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 [2469 journals] 
 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

 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

 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

 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

 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

 Optimal linear interpolation of multiple missing values

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Abstract: Abstract The problem of linear interpolation in the context of a multivariate time series having multiple (possibly nonconsecutive) missing values is studied. A concise formula for the optimal interpolating filter is derived, and illustrations using two simple models are provided.
PubDate: 20221001

 A Lepskiĭtype stopping rule for the covariance estimation of
multidimensional Lévy processes
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Abstract: 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 datadriven parameter choice for the frequency of estimating the covariance. We investigate a Lepskiĭtype stopping rule for the adaptive procedure. Consequently, we use a balancing principle for the best possible datadriven parameter. The adaptive estimator achieves almost the optimal rate. Numerical experiments with the proposed selection rule are also presented.
PubDate: 20221001

 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

 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

 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

 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

 Estimation of stationary probability of semiMarkov Chains

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Abstract: Abstract This paper concerns the estimation of stationary probability of ergodic semiMarkov 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 discretetime semiMarkov random evolutions.
PubDate: 20220701
DOI: 10.1007/s11203021092553

 Martingale estimation functions for Bessel processes

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Abstract: Abstract In this paper we derive martingale estimating functions for the dimensionality parameter of a Bessel process based on the eigenfunctions of the diffusion operator. Since a Bessel process is nonergodic and the theory of martingale estimating functions is developed for ergodic diffusions, we use the spacetime transformation of the Bessel process and formulate our results for a modified Bessel process. We deduce consistency, asymptotic normality and discuss optimality. It turns out that the martingale estimating function based of the first eigenfunction of the modified Bessel process coincides with the linear martingale estimating function for the Cox Ingersoll Ross process. Furthermore, our results may also be applied to estimating the multiplicity parameter of a onedimensional Dunkl process and some related polynomial processes.
PubDate: 20220701
DOI: 10.1007/s11203021092508

 Detection and identification of changes of hidden Markov chains:
asymptotic theory
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Abstract: Abstract This paper revisits a unified framework of sequential changepoint detection and hypothesis testing modeled using hidden Markov chains and develops its asymptotic theory. Given a sequence of observations whose distributions are dependent on a hidden Markov chain, the objective is to quickly detect critical events, modeled by the first time the Markov chain leaves a specific set of states, and to accurately identify the class of states that the Markov chain enters. We propose computationally tractable sequential detection and identification strategies and obtain sufficient conditions for the asymptotic optimality in two Bayesian formulations. Numerical examples are provided to confirm the asymptotic optimality.
PubDate: 20220701
DOI: 10.1007/s11203021092535

 Adaptive tests for parameter changes in ergodic diffusion processes from
discrete observations
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Abstract: Abstract We consider the adaptive test for the parameter change in discretely observed ergodic diffusion processes based on the cusum test. Using two test statistics based on the two quasilog likelihood functions of the diffusion parameter and the drift parameter, we perform the change point tests for both diffusion and drift parameters of the diffusion process. It is shown that the test statistics have the limiting distribution of the sup of the norm of a Brownian bridge. Simulation results are illustrated for the 1dimensional OrnsteinUhlenbeck process.
PubDate: 20220701
DOI: 10.1007/s11203021092491

 Contrast estimation for noisy observations of diffusion processes via
closedform density expansions
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Abstract: Abstract When a continuoustime diffusion is observed only at discrete times with measurement noise, in most cases the transition density is not known and the likelihood is in the form of a highdimensional integral that does not have a closedform solution and is difficult to compute accurately. Using Hermite expansions and deconvolution strategy, we provide a general explicit sequence of closedform contrast for noisy and discretely observed diffusion processes. This work allows the estimation of many diffusion processes. We show that the approximation is very accurate and prove that minimizing the sequence results in a consistent and asymptotically normal estimator. Monte Carlo evidence for the Ornstein–Uhlenbeck process reveals that this method works well and outperforms the Euler expansion of the transition density in situations relevant for financial models.
PubDate: 20220701
DOI: 10.1007/s11203021092562

 Quasilikelihood analysis for marked point processes and application to
marked Hawkes processes
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Abstract: Abstract We develop a quasilikelihood analysis procedure for a general class of multivariate marked point processes. As a byproduct of the general method, we establish under stability and ergodicity conditions the local asymptotic normality of the quasilog likelihood, along with the convergence of moments of quasilikelihood and quasiBayesian estimators. To illustrate the general approach, we then turn our attention to a class of multivariate marked Hawkes processes with generalized exponential kernels, comprising among others the socalled Erlang kernels. We provide explicit conditions on the kernel functions and the mark dynamics under which a certain transformation of the original process is Markovian and Vgeometrically ergodic. We finally prove that the latter result, which is of interest in its own right, constitutes the key ingredient to show that the generalized exponential Hawkes process falls under the scope of application of the quasilikelihood analysis.
PubDate: 20220701
DOI: 10.1007/s11203021092517

 Likelihood theory for the graph OrnsteinUhlenbeck process

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Abstract: Abstract We consider the problem of modelling restricted interactions between continuouslyobserved time series as given by a known static graph (or network) structure. For this purpose, we define a parametric multivariate Graph OrnsteinUhlenbeck (GrOU) process driven by a general Lévy process to study the momentum and network effects amongst nodes, effects that quantify the impact of a node on itself and that of its neighbours, respectively. We derive the maximum likelihood estimators (MLEs) and their usual properties (existence, uniqueness and efficiency) along with their asymptotic normality and consistency. Additionally, an Adaptive Lasso approach, or a penalised likelihood scheme, infers both the graph structure along with the GrOU parameters concurrently and is shown to satisfy similar properties. Finally, we show that the asymptotic theory extends to the case when stochastic volatility modulation of the driving Lévy process is considered.
PubDate: 20220701
DOI: 10.1007/s11203021092571

 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

 Calibration for multivariate Lévydriven OrnsteinUhlenbeck processes
with applications to weak subordination
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Abstract: Abstract Consider a multivariate Lévydriven OrnsteinUhlenbeck process where the stationary distribution or background driving Lévy process is from a parametric family. We derive the likelihood function assuming that the innovation term is absolutely continuous. Two examples are studied in detail: the process where the stationary distribution or background driving Lévy process is given by a weak variance alphagamma process, which is a multivariate generalisation of the variance gamma process created using weak subordination. In the former case, we give an explicit representation of the background driving Lévy process, leading to an innovation term which is a discrete and continuous mixture, allowing for the exact simulation of the process, and a separate likelihood function. In the latter case, we show the innovation term is absolutely continuous. The results of a simulation study demonstrate that maximum likelihood numerically computed using Fourier inversion can be applied to accurately estimate the parameters in both cases.
PubDate: 20211102
DOI: 10.1007/s11203021092544
