneous Poisson process whose intensity function is the sum of signals arriving from the sources and of a constant Poisson noise of known intensity. The time of the beginning of emission of the sources is known, and the main problem is the estimation of the positions of the sources. The properties of the maximum likelihood and Bayesian estimators are described in the asymptotics of large signals in three situations of different regularities of the fronts of the signals: smooth, cusp-type and change-point type. PubDate: 2023-09-19 DOI: 10.1007/s11203-023-09297-9

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Abstract: Abstract In this paper, we consider an inference problem in generalized exponential Ornstein–Uhlenbeck processes with change-point in the context where the dimensions of the drift parameter are unknown. The proposed method generalizes the work in recent literature for which the change-point has never been considered. Thus, in addition to taking care of possible chock, we study the asymptotic properties of the unrestricted estimator, the restricted estimator, and shrinkage estimators for the drift parameters. We also derive an asymptotic test for change-point detection and we establish the asymptotic distributional risk of the proposed estimators as well as their relative efficiency. Further, we prove that the proposed methods improve the goodness-of-fit. Finally, we present the simulation results which corroborate the theoretical findings and we analyze a financial market data set. PubDate: 2023-09-01 DOI: 10.1007/s11203-023-09293-z

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Abstract: Abstract We derive a Cramér–von Mises test for testing a class of time dependent coefficients Coditional Heteroscedastic AutoRegressive Non Linear (CHARN) models. The test statistic is based on the log-likelihood ratio process whose weak convergence in a suitable Fréchet space is studied under the null hypothesis and under the sequence of local alternatives considered. This study makes use of the locally asymptotically normal (LAN) result previously established. Using the Karhunen–Loève expansion of the limiting process of the log-likelihood ratio process, the asymptotic null distribution and the power of the test statistic are accurately approximated. These results are applied to change-point analysis. An empirical study is done for evaluating the performance of the methodology proposed. PubDate: 2023-08-23 DOI: 10.1007/s11203-023-09295-x

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Abstract: Abstract In this paper, we consider an inference problem in generalized exponential Ornstein–Uhlenbeck processes. Salient features of this paper consists in the fact that, first, we generalized the classical exponential Ornstein–Uhlenbeck processes to the case where the drift coefficient is driven by a period function of time. Second, as opposed to the results in recent literature, the dimension of the drift parameter is considered as unknown. Third, we weaken some assumptions, in recent literature, underlying the asymptotic optimality of some estimators of the drift parameter. We propose the unrestricted maximum likelihood estimator, the restricted maximum likelihood estimator and some shrinkage estimators for the drift parameters. We also derive asymptotic distributional risk of the proposed estimators as well as their relative efficiency. Finally, we present the simulation results which corroborate the theoretical findings. PubDate: 2023-08-12 DOI: 10.1007/s11203-023-09291-1

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Abstract: Abstract We study the asymptotic behaviour of a stationnary shot noise random field. We use the notion of association to prove the asymptotic normality of the moments and a multidimensional version for the correlation functions. The variance of the moment estimates is detailed as well as their correlation. When the field is isotropic, the estimators are improved by reducing the variance. These results will be applied to the estimation of the model parameters in the case of a Gaussian kernel, with a focus on the correlation parameter. The asymptotic normality is proved and a simulation study is carried out. PubDate: 2023-07-07 DOI: 10.1007/s11203-023-09294-y

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Abstract: Abstract In this paper we consider continuous-time hidden Markov processes (CTHMM). The model considered is a two-dimensional stochastic process \((X_t,Y_t)\) , with \(X_t\) an unobserved (hidden) Markov chain defined by its generating matrix and \(Y_t\) an observed process whose distribution law depends on \(X_t\) and is called the emission function. In general, we allow the process \(Y_t\) to take values in a subset of the q-dimensional real space, for some q. The coupled process \((X_t,Y_t)\) is a continuous-time Markov chain whose generator is constructed from the generating matrix of X and the emission distribution. We study the theoretical properties of this two-dimensional process using a formulation based on semi-Markov processes. Observations of the CTHMM are obtained by discretization considering two different scenarii. In the first case we consider that observations of the process Y are registered regularly in time, while in the second one, observations arrive at random. Maximum-likelihood estimators of the characteristics of the coupled process are obtained in both scenarii and the asymptotic properties of these estimators are shown, such as consistency and normality. To illustrate the model a real-data example and a simulation study are considered. PubDate: 2023-06-23 DOI: 10.1007/s11203-023-09292-0

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Abstract: Abstract This paper deals with inference in a class of stable but nearly-unstable processes. Autoregressive processes are considered, in which the bridge between stability and instability is expressed by a time-varying companion matrix \(A_{n}\) with spectral radius \(\rho (A_{n}) < 1\) satisfying \(\rho (A_{n}) \rightarrow 1\) . This framework is particularly suitable to understand unit root issues by focusing on the inner boundary of the unit circle. Consistency is established for the empirical covariance and the OLS estimation together with asymptotic normality under appropriate hypotheses when A, the limit of \(A_n\) , has a real spectrum, and a particular case is deduced when A also contains complex eigenvalues. The asymptotic process is integrated with either one unit root (located at 1 or \(-1\) ), or even two unit roots located at 1 and \(-1\) . Finally, a set of simulations illustrate the asymptotic behavior of the OLS. The results are essentially proved by \(L^2\) computations and the limit theory of triangular arrays of martingales. PubDate: 2023-06-02 DOI: 10.1007/s11203-023-09290-2

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Abstract: Abstract In this paper, we propose a new type of univariate and bivariate Integer-valued autoregressive model of order one (INAR(1)) to approximate univariate and bivariate linear birth and death process with constant rates. Under a specific parametric setting, the dynamic of transition probabilities and probability generating function of INAR(1) will converge to that of birth and death process as the length of subintervals goes to 0. Due to the simplicity of Markov structure, maximum likelihood estimation is feasible for INAR(1) model, which is not the case for bivariate and multivariate birth and death process. This means that the statistical inference of bivariate birth and death process can be achieved via the maximum likelihood estimation of a bivariate INAR(1) model. PubDate: 2023-05-15 DOI: 10.1007/s11203-023-09289-9

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Abstract: Abstract In this paper, we prove a result of equivalence in law between a diffusion conditioned with respect to partial observations and an auxiliary process. By partial observations we mean coordinates (or linear transformation) of the process at a finite collection of deterministic times. Apart from the theoretical interest, this result allows to simulate the conditional diffusion through Monte Carlo methods, using the fact that the auxiliary process is easy to simulate. PubDate: 2023-05-11 DOI: 10.1007/s11203-023-09287-x