This paper revisits the state and parameter estimation problems for a system of partially observed linear stochastic differential equations. An asymptotically optimal adaptive filter of the hidden state process is constructed using a three stage procedure. First, the unknown parameter is estimated by means of the method of moments. Then this preliminary estimator is used to define the One-step MLE process by applying the scoring technique, and, finally, the improved estimator is plugged into Kalman-Bucy filter. The obtained parameter estimator and the adaptive filter are proved to be asymptotically efficient in the long-time regime.

The purpose of this paper is to study the convergence of the quasi-maximum likelihood (QML) estimator for long memory linear processes. We first establish a correspondence between the long-memory linear process representation and the long-memory AR $$(\infty )$$ process representation. We then establish the almost sure consistency and asymptotic normality of the QML estimator. Numerical simulations illustrate the theoretical results and confirm the good performance of the estimator.

We consider the problem of robust and adaptive time series forecasting in an uncertain environment. We focus on the inference in state-space models under unknown time-varying noise variances and potential misspecification (violation of the state-space data generation assumption). We introduce an augmented model in which the variances are represented by auxiliary Gaussian latent variables in a tracking mode. The inference relies on the online variational Bayesian methodology, which minimizes a Kullback–Leibler divergence at each time step. We observe that optimizing the Kullback–Leibler divergence leads to an extension of the Kalman filter. We design a novel algorithm named Viking, using second-order bounds for the auxiliary latent variables, whose minima admit closed-form solutions. The main step of Viking does not coincide with the standard Kalman filter when the variances of the state-space model are uncertain. Experiments on synthetic and real data show that Viking behaves well and is robust to misspecification.

We investigate the problem of joint statistical estimation of several parameters for a stochastic differential equation driven by an additive fractional Brownian motion. Based on discrete-time observations of the model, we construct an estimator of the Hurst parameter, the diffusion parameter and the drift, which lies in a parametrised family of coercive drift coefficients. Our procedure is based on the assumption that the stationary distribution of the SDE and of its increments permits to identify the parameters of the model. Under this assumption, we prove consistency results and derive a rate of convergence for the estimator. Finally, we show that the identifiability assumption is satisfied in the case of a family of fractional Ornstein–Uhlenbeck processes and illustrate our results with some numerical experiments.

Consider a diffusion process $$X=(X_t)_{t\in [0,1]}$$ observed at discrete times and high frequency, solution of a stochastic differential equation whose drift and diffusion coefficients are assumed to be unknown. In this article, we focus on the nonparametric estimation of the diffusion coefficient. We propose ridge estimators of the square of the diffusion coefficient from discrete observations of X that are obtained by minimization of the least squares contrast. We prove that the estimators are consistent and derive rates of convergence as the number of observations tends to infinity. Two observation schemes are considered in this paper. The first scheme consists in one diffusion path observed at discrete times, where the discretization step of the time interval [0, 1] tends to zero. The second scheme consists in repeated observations of the diffusion process X, where the number of the observed paths tends to infinity. The theoretical results are completed with a numerical study over synthetic data.

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.

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 $${\widehat{b}}_m$$ .

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.

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.

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.

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.

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.

AbstractIn this paper we consider continuous-time hidden Markov processes (CTHMM). The model considered is a two-dimensional stochastic process $$(X_t,Y_t)$$ ( X t , Y t ) , with $$X_t$$ X t an unobserved (hidden) Markov chain defined by its generating matrix and $$Y_t$$ Y t an observed process whose distribution law depends on $$X_t$$ X t and is called the emission function. In general, we allow the process $$Y_t$$ Y t to take values in a subset of the q-dimensional real space, for some q. The coupled process $$(X_t,Y_t)$$ ( 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.

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.

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.

We consider the model selection problem for a large class of time series models, including, multivariate count processes, causal processes with exogenous covariates. A procedure based on a general penalized contrast is proposed. Some asymptotic results for weak and strong consistency are established. The non consistency issue is addressed, and a class of penalty term, that does not ensure consistency is provided. Examples of continuous valued and multivariate count autoregressive time series are considered.

In this paper, we construct the wavelet eigenvalue regression methodology (Abry and Didier in J Multivar Anal 168:75–104, 2018a; in Bernoulli 24(2):895–928, 2018b) in high dimensions. We assume that possibly non-Gaussian, finite-variance p-variate measurements are made of a low-dimensional r-variate ( $$r \ll p$$ ) fractional stochastic process with non-canonical scaling coordinates and in the presence of additive high-dimensional noise. The measurements are correlated both time-wise and between rows. Building upon the asymptotic and large scale properties of wavelet random matrices in high dimensions, the wavelet eigenvalue regression is shown to be consistent and, under additional assumptions, asymptotically Gaussian in the estimation of the fractal structure of the system. We further construct a consistent estimator of the effective dimension r of the system that significantly increases the robustness of the methodology. The estimation performance over finite samples is studied by means of simulations.

U-statistics represent a fundamental class of statistics arising from modeling quantities of interest defined by multi-subject responses. U-statistics generalize the empirical mean of a random variable X to sums over every m-tuple of distinct observations of X. W. Stute [Ann. Probab. 19 (1991) 812–825] introduced a class of so-called conditional U-statistics, which may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to : $$\begin{aligned} m(\mathbf { t}):=\mathbb {E}[\varphi (Y_{1},\ldots ,Y_{m})|(X_{1},\ldots ,X_{m})=\mathbf {t}], ~~\text{ for }~~\mathbf { t}\in \mathcal {X}^{m}. \end{aligned}$$ In this paper we are mainly interested in establishing weak convergence of conditional U-processes in a functional mixing data framework. More precisely, we investigate the weak convergence of the conditional empirical process indexed by a suitable class of functions and of conditional U-processes when the explicative variable is functional. We treat the weak convergence in both cases when the class of functions is bounded or unbounded satisfying some moment conditions. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis classes of functions and some mild conditions on the model. The theoretical results established in this paper are (or will be) key tools for many further developments in functional data analysis.

In this paper we consider the problem of comparison of two strictly stationary processes. The novelty of our approach is that we consider all their d-dimensional joint distributions, for $$d\geqslant 1$$ . Our procedure consists in expanding their densities in a multivariate orthogonal basis and comparing their k first coefficients. The dimension d to consider and the number k of coefficients to compare in view of performing the test can growth with the sample size and are automatically selected by a two-step data-driven procedure. The method works for possibly paired, short or long range dependent processes. A simulation study shows the good behavior of the test procedure. In particular, we apply our method to compare ARFIMA processes. Some real-life applications also illustrate this approach.

This paper deals with the weak convergence of nonparametric estimators of the multidimensional and multidimensional-multivariate renewal functions on Skorohod topology spaces. It is an extension of Harel et al. (J Math Anal Appl 189:240–255, 1995) from the one-dimensional case to the multivariate and multidimensional case. The estimators are based on a sequence of non-negative independent and identically distributed (iid) random vectors. They are expressed as infinite sums of k-folds convolutions of the empirical distribution function. Their weak convergence study heavily rests on that of the empirical distribution function.

We consider the problem of the estimation of the position on the plane and of the moment of beginning of emission of a source by observations from K detectors on the plane. We propose the conditions of regularity and identifiability, which allow us to use two general theorems by Ibragimov and Khasminskii and to describe the asymptotic behavior of the maximum likelihood and Bayes estimators. Then we propose the construction of a linear estimator of unknown parameters and study its properties in slightly more general situation. Special attention is payed to condition of identifiability.

In this paper we study a high dimension (Big Data) regression model in continuous time observed in the discrete time moments with dependent noises defined by semimartingale processes. To this end an improved (shrinkage) estimation method is developed and the non-asymptotic comparison between shrinkage and least squares estimates is studied. The improvement effect for the shrinkage estimates showing the significant advantage with respect to the "small" dimension case is established. It turns out that obtained improvement effect holds true uniformly over observation frequency. Then, a model selection method based on these estimates is developed. Non-asymptotic sharp oracle inequalities for the constructed model selection procedure are obtained. Constructive sufficient conditions for the observation frequency providing the robust efficiency property in adaptive setting without using any sparsity assumption are found. A special stochastic calculus tool to guarantee these conditions for non-Gaussian Ornstein–Uhlenbeck processes is developed. Monte-Carlo simulations for the numeric confirmation of the obtained theoretical results are given.