Statistical Inference for Stochastic Processes
Journal Prestige (SJR): 0.322 Citation Impact (citeScore): 1 Number of Followers: 3 Hybrid journal (It can contain Open Access articles) ISSN (Print) 15729311  ISSN (Online) 13870874 Published by SpringerVerlag [2467 journals] 
 Testing the equality of the laws of two strictly stationary processes

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Abstract: Abstract 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 ddimensional 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 twostep datadriven 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 reallife applications also illustrate this approach.
PubDate: 20230401

 A functional central limit theorem on nonstationary random fields with
nested spatial structure
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Abstract: Abstract In this paper, we establish a functional central limit theorem on high dimensional random fields in the context of modelbased survey analysis. For stronglymixing nonstationary random fields, we provide an upper bound for the fourth moment of the finite population total. This inequality is the generalization of a key tool for proving functional central limit theorems in Rio (Asymptotic theory of weakly dependent random processes, Springer, Berlin, 2017). Under the nested sampling strategy, we introduce assumptions on stronglymixing coefficients and quantile functions to show that a functional stochastic process asymptotically approaches to a Gaussian process.
PubDate: 20230401

 Sparse estimation for generalized exponential marked Hawkes process

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Abstract: Abstract We established a sparse estimation method for the generalized exponential marked Hawkes process by the penalized method to ordinary method (P–O) estimator. Furthermore, we evaluated the probability of the correct variable selection. In the course of this, we established a framework for a likelihood analysis and the P–O estimation when there might be nuisance parameters, and the true value of the parameter might be at the boundary of the parameter space. Finally, numerical simulations are given for several important examples.
PubDate: 20230401

 Threshold estimation for jumpdiffusions under small noise asymptotics

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Abstract: 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 thresholdtype quasilikelihood 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: 20230220

 A portmanteautype test for detecting serial correlation in locally
stationary functional time series
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Abstract: 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: 20230117

 On consistency for time series model selection

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Abstract: 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: 20221227
DOI: 10.1007/s11203022092846

 Highdimensional estimation of quadratic variation based on penalized
realized variance
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Abstract: Abstract In this paper, we develop a penalized realized variance (PRV) estimator of the quadratic variation (QV) of a highdimensional continuous Itô semimartingale. We adapt the principle idea of regularization from linear regression to covariance estimation in a continuoustime highfrequency setting. We show that under a nuclear norm penalization, the PRV is computed by softthresholding 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 nonnegligible eigenvalues of the QV. Moreover, we also provide the associated nonasymptotic analysis for the spot variance. We suggest an intuitive datadriven 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 crosssection of stock returns are perturbed by idiosyncratic errors, rendering the QV—and also RV—of full rank.
PubDate: 20221205
DOI: 10.1007/s11203022092828

 On the $$\alpha $$ lazy version of Markov chains in estimation and
testing problems
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Abstract: 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: 20221205
DOI: 10.1007/s11203022092837

 On the integrated mean squared error of wavelet density estimation for
linear processes
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Abstract: 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 chisquared distributions.
PubDate: 20221117
DOI: 10.1007/s11203022092819

 Large deviation inequalities of Bayesian estimator in nonlinear regression
models
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Abstract: 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: 20221029
DOI: 10.1007/s1120302209280w

 Finitesample properties of estimators for first and second order
autoregressive processes
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Abstract: 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 finitesample properties of wellknown estimators for the coefficients of stationary AR(1) and AR(2) processes and provides biascorrected 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 finitesample distributions of the new biascorrected estimators are approximated using transformations of skewnormal 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 Rpackage for AR(1) and AR(2) processes of length 10–50, both giving biascorrected coefficient estimates and corresponding confidence intervals.
PubDate: 20221001
DOI: 10.1007/s11203021092624

 Improved estimation method for high dimension semimartingale regression
models based on discrete data
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Abstract: 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 nonasymptotic 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. Nonasymptotic 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 nonGaussian Ornstein–Uhlenbeck processes is developed. MonteCarlo simulations for the numeric confirmation of the obtained theoretical results are given.
PubDate: 20221001
DOI: 10.1007/s11203021092580

 On minimax robust testing of composite hypotheses on Poisson process
intensity
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Abstract: 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 “LikelihoodRatio” 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 1st kind error probability \(\alpha \) is fixed and we are interested in the minimal possible 2nd 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: 20221001
DOI: 10.1007/s11203021092651

 Randomized consistent statistical inference for random processes and
fields
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Abstract: 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: 20221001
DOI: 10.1007/s1120302209270y

 Weak convergence of nonparametric estimators of the multidimensional and
multidimensionalmultivariate renewal functions on Skorohod topology
spaces
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Abstract: Abstract This paper deals with the weak convergence of nonparametric estimators of the multidimensional and multidimensionalmultivariate renewal functions on Skorohod topology spaces. It is an extension of Harel et al. (J Math Anal Appl 189:240–255, 1995) from the onedimensional case to the multivariate and multidimensional case. The estimators are based on a sequence of nonnegative independent and identically distributed (iid) random vectors. They are expressed as infinite sums of kfolds convolutions of the empirical distribution function. Their weak convergence study heavily rests on that of the empirical distribution function.
PubDate: 20221001
DOI: 10.1007/s11203021092633

 Wavelet eigenvalue regression in high dimensions

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Abstract: 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 nonGaussian, finitevariance pvariate measurements are made of a lowdimensional rvariate ( \(r \ll p\) ) fractional stochastic process with noncanonical scaling coordinates and in the presence of additive highdimensional noise. The measurements are correlated both timewise 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: 20220918
DOI: 10.1007/s11203022092793

 Robust and efficient specification tests in Markovswitching
autoregressive models
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Abstract: Abstract This study develops two types of robust test statistics applicable to Markovswitching 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 finitesample size and power properties. The tests are applied to investigate the fluctuations in real GNP growth in the U.S.
PubDate: 20220830
DOI: 10.1007/s11203022092775

 On Stein’s lemma in hypotheses testing in general nonasymptotic
case
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Abstract: Abstract The problem of testing two simple hypotheses in a general probability space is considered. For a fixed typeI error probability, the best exponential decay rate of the typeII 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 nonasymptotic 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: 20220824
DOI: 10.1007/s11203022092784

 Weakconvergence of empirical conditional processes and conditional
Uprocesses involving functional mixing data
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Abstract: Abstract Ustatistics represent a fundamental class of statistics arising from modeling quantities of interest defined by multisubject responses. Ustatistics generalize the empirical mean of a random variable X to sums over every mtuple of distinct observations of X. W. Stute [Ann. Probab. 19 (1991) 812–825] introduced a class of socalled conditional Ustatistics, which may be viewed as a generalization of the NadarayaWatson 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 Uprocesses 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 Uprocesses 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 VapnikChervonenkis 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: 20220725
DOI: 10.1007/s11203022092766

 A chisquare type test for timeinvariant fiber pathways of the brain

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Abstract: 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 chisquare 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: 20220411
DOI: 10.1007/s11203022092686
