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- Projection-based white noise and goodness-of-fit tests for functional time
series-
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Abstract: Abstract We develop two significance tests in the setting of functional time series. The null hypothesis of the first test is that the observed data are sampled from a general weak white noise sequence. The null hypothesis of the second test is that the observed data are sampled from a functional autoregressive model of order one (FAR(1)), which can be used as a goodness-of-fit test. Both tests are based on projections on functional principal components. Such projections are used in a great many functional data analysis (FDA) procedures, so our tests are practically relevant. We derive test statistics for each test that are quadratic forms of lagged autocovariance estimates constructed from principal component projections, and establish the requisite asymptotic theory. A simulation study shows that the tests have complimentary advantages against relevant benchmarks.
PubDate: 2024-07-24
- Quasi-maximum likelihood estimation of long-memory linear processes
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Abstract: Abstract 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.
PubDate: 2024-07-13
- Viking: variational Bayesian variance tracking
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Abstract: Abstract 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.
PubDate: 2024-05-30
DOI: 10.1007/s11203-024-09312-7
- Estimation of several parameters in discretely-observed stochastic
differential equations with additive fractional noise-
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Abstract: Abstract 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.
PubDate: 2024-05-29
DOI: 10.1007/s11203-024-09311-8
- Nonparametric estimation of the diffusion coefficient from i.i.d. S.D.E.
paths-
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Abstract: Abstract 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.
PubDate: 2024-04-29
DOI: 10.1007/s11203-024-09310-9
- Nonparametric spectral density estimation under local differential privacy
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Abstract: Abstract We consider nonparametric estimation of the spectral density of a stationary time series under local differential privacy. In this framework only an appropriately anonymised version of a finite snippet from the time series can be observed and used for inference. The anonymisation procedure can be chosen in advance among all mechanisms satisfying the condition of local differential privacy, and we propose truncation followed by Laplace perturbation for this purpose. Afterwards, the anonymised time series snippet is used to define a surrogate of the classical periodogram and our estimator is obtained by projection of this privatised periodogram to a model given by a finite dimensional subspace of \(L^2([-\pi ,\pi ])\) . This estimator attains nearly the same convergence rate as in the case where the original time series can be observed. However, a reduction of the effective sample size in contrast to the non-privacy framework is shown to be unavoidable. We also consider adaptive estimation and suggest to select an estimator from a set of candidate estimators by means of a penalised contrast criterion. We derive an oracle inequality which shows that the adaptive estimator attains nearly the same rate of convergence as the best estimator from the candidate set. Concentration inequalities for quadratic forms in terms of sub-exponential random variables, which have been recently derived in Götze et al. (Electron J Probab 26:1–22, 2021), turn out to be essential for our proof. Finally, we illustrate our findings in a small simulation study.
PubDate: 2024-03-25
DOI: 10.1007/s11203-024-09308-3
- A pseudo-likelihood estimator of the Ornstein–Uhlenbeck parameters
from suprema observations-
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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
DOI: 10.1007/s11203-024-09307-4
- On a calculable Skorokhod’s integral based projection estimator of the
drift function in fractional SDE-
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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
DOI: 10.1007/s11203-024-09306-5
- The distribution of the maximum likelihood estimates of the change point
and their relation to random walks-
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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
DOI: 10.1007/s11203-023-09304-z
- Weak convergence of the conditional U-statistics for locally stationary
functional time series-
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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
DOI: 10.1007/s11203-023-09305-y
- Statistical inference for discretely sampled stochastic functional
differential equations with small noise-
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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
- Nonparametric estimation for random effects models driven by fractional
Brownian motion using Hermite polynomials-
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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
- Statistical estimation and nonlinear filtering in environmental pollution
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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
- Parameter estimation for a linear parabolic SPDE model in two space
dimensions with a small noise-
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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
- Asymptotically efficient estimation of Ergodic rough fractional
Ornstein-Uhlenbeck process under continuous observations-
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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
- Asymptotic expansion of an estimator for the Hurst coefficient
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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
- Second-order robustness for time series inference
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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
- Localization of two radioactive sources on the plane
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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
- Inference in generalized exponential O–U processes with change-point
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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
- A Cramér–von Mises test for a class of mean time dependent CHARN models
with application to change-point detection-
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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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