This paper investigates the nonparametric estimation problem for some periodic stochastic differential equation driven by fractional G-Brownian motion (fGBm), which generalizes the concepts of the standard Brownian motion, fractional Brownian motion and G-Brownian motion in the framework of sublinear expectation. The fGBm can exhibit long-range dependence and feature the volatility uncertainty simultaneously. Thus it can be a better alternative stochastic process in real applications. First, some probability density function (pdf) called H-G-normal pdf associated with the fGBm is defined, the Volterra representation for fGBm and its Wiener integral are established. Then, the nonparametric estimator for the drift function of the periodic stochastic differential equation is defined by some kernel function, and its consistency and asymptotic distribution are investigated. Finally, some numerical experiments are carried out to illustrate the theoretical results. This study generalizes some well-known existing results of nonparametric estimation.
This paper addresses the problem of global parameter estimation for the AD (1,n) model, where n is a positive integer. The AD (1,n) model is a subclass of affine diffusions introduced by Duffie, Filipovi?, and Schachermayer in Duffie et al. (2003). Affine diffusion models are widely used in the pricing of bonds and stock options, including the Vasicek, Cox-Ingersoll-Ross, and Heston models. Our main results concern the conditional least squares estimation of the drift parameters of the AD (1,n) model, based on high-frequency discrete-time observations over an infinite horizon. We then analyze the asymptotic properties of the estimators in both ergodic and non-ergodic cases. Additionally, this paper presents some moment results related to the AD (1,n) model.
We consider the estimation of the marginal expected shortfall 𝔼( X_h | Y_0>U_Y(1/p)) at extreme levels, when ((X_t, Y_t))_t∈ℤ is a strictly stationary β - mixing time series with marginal distributions of Pareto-type, U_Y is the tail quantile function associated to Y_t , h is a positive integer and p∈ (0, 1) is such that p→ 0 . We propose an estimator for this risk measure based on a Weissman-type construction. First, in case of a non-negative time series, we establish the weak convergence of our estimator by using empirical processes arguments combined with the cluster method of Drees and Rootzén (2010). Then, we extend our result to the case of real-valued time series by using the decomposition of the original time series into the positive and negative parts, and we also propose a bootstrap procedure. The performance of our estimator is illustrated on a simulation experiment. Finally, the method is applied on river flow data.
This study examines the intricate process of estimating nonparametrically in expectile regression models for functional time series data that exhibit strong mixing properties within the context of a random right-censoring model. Specifically, we establish the almost complete consistency and asymptotic normality of the kernel-based expectile regression estimator. Notably, these results are derived in an asymptotic setting and are applicable under reasonably broad assumptions about the underlying model. Furthermore, we explore the practical implications of our theoretical findings in analyzing financial time series. To evaluate the performance of the proposed estimator on finite samples, we conducted comprehensive Monte Carlo simulations. These simulations provide a quantitative assessment of the estimator’s accuracy and efficiency under various scenarios, allowing for a thorough understanding of its practical utility.
We deal with a model selection problem for structural equation modeling (SEM) for diffusion processes. Based on the asymptotic expansion of the marginal quasi-log likelihood, we propose two types of quasi-Bayesian information criteria of the SEM. It is shown that the information criteria have model selection consistency. Furthermore, we examine the finite-sample performance of the proposed information criteria by numerical experiments.
We study structural equation modeling (SEM) for diffusion processes with jumps. Based on high-frequency data, we consider the parameter estimation and the goodness-of-fit test in the SEM. Using a threshold method, we propose the quasi-likelihood of the SEM and prove that the quasi-maximum likelihood estimator has consistency and asymptotic normality. To examine whether a specified parametric model is correct or not, we also construct the quasi-likelihood ratio test statistics and investigate the asymptotic properties. Furthermore, numerical simulations are conducted.
This article generalizes the maximum spacing (MSP) method to dependent observations by considering hidden Markov models. The MSP method for estimating the model parameters is applied in two steps: at first the parameters of the marginal distribution of observations are estimated, in the second step the transition probabilities of the underlying Markov chain are estimated using the obtained marginal parameter estimates. We prove that the proposed MSP estimation procedure gives consistent estimators. The possibility of using the proposed estimation procedure in the context of model validation is investigated in simulation examples. It is demonstrated that when the observations are dependent, then taking into account the dependence structure by considering two-dimensional spacings provides additional information about a suitable number of mixture components in the model. The proposed estimation method is also applied in a real data example.
In this article, we consider a jump diffusion process (X_t)_t ≥ 0 with drift function b, diffusion coefficient σ and jump coefficient ξ . This process is supposed to be ergodic, exponentially β -mixing and stationary. It is observed at discrete times t=0,Δ ,… ,nΔ . The sampling interval Δ tends to 0 and the time interval nΔ tends to infinity. We construct a robust, adaptive non-parametric estimator of the function ξ ^4 thanks to a penalized least-square approach. We provide bounds of the empirical and L^2 -risk of our estimator.
Statistical inference for epidemic outbreaks is often complicated by only partial observation of the epidemic process. Recently in Ball and Neal (Adv Appl Probab 55:895-926, 2023) the distribution of the number of infectives (individuals alive) given only the times of removals (death) in a Markovian SIR epidemic (time-inhomogeneous birth–death process) was derived. We show that this allows us to derive an explicit expression for the likelihood of the observed inter-removal times of the epidemic without recourse to data augmentation techniques. Moreover, the time-inhomogeneous birth–death process provides a good approximation for the SIR epidemic model for which we are able to obtain both, the exact likelihood of the inter-arrival death times, and a fast to compute Gaussian-based approximation of the likelihood. The explicit expressions for the likelihood enable us to reveal bi-modality in the likelihood of the ongoing Markovian SIR epidemic model and to devise scaleable MCMC algorithms which are applied to the emergence of the Covid-19 epidemic in Europe (March–May 2020).
Consider a diffusion process X, solution of a time-homogeneous stochastic differential equation. We assume that the diffusion process X is observed at discrete times, at high frequency, which means that the time step tends toward zero. In addition, the drift and diffusion coefficients of the process X are assumed to be unknown. In this paper, we study the minimax rates of convergence of the nonparametric estimators of the square of the diffusion coefficient. Two observation schemes are considered depending on the estimation interval. The square of the diffusion coefficient is estimated on the real line from repeated observations of the process X, where the number of diffusion paths tends to infinity. For the case of a compact estimation interval, we study the nonparametric estimation of the square of the diffusion coefficient constructed from a single diffusion path on one side and from repeated observations on the other side, where the number of trajectories tends to infinity. In each of these cases, we establish minimax convergence rates of the risk of estimation of the diffusion coefficient over a space of Holder functions.
Penalized estimation methods for diffusion processes and dependent data have recently gained significant attention due to their effectiveness in handling stochastic systems. In this work, we introduce an adaptive Elastic-Net estimator for ergodic diffusion processes observed under high-frequency sampling schemes. Our method combines the least squares approximation of the quasi-likelihood with adaptive $$\ell _1$$ ℓ 1 and $$\ell _2$$ ℓ 2 regularization. This approach allows to enhance prediction accuracy and interpretability while effectively recovering the sparse underlying structure of the model. In the spirit of recent research trends, we provide finite-sample guarantees for the (block-diagonal) estimator’s performance by deriving high-probability non-asymptotic bounds for the $$\ell _2$$ ℓ 2 estimation error. These results complement the established oracle properties in the high-frequency asymptotic regime with mixed convergence rates, ensuring consistent selection of the relevant interactions and achieving optimal rates of convergence. Furthermore, we utilize our results to analyze one-step-ahead predictions, offering non-asymptotic control over the $$\ell _1$$ ℓ 1 prediction error. The performance of our method is evaluated through simulations and real data applications, demonstrating its effectiveness, particularly in scenarios with strongly correlated variables.
The quantile-crossing spectrum is the spectrum of quantile-crossing processes created from a time series by the indicator function that shows whether or not the time series lies above or below a given quantile at a given time. This bivariate function of frequency and quantile level provides a richer view of serial dependence than that offered by the ordinary spectrum. A new estimator is proposed in this paper for the quantile-crossing spectrum as a bivariate function of frequency and quantile level. The proposed estimator is derived from a method called spline autoregression (SAR). It jointly fits an autoregressive (AR) model to the quantile-crossing series across multiple quantiles, where the functional AR coefficients are represented as spline functions of the quantile level and penalized for their roughness. Numerical experiments show that when the underlying spectrum is smooth in quantile level the proposed method is able to produce more accurate estimates in comparison with the alternative that ignores the smoothness.
We observe an unknown function of d variables f(t) , t∈ [0,1]^d , in the Gaussian white noise model of intensity ε >0 . We assume that the function f is regular and that it is a sum of k-variate functions, where k varies from 1 to s ( 1≤ s≤ d ). These functions are unknown to us and only a few of them are nonzero. In this article, we address the problem of identifying the nonzero components of f in the case when d=d_ε→∞ as ε→ 0 and s is either fixed or s=s_ε→∞ , s=o(d) as ε→∞ . This may be viewed as a variable selection problem. We derive the conditions when exact variable selection in the model at hand is possible and provide a selection procedure that achieves this type of selection. The procedure is adaptive to a degree of model sparsity described by the sparsity parameter β∈ (0,1) . We also derive conditions that make the exact variable selection impossible. Our results augment previous work in this area.
The purpose of the present work is to construct estimators for the random effects in a fractional diffusion model using a hybrid estimation method where we combine parametric and nonparametric techniques. We precisely consider n stochastic processes { X_t^j, 0≤ t≤ T} , j=1,… , n continuously observed over the time interval [0, T], where the dynamics of each process are described by fractional stochastic differential equations with drifts depending on random effects. We first construct a parametric estimator for random effects using maximum likelihood estimation techniques and study its asymptotic properties when the time horizon T is sufficiently large. Then, on the basis of the obtained estimator for the random effects, we build a nonparametric estimator for their common unknown density function using Bernstein polynomials approximation. Some asymptotic properties of the density estimator, such as its asymptotic bias, variance, and mean integrated squared error, are studied for an infinite time horizon T and a fixed sample size n. The asymptotic normality of the estimator is established for a fixed T, a high frequency, and as long as the order of Bernstein polynomials is sufficiently large. We also investigate a non-asymptotic bound for the expected uniform error between the density function and its estimator. A numerical study is then presented in order to evaluate both qualitative and quantitative performance of the Bernstein estimator compared with the standard kernel estimator within and at boundaries of the support of the density function.
We study the problem of parametric estimation for continuously observed stochastic differential equation driven by fractional Brownian motion. Under some assumptions on drift and diffusion coefficients, we construct maximum likelihood estimator and establish its the asymptotic normality and moment convergence of the drift parameter when a small dispersion coefficient vanishes.
We consider the problem of estimating the density of the process associated with the small jumps of a pure jump Lévy process, possibly of infinite variation, from discrete observations of one trajectory. The interest of such a question lies on the observation that even when the Lévy measure is known, the density of the increments of the small jumps of the process cannot be computed in closed-form. We discuss results both from low and high-frequency observations. In a low frequency setting, assuming the Lévy density associated with the jumps larger than ε∈ (0,1] in absolute value is known, a spectral estimator relying on the convolution structure of the problem achieves a parametric rate of convergence with respect to the integrated L_2 loss, up to a logarithmic factor. In a high-frequency setting, we remove the assumption on the knowledge of the Lévy measure of the large jumps and show that the rate of convergence depends both on the sampling scheme and on the behaviour of the Lévy measure in a neighborhood of zero. We show that the rate we find is minimax up to a logarithmic factor. An adaptive penalized procedure is studied to select the cutoff parameter. These results are extended to encompass the case where a Brownian component is present in the Lévy process. Furthermore, we numerically illustrate the performances of our procedures.
We consider parameter estimation of the reaction term for a second order linear parabolic stochastic partial differential equation in two space dimensions driven by a Q-Wiener process under small diffusivity. We first construct an estimator of the reaction parameter based on continuous spatio-temporal data, and then derive an estimator of the reaction parameter based on high frequency spatio-temporal data by discretizing the estimator based on the continuous data. We show that the estimators have consistency and asymptotic normality. Furthermore, we give simulation results of the estimator based on high frequency data.
We propose a unified stochastic SIR model driven by Lévy noise. The model is structural enough to allow for time-dependency, nonlinearity, discontinuity, demography and environmental disturbances. We present concise results on the existence and uniqueness of positive global solutions and investigate the extinction and persistence of the novel model. Examples and simulations are provided to illustrate the main results.
Let $X$ be a chemical reaction process, modeled as a multi-dimensional continuous-time jump process. Assume that at given times $0
The model of partially observed linear stochastic differential equations depending on some unknown parameters is considered. An proximation of the unobserved component is proposed. This approximation is realized in three steps. First an estimator of the thod of moments of unknown parameter is constructed. Then this estimator is used for defining the One-step MLE-process and nally the last estimator is substituted to the equations of Kalman-Bucy (K-B) filter. The solution of obtained K-B equations ovide us the approximation (adaptive K-B filter). The asymptotic properties of all mentioned estimators and MLE and Bayesian timators of the unknown parameters are described. The asymptotic efficiency of the proposed adaptive filter is shown.
