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Abstract: We consider parameter estimation of stochastic differential equations driven by a Wiener process and a compound Poisson process as small noises. The goal is to give a threshold-type quasi-likelihood estimator and show its consistency and asymptotic normality under new asymptotics. One of the novelties of the paper is that we give a new localization argument, which enables us to avoid truncation in the contrast function that has been used in earlier works and to deal with a wider class of jumps in threshold estimation than ever before. PubDate: 2023-02-20

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Abstract: The portmanteau test provides the vanilla method for detecting serial correlations in classical univariate time series analysis. The method is extended to the case of observations from a locally stationary functional time series. Asymptotic critical values are obtained by a suitable block multiplier bootstrap procedure. The test is shown to asymptotically hold its level and to be consistent against general alternatives. PubDate: 2023-01-17

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Abstract: 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. PubDate: 2022-12-27 DOI: 10.1007/s11203-022-09284-6

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Abstract: In this paper, we develop a penalized realized variance (PRV) estimator of the quadratic variation (QV) of a high-dimensional continuous Itô semimartingale. We adapt the principle idea of regularization from linear regression to covariance estimation in a continuous-time high-frequency setting. We show that under a nuclear norm penalization, the PRV is computed by soft-thresholding the eigenvalues of realized variance (RV). It therefore encourages sparsity of singular values or, equivalently, low rank of the solution. We prove our estimator is minimax optimal up to a logarithmic factor. We derive a concentration inequality, which reveals that the rank of PRV is—with a high probability—the number of non-negligible eigenvalues of the QV. Moreover, we also provide the associated non-asymptotic analysis for the spot variance. We suggest an intuitive data-driven subsampling procedure to select the shrinkage parameter. Our theory is supplemented by a simulation study and an empirical application. The PRV detects about three–five factors in the equity market, with a notable rank decrease during times of distress in financial markets. This is consistent with most standard asset pricing models, where a limited amount of systematic factors driving the cross-section of stock returns are perturbed by idiosyncratic errors, rendering the QV—and also RV—of full rank. PubDate: 2022-12-05 DOI: 10.1007/s11203-022-09282-8

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Abstract: Given access to a single long trajectory generated by an unknown irreducible Markov chain M, we simulate an \(\alpha \) -lazy version of M which is ergodic. This enables us to generalize recent results on estimation and identity testing, that were stated for ergodic Markov chains, in a way that allows fully empirical inference. In particular, our approach shows that the pseudo spectral gap introduced by Paulin (Electron J Probab 20:32, 2015) and defined for ergodic Markov chains may be given a meaning already in the case of irreducible but possibly periodic Markov chains. PubDate: 2022-12-05 DOI: 10.1007/s11203-022-09283-7

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Abstract: Let \(\{X_n: n\in {{\mathbb {N}}}\}\) be a linear process with density function \(f(x)\in L^2({{\mathbb {R}}})\) . We study wavelet density estimation of f(x). Under some regular conditions on the characteristic function of innovations, we achieve, based on the number of nonzero coefficients in the linear process, the minimax optimal convergence rate of the integrated mean squared error of density estimation. Considered wavelets have compact support and are twice continuously differentiable. The number of vanishing moments of mother wavelet is proportional to the number of nonzero coefficients in the linear process and to the rate of decay of characteristic function of innovations. Theoretical results are illustrated by simulation studies with innovations following Gaussian, Cauchy and chi-squared distributions. PubDate: 2022-11-17 DOI: 10.1007/s11203-022-09281-9

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Abstract: In the present paper, we establish some large deviation inequalities of the Bayesian estimator for the nonlinear regression model under the conditions of dependent errors which extend the results in Jeganathan (J Multivar Anal 30(2):227–240, 1989) from independent errors and dependent sequences. As an application, we give an large deviation inequality for the Michaelis–Menten model. PubDate: 2022-10-29 DOI: 10.1007/s11203-022-09280-w

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Abstract: The class of autoregressive (AR) processes is extensively used to model temporal dependence in observed time series. Such models are easily available and routinely fitted using freely available statistical software like R. A potential problem is that commonly applied estimators for the coefficients of AR processes are severely biased when the time series are short. This paper studies the finite-sample properties of well-known estimators for the coefficients of stationary AR(1) and AR(2) processes and provides bias-corrected versions of these estimators which are quick and easy to apply. The new estimators are constructed by modeling the relationship between the true and originally estimated AR coefficients using weighted orthogonal polynomial regression, taking the sampling distribution of the original estimators into account. The finite-sample distributions of the new bias-corrected estimators are approximated using transformations of skew-normal densities, combined with a Gaussian copula approximation in the AR(2) case. The properties of the new estimators are demonstrated by simulations and in the analysis of a real ecological data set. The estimators are easily available in our accompanying R-package for AR(1) and AR(2) processes of length 10–50, both giving bias-corrected coefficient estimates and corresponding confidence intervals. PubDate: 2022-10-01 DOI: 10.1007/s11203-021-09262-4

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Abstract: 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. PubDate: 2022-10-01 DOI: 10.1007/s11203-021-09258-0

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Abstract: The problem on the minimax testing of a Poisson process intensity is considered. For a given disjoint sets \({{\mathcal {S}}}_T\) and \({{\mathcal {V}}}_T\) of possible intensities \({{\mathbf {s}}}_{T}\) and \({{\mathbf {v}}}_{T}\) , respectively, the minimax testing of the composite hypothesis \(H_{0}: {{\mathbf {s}}_T} \in {{\mathcal {S}}}_T\) against the composite alternative \(H_{1}: {{\mathbf {v}}_T} \in {{\mathcal {V}}}_T\) is investigated. It is assumed that a pair of intensities \({{\mathbf {s}}_T^{0}} \in {{\mathcal {S}}}_T\) and \({{\mathbf {v}}_T^{0}} \in {{\mathcal {V}}}_T\) are chosen, and the “Likelihood-Ratio” test for intensities \({{\mathbf {s}}_T^{0}}\) and \({{\mathbf {v}}_T^{0}}\) is used for testing composite hypotheses \(H_{0}\) and \(H_{1}\) . The case, when the 1-st kind error probability \(\alpha \) is fixed and we are interested in the minimal possible 2-nd kind error probability \(\beta ({{\mathcal {S}}}_T,{{\mathcal {V}}}_T)\) , is considered. What are the maximal sets \({{\mathcal {S}}}({{\mathbf {s}}}_{T}^{0},{{\mathbf {v}}}_{T}^{0})\) and \({{\mathcal {V}}}({{\mathbf {s}}}_{T}^{0},{{\mathbf {v}}}_{T}^{0})\) , which can be replaced by the pair of intensities \(({{\mathbf {s}}_T^{0}},{{\mathbf {v}}_T^{0}})\) without essential loss for testing performance ' In the asymptotic case ( \(T\rightarrow \infty \) ) those maximal sets \({{\mathcal {S}}}({{\mathbf {s}}}_{T}^{0},{{\mathbf {v}}}_{T}^{0})\) and \({{\mathcal {V}}}({{\mathbf {s}}}_{T}^{0},{{\mathbf {v}}}_{T}^{0})\) are described. PubDate: 2022-10-01 DOI: 10.1007/s11203-021-09265-1

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Abstract: We propose a randomized approach to the consistent statistical analysis of random processes and fields on \({\mathbb {R}}^m\) and \({\mathbb {Z}}^m, m=1,2,...\) , which is valid in the case of strong dependence: the parameter of interest \(\theta \) only has to possesses a consistent sequence of estimators \({\hat{\theta }}_n\) . The limit theorem is related to consistent sequences of randomized estimators \({\hat{\theta }}_n^*\) ; it is used to construct consistent asymptotically efficient sequences of confidence intervals and tests of hypotheses related to the parameter \(\theta \) . Upper bounds for “admissible” sequences of normalizing coefficients in the limit theorem are established for some statistical models in Part 2. PubDate: 2022-10-01 DOI: 10.1007/s11203-022-09270-y

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Abstract: 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. PubDate: 2022-10-01 DOI: 10.1007/s11203-021-09263-3

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Abstract: The problem of linear interpolation in the context of a multivariate time series having multiple (possibly non-consecutive) missing values is studied. A concise formula for the optimal interpolating filter is derived, and illustrations using two simple models are provided. PubDate: 2022-10-01 DOI: 10.1007/s11203-022-09269-5

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Abstract: We suppose that a Lévy process is observed at discrete time points. Starting from an asymptotically minimax family of estimators for the continuous part of the Lévy Khinchine characteristics, i.e., the covariance, we derive a data-driven parameter choice for the frequency of estimating the covariance. We investigate a Lepskiĭ-type stopping rule for the adaptive procedure. Consequently, we use a balancing principle for the best possible data-driven parameter. The adaptive estimator achieves almost the optimal rate. Numerical experiments with the proposed selection rule are also presented. PubDate: 2022-10-01 DOI: 10.1007/s11203-021-09264-2

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Abstract: 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. PubDate: 2022-09-18 DOI: 10.1007/s11203-022-09279-3

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Abstract: This study develops two types of robust test statistics applicable to Markov-switching autoregressive models. The test statistics can be constructed by sum functionals of the “smoothed” probabilities that a given observation came from a particular regime and do not require the estimation of additional parameters. Monte Carlo experiments show that the tests have good finite-sample size and power properties. The tests are applied to investigate the fluctuations in real GNP growth in the U.S. PubDate: 2022-08-30 DOI: 10.1007/s11203-022-09277-5

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Abstract: The problem of testing two simple hypotheses in a general probability space is considered. For a fixed type-I error probability, the best exponential decay rate of the type-II error probability is investigated. In regular asymptotic cases (i.e., when the length of the observation interval grows without limit) the best decay rate is given by Stein’s exponent. In the paper, for a general probability space, some non-asymptotic lower and upper bounds for the best rate are derived. These bounds represent pure analytic relations without any limiting operations. In some natural cases, these bounds also give the convergence rate for Stein’s exponent. Some illustrating examples are also provided. PubDate: 2022-08-24 DOI: 10.1007/s11203-022-09278-4

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Abstract: 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. PubDate: 2022-07-25 DOI: 10.1007/s11203-022-09276-6

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Abstract: This paper concerns the estimation of stationary probability of ergodic semi-Markov chains based on an observation over a time interval. We derive asymptotic properties of the proposed estimator, when the time of observation goes to infinity, as consistency, asymptotic normality, law of iterated logarithm and rate of convergence in a functional setting. The proofs are based on asymptotic results on discrete-time semi-Markov random evolutions. PubDate: 2022-07-01 DOI: 10.1007/s11203-021-09255-3

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Abstract: A longitudinal diffusion tensor imaging (DTI) study on a single brain can be remarkably useful to probe white matter fiber connectivity that may or may not be stable over time. We consider a novel testing problem where the null hypothesis states that the trajectories of a coherently oriented fiber population remain the same over a fixed period of time. Compared to other applications that use changes in DTI scalar metrics over time, our test is focused on the partial derivative of the continuous ensemble of fiber trajectories with respect to time. The test statistic is shown to have the limiting chi-square distribution under the null hypothesis. The power of the test is demonstrated using Monte Carlo simulations based on both the theoretical and empirical critical values. The proposed method is applied to a longitudinal DTI study of a normal brain. PubDate: 2022-04-11 DOI: 10.1007/s11203-022-09268-6