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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: 2022-04-13
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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 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
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Abstract: We consider drift estimation problems for high dimension ergodic diffusion processes in nonparametric setting based on observations at discrete fixed time moments in the case when diffusion coefficients are unknown. To this end on the basis of sequential analysis methods we develop model selection procedures, for which we show non asymptotic sharp oracle inequalities. Through the obtained inequalities we show that the constructed model selection procedures are asymptotically efficient in adaptive setting, i.e. in the case when the model regularity is unknown. For the first time for such problem, we found in the explicit form the celebrated Pinsker constant which provides the sharp lower bound for the minimax squared accuracy normalized with the optimal convergence rate. Then we show that the asymptotic quadratic risk for the model selection procedure asymptotically coincides with the obtained lower bound, i.e this means that the constructed procedure is efficient. Finally, on the basis of the constructed model selection procedures in the framework of the big data models we provide the efficient estimation without using the parameter dimension or any sparse conditions. PubDate: 2022-04-01
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Abstract: We investigate the mixed fractional Brownian motion with trend of the form \(X_t = \theta t + \sigma W_t + \kappa B^H_t\) , driven by a standard Brownian motion W and a fractional Brownian motion \(B^H\) with Hurst parameter H. We develop and compare two approaches to estimation of four unknown parameters \(\theta \) , \(\sigma \) , \(\kappa \) and H by discrete observations. The first algorithm is more traditional: we estimate \(\sigma \) , \(\kappa \) and H using the quadratic variations, while the estimator of \(\theta \) is obtained as a discretization of a continuous-time estimator of maximum likelihood type. This approach has several limitations, in particular, it assumes that \(H<\frac{3}{4}\) , moreover, some estimators have too low rate of convergence. Therefore, we propose a new method for simultaneous estimation of all four parameters, which is based on the ergodic theorem. Finally, we compare two approaches by Monte Carlo simulations. PubDate: 2022-04-01
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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 model-based survey analysis. For strongly-mixing non-stationary 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 strongly-mixing coefficients and quantile functions to show that a functional stochastic process asymptotically approaches to a Gaussian process. PubDate: 2022-03-26
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Abstract: Abstract The paper focuses on the statistical analysis of Bayes and Maximum Posterior Probability (MAP) tests in the problem of detecting sparse vectors. Our main goal is to test the simple hypothesis \({\mathcal {H}}_0: \Vert \theta \Vert =0\) versus the composite alternative \({\mathcal {H}}_1: \Vert \theta \Vert >0 \) based on the observations $$\begin{aligned} Y_k=\sum _{s=1}^p\ {\mathbf {1}}\bigl \{ I_s=k\bigr \} \theta _s+\sigma \xi _k, \quad k=1, 2,\ldots , \end{aligned}$$ where \(\theta =(\theta _1,\ldots , \theta _p)^\top \in {\mathbb {R}}^p\) is an unknown vector, \(\xi _k\) are i.i.d. \({\mathcal {N}}(0,1)\) , and \(\mathbf { I}=\{I_1,\ldots ,I_p\}\) is an unknown random multi-index with differing components and the probability distribution $$\begin{aligned} {\mathbf {P}}\bigl (\mathbf { I}\bigr )\propto \prod _{k\in \mathbf { I}}{\bar{\pi }}_{k}. \end{aligned}$$ It is assumed that the known prior distribution \({\bar{\pi }}=({\bar{\pi }}_1,\ldots )\) has a large entropy. In the case of a known p, we find the limiting distributions of MAP and Bayes tests statistics under \({\mathcal {H}}_0\) . Under the alternative, we characterize the sensitivity of these tests by computing detectable sets of \(\theta \) . Finally, the case of unknown p is considered. We construct a multiple MAP test and show that it adapts to p under mild assumptions. This test is based on the non-asymptotic law of the three times iterated logarithm for the cumulative mean of the Wiener process. PubDate: 2022-03-08
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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: 2022-03-07
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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 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. PubDate: 2022-02-23 DOI: 10.1007/s11203-022-09272-w
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Abstract: Abstract The Ibragimov–Khasminskii theory established a scheme that gives asymptotic properties of the likelihood estimators through the convergence of the likelihood ratio random field. This scheme is extending to various nonlinear stochastic processes, combined with a polynomial type large deviation inequality proved for a general locally asymptotically quadratic quasi-likelihood random field. We give an overview of the quasi-likelihood analysis and its applications to ergodic/non-ergodic statistics for stochastic processes. PubDate: 2022-02-23 DOI: 10.1007/s11203-021-09266-0
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Abstract: 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: 2022-02-04 DOI: 10.1007/s11203-022-09269-5
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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 “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: 2022-01-29 DOI: 10.1007/s11203-021-09265-1
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Abstract: Abstract This work is devoted to numerical solutions of controlled stochastic Kolmogorov systems with regime switching and random jumps. Markov chain approximation methods are used to design numerical algorithms to approximate the controlled switching jump diffusions, the cost functions, and the value functions. Under suitable conditions, the convergence of the algorithms is proved. Numerical examples are provided to demonstrate the performance of the algorithms. PubDate: 2022-01-21 DOI: 10.1007/s11203-021-09267-z
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Abstract: Abstract We are considering the asymptotic behavior as \(t\rightarrow \infty \) of solutions of the Cauchy problem for parabolic second order equations with time periodic coefficients. The problem is reduced to considering degenerate time-homogeneous diffusion processes on the product of a unit circle and Euclidean space. PubDate: 2022-01-18 DOI: 10.1007/s11203-021-09259-z
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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 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: 2022-01-08 DOI: 10.1007/s11203-021-09264-2
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Abstract: Abstract It is shown that in the problem of cardinal interpolation, spline interpolants of various degrees are R-minimax, with respect to corresponding Sobolev and Hardy functional classes, under restrictions determined by the interference between their oscillating variance and bias. The results raise a natural question: what degrees of interpolating splines are more appropriate, for given Sobolev or Hardy classes' It turns out that the scales of such functional classes can be divided into “very smooth” and “not-so-smooth” subfamilies, whereby “very smooth” classes can benefit from higher degrees of cardinal splines, and vice versa. PubDate: 2022-01-03 DOI: 10.1007/s11203-021-09261-5
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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 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: 2021-12-05 DOI: 10.1007/s11203-021-09262-4
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Abstract: 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: 2021-11-29 DOI: 10.1007/s11203-021-09263-3
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Abstract: Abstract 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. PubDate: 2021-11-10 DOI: 10.1007/s11203-021-09260-6
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Abstract: Abstract Consider a multivariate Lévy-driven Ornstein-Uhlenbeck 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 alpha-gamma 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: 2021-11-02 DOI: 10.1007/s11203-021-09254-4
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