Abstract: In this paper we address the problem of estimating the posterior distribution of the static parameters of a continuous-time state space model with discrete-time observations by an algorithm that combines the Kalman filter and a particle filter. The proposed algorithm is semi-recursive and has a two layer structure, in which the outer layer provides the estimation of the posterior distribution of the unknown parameters and the inner layer provides the estimation of the posterior distribution of the state variables. This algorithm has a similar structure as the so-called recursive nested particle filter, but unlike the latter filter, in which both layers use a particle filter, our algorithm introduces a dynamic kernel to sample the parameter particles in the outer layer to obtain a higher convergence speed. Moreover, this algorithm also implements the Kalman filter in the inner layer to reduce the computational time. This algorithm can also be used to estimate the parameters that suddenly change value. We prove that, for a state space model with a certain structure, the estimated posterior distribution of the unknown parameters and the state variables converge to the actual distribution in \(L^p\) with rate of order \({\mathcal {O}}(N^{-\frac{1}{2}}+\varDelta ^{\frac{1}{2}})\) , where N is the number of particles for the parameters in the outer layer and \(\varDelta \) is the maximum time step between two consecutive observations. We present numerical results of the implementation of this algorithm, in particularly we implement this algorithm for affine interest models, possibly with stochastic volatility, although the algorithm can be applied to a much broader class of models. PubDate: 2021-03-03

Abstract: Let \(X=(X_t)_{t\ge 0}\) be a known process and T an unknown random time independent of X. Our goal is to derive the distribution of T based on an iid sample of \(X_T\) . Belomestny and Schoenmakers (Stoch Process Appl 126(7):2092–2122, 2015) propose a solution based the Mellin transform in case where X is a Brownian motion. Applying their technique we construct a non-parametric estimator for the density of T for a self-similar one-dimensional process X. We calculate the minimax convergence rate of our estimator in some examples with a particular focus on Bessel processes where we also show asymptotic normality. PubDate: 2021-03-01

Abstract: In this article, the maximum spacing (MSP) method is extended to continuous time Markov chains and semi-Markov processes and consistency of the MSP estimator is proved. For independent and identically distributed univariate observations the idea behind the MSP method is to approximate the Kullback–Leibler information so that each contribution is bounded from above. Following the same idea, the MSP function in this article is defined as an approximation of the relative entropy rate for semi-Markov processes and continuous time Markov chains. The MSP estimator is defined as the parameter value that maximizes the MSP function. Consistency of the MSP estimator is also studied when the assigned model is incorrect. PubDate: 2021-02-15

Abstract: We provide a rigorous mathematical foundation of the theory for the higher-order asymptotic behavior of the one-dimensional Hawkes process with an exponential kernel. As an important application, we give the second-order asymptotic distribution for the maximum likelihood estimator of the exponential Hawkes process. PubDate: 2021-01-18 DOI: 10.1007/s11203-021-09237-5

Abstract: Let the Ornstein–Uhlenbeck process \((X_t)_{t\ge 0}\) driven by a fractional Brownian motion \(B^{H }\) described by \(dX_t = -\theta X_t dt + \sigma dB_t^{H }\) be observed at discrete time instants \(t_k=kh\) , \(k=0, 1, 2, \ldots , 2n+2 \) . We propose an ergodic type statistical estimator \({\hat{\theta }}_n \) , \({\hat{H}}_n \) and \({\hat{\sigma }}_n \) to estimate all the parameters \(\theta \) , H and \(\sigma \) in the above Ornstein–Uhlenbeck model simultaneously. We prove the strong consistence and the rate of convergence of the estimator. The step size h can be arbitrarily fixed and will not be forced to go zero, which is usually a reality. The tools to use are the generalized moment approach (via ergodic theorem) and the Malliavin calculus. PubDate: 2021-01-13 DOI: 10.1007/s11203-020-09235-z

Abstract: We show asymptotic distributions of the residual process in Ornstein–Uhlenbeck model, when the model is true. These distributions are of Brownian motion and of Brownian bridge, depending on whether we estimate one parameter or two. This leads to seemingly simple asymptotic theory of goodness of fit tests based on this process. However, next we show that the residual process would lead to a deficient testing procedures, unless a transformed form of it is introduced. The transformed process is introduced and their role is explained through connection with what is known for the so called chimeric alternatives in testing problems for samples. PubDate: 2021-01-13 DOI: 10.1007/s11203-020-09233-1

Abstract: We develop a nonparametric technique for the estimation of curve trajectories using HARDI data. For various regions of the brain, we consider the imaging signal process and apply multivariate kernel smoothing techniques to a general function f describing the signal process obtained from the MRI image. At each location in the brain we search for the direction of maximum diffusion on the unit sphere, and then trace the integral curve driven by the vector field to obtain the estimates of curve trajectories. We establish the convergence of the properly normalized curve estimators to a Gaussian process. This method is computationally efficient as with each step of the curve tracing we construct a pointwise confidence ellipsoid region as opposed to exhaustive iterative sampling methods. These curve trajectories are models of axonal fibers whose location and geometry are important in neuroscience. PubDate: 2021-01-11 DOI: 10.1007/s11203-020-09236-y

Abstract: A stochastic hybrid system, also known as a switching diffusion, is a continuous-time Markov process with state space consisting of discrete and continuous parts. We consider parametric estimation of the Q matrix for the discrete state transitions and of the drift coefficient for the diffusion part. First, we derive the likelihood function under the complete observation of a sample path in continuous-time. Then, extending a finite-dimensional filter for hidden Markov models developed by Elliott et al. (Hidden Markov Models, Springer, 1995) to stochastic hybrid systems, we derive the likelihood function and the EM algorithm under a partial observation where the continuous state is monitored continuously in time, while the discrete state is unobserved. PubDate: 2021-01-07 DOI: 10.1007/s11203-020-09231-3

Abstract: Max-stable processes have been expanded to quantify extremal dependence in spatiotemporal data. Due to the interaction between space and time, spatiotemporal data are often complex to analyze. So, characterizing these dependencies is one of the crucial challenges in this field of statistics. This paper suggests a semiparametric inference methodology based on the spatiotemporal F-madogram for estimating the parameters of a space-time max-stable process using gridded data. The performance of the method is investigated through various simulation studies. Finally, we apply our inferential procedure to quantify the extremal behavior of radar rainfall data in a region in the State of Florida. PubDate: 2021-01-07 DOI: 10.1007/s11203-020-09232-2

Abstract: The problem of the minimax testing of the Poisson process intensity \({\mathbf{s}}\) is considered. For a given intensity \({\mathbf{p}}\) and a set \(\mathcal{Q}\) , the minimax testing of the simple hypothesis \(H_{0}: {\mathbf{s}} = {\mathbf{p}}\) against the composite alternative \(H_{1}: {\mathbf{s}} = {\mathbf{q}},\,{\mathbf{q}} \in \mathcal{Q}\) is investigated. 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 \(\beta ({\mathbf{p}},\mathcal{Q})\) , is considered. What is the maximal set \(\mathcal{Q}\) , which can be replaced by an intensity \({\mathbf{q}} \in \mathcal{Q}\) without any loss of testing performance' In the asymptotic case ( \(T\rightarrow \infty \) ) that maximal set \(\mathcal{Q}\) is described. PubDate: 2021-01-06 DOI: 10.1007/s11203-020-09230-4

Abstract: The conditional density of Brownian motion is considered given the max, \(B(t \max )\) , as well as those with additional information: \(B(t close, \max )\) , \(B(t close, \max , \min )\) where the close is the final value: \(B(t=1)=c\) and \(t \in [0,1]\) . The conditional expectation and conditional variance of Brownian motion are evaluated subject to one or more of the statistics: the close (final value), the high (maximum), the low (minimum). Computational results displaying both the expectation and variance in time are presented and compared with the theoretical values. We tabulate the time averaged variance of Brownian motion conditional on knowing various extremal properties of the motion. The final table shows that knowing the high is more useful than knowing the final value among other results. Knowing the open, high, low and close reduces the time averaged variance to \(42\%\) of the value of knowing only the open and close (Brownian bridge). PubDate: 2020-12-04 DOI: 10.1007/s11203-020-09229-x

Abstract: Claeskens and Hjort (J Am Stat Assoc 98(464):900–916, 2003) constructed the focused information criterion (FIC) using maximum likelihood estimators to facilitate the contextual selection of probability models for independently distributed observations. We generalize these results to the case of stationary, strong mixing stochastic processes exhibiting outliers when the “true” finite dimensional distribution of the process lies in the contamination neighbourhood of the assumed parametric model for the process. Given the natural filtration of a stochastic process, we obtain the FIC using robust M-estimators having bounded conditional influence functions. We utilize Le Cam’s contiguity lemmas to tract the parametric form of the asymptotic bias of M-estimators under model misspecification induced by additive outliers. The local asymptotic normality is established assuming the finite dimensional parametric density of the process to be \(\mathscr {L}^{2}\) -differentiable (differentiable in quadratic mean). As a result, our theory is also applicable for constructing FIC for moderately irregular models outside the exponential family such as Laplace and related densities. We apply our results to derive the robust FIC for simultaneous selection of the order and the innovation density of asymmetric Laplace autoregressive processes observed with outliers. We demonstrate our theory with the focused modeling of United States Dollar to Indian Rupee currency exchange rates exhibiting outliers post Indian demonetization. PubDate: 2020-10-01 DOI: 10.1007/s11203-020-09208-2

Abstract: We consider the problem of estimation of the drift parameter of an ergodic Ornstein–Uhlenbeck type process driven by a Lévy process with heavy tails. The process is observed continuously on a long time interval [0, T], \(T\rightarrow \infty \) . We prove that the statistical model is locally asymptotic mixed normal and the maximum likelihood estimator is asymptotically efficient. PubDate: 2020-10-01 DOI: 10.1007/s11203-020-09210-8

Abstract: In this paper, we propose a new threshold-kernel jump-detection method for jump-diffusion processes, which iteratively applies thresholding and kernel methods in an approximately optimal way to achieve improved finite-sample performance. As in Figueroa-López and Nisen (Stoch Process Appl 123(7):2648–2677, 2013), we use the expected number of jump misclassifications as the objective function to optimally select the threshold parameter of the jump detection scheme. We prove that the objective function is quasi-convex and obtain a new second-order infill approximation of the optimal threshold in closed form. The approximate optimal threshold depends not only on the spot volatility \(\sigma _t\) , but also the jump intensity and the value of the jump density at the origin. Estimation methods for these quantities are then developed, where the spot volatility is estimated by a kernel estimator with thresholding and the value of the jump density at the origin is estimated by a density kernel estimator applied to those increments deemed to contain jumps by the chosen thresholding criterion. Due to the interdependency between the model parameters and the approximate optimal estimators built to estimate them, a type of iterative fixed-point algorithm is developed to implement them. Simulation studies for a prototypical stochastic volatility model show that it is not only feasible to implement the higher-order local optimal threshold scheme but also that this is superior to those based only on the first order approximation and/or on average values of the parameters over the estimation time period. PubDate: 2020-10-01 DOI: 10.1007/s11203-020-09211-7

Abstract: The problem of hypothesis testing is considered in the case of observation of an inhomogeneous Poisson process with an intensity function depending on two parameters. It is supposed that the dependence on the first of them is sufficiently regular, while the second one is a change-point location. Under the null hypothesis the parameters take some known values, while under the alternative they are greater (with at least one of the inequalities being strict). Four test are studied: the general likelihood ratio test (GLRT), the Wald’s test and two Bayesian tests (BT1 and BT2). For each of the tests, expressions allowing to approximate its threshold and its limit power function by Monte Carlo numerical simulations are derived. Moreover, for the GLRT, an analytic equation for the threshold and an analytic expression of the limit power function are obtained. Finally, numerical simulations are carried out and the performance of the tests is discussed. PubDate: 2020-10-01 DOI: 10.1007/s11203-020-09207-3

Abstract: We consider a Gaussian continuous time moving average model \(X(t)=\int _0^t a(t-s)dW(s)\) where W is a standard Brownian motion and a(.) a deterministic function locally square integrable on \({{\mathbb {R}}}^+\) . Given N i.i.d. continuous time observations of \((X_i(t))_{t\in [0,T]}\) on [0, T], for \(i=1, \dots , N\) distributed like \((X(t))_{t\in [0,T]}\) , we propose nonparametric projection estimators of \(a^2\) under different sets of assumptions, which authorize or not fractional models. We study the asymptotics in T, N (depending on the setup) ensuring their consistency, provide their nonparametric rates of convergence on functional regularity spaces. Then, we propose a data-driven method corresponding to each setup, for selecting the dimension of the projection space. The findings are illustrated through a simulation study. PubDate: 2020-09-25 DOI: 10.1007/s11203-020-09228-y

Abstract: In the present paper, we extend the work of Slaoui (Stat Sin 30:417–437, 2020) in the case of strong mixing data. Since, we are interested in nonparametric regression estimation, we focus on well adapted dependence structures based on mixing type conditions. We study the properties of these regression estimators and compare them with the nonparametric non-recursive regression estimator. The bias, variance and mean squared error are computed explicitly. We showed that using a selected wild bootstrap bandwidth procedure and a special stepsize, our proposed recursive regression estimators allowed us to obtain quite similar results compared to the non-recursive regression estimator under \(\alpha \) -mixing condition in terms of estimation error and much better in terms of computational costs. PubDate: 2020-07-27 DOI: 10.1007/s11203-020-09223-3

Abstract: Multidimensional hypoelliptic diffusions arise naturally in different fields, for example to model neuronal activity. Estimation in those models is complex because of the degenerate structure of the diffusion coefficient. In this paper we consider hypoelliptic diffusions, given as a solution of two-dimensional stochastic differential equations, with the discrete time observations of both coordinates being available on an interval \(T = n\varDelta _n\) , with \(\varDelta _n\) the time step between the observations. The estimation is studied in the asymptotic setting, with \(T\rightarrow \infty \) as \(\varDelta _n\rightarrow 0\) . We build a consistent estimator of the drift and variance parameters with the help of a discretized log-likelihood of the continuous process. We discuss the difficulties generated by the hypoellipticity and provide a proof of the consistency and the asymptotic normality of the estimator. We test our approach numerically on the hypoelliptic FitzHugh–Nagumo model, which describes the firing mechanism of a neuron. PubDate: 2020-07-17 DOI: 10.1007/s11203-020-09222-4

Abstract: We consider a 1-dimensional diffusion process X with jumps. The particularity of this model relies in the jumps which are driven by a multidimensional Hawkes process denoted N. This article is dedicated to the study of a nonparametric estimator of the drift coefficient of this original process. We construct estimators based on discrete observations of the process X in a high frequency framework with a large horizon time and on the observations of the process N. The proposed nonparametric estimator is built from a least squares contrast procedure on subspace spanned by trigonometric basis vectors. We obtain adaptive results that are comparable with the one obtained in the nonparametric regression context. We finally conduct a simulation study in which we first focus on the implementation of the process and then on showing the good behavior of the estimator. PubDate: 2020-05-07 DOI: 10.1007/s11203-020-09213-5

Abstract: In this article we introduce and study oscillating Gaussian processes defined by \(X_t = \alpha _+ Y_t \mathbf{1}_{Y_t >0} + \alpha _- Y_t\mathbf{1}_{Y_t<0}\), where \(\alpha _+,\alpha _->0\) are free parameters and Y is either stationary or self-similar Gaussian process. We study the basic properties of X and we consider estimation of the model parameters. In particular, we show that the moment estimators converge in \(L^p\) and are, when suitably normalised, asymptotically normal. PubDate: 2020-04-29 DOI: 10.1007/s11203-020-09212-6