In this study, we develop an asymptotic theory of nonparametric regression for a locally stationary functional time series. First, we introduce the notion of a locally stationary functional time series (LSFTS) that takes values in a semi-metric space. Then, we propose a nonparametric model for LSFTS with a regression function that changes smoothly over time. We establish the uniform convergence rates of a class of kernel estimators, the Nadaraya-Watson (NW) estimator of the regression function, and a central limit theorem of the NW estimator.
In this paper we consider multivariate Hawkes processes with baseline hazard and kernel functions that depend on time. This defines a class of locally stationary processes. We discuss estimation of the time-dependent baseline hazard and kernel functions based on a localized criterion. Theory on stationary Hawkes processes is extended to develop asymptotic theory for the estimator in the locally stationary model.
This article considers a nonparametric method for detecting change points in non-stationary time series. The proposed method will divide the time series into several segments so that between two adjacent segments, the normalized spectral density functions are different. The theory is based on the assumption that within each segment, time series is a linear process, which means that our method works not only for classic time series models, e.g., causal and invertible ARMA process, but also preserves good performance for non-invertible moving average process. We show that our estimations for cha…
In this paper, we study first the problem of nonparametric estimation of the stationary density $f$ of a discrete-time Markov chain $(X_i)$. We consider a collection of projection estimators on finite dimensional linear spaces. We select an estimator among the collection by minimizing a penalized contrast. The same technique enables to estimate the density $g$ of $(X_i, X_{i+1})$ and so to provide an adaptive estimator of the transition density $π=g/f$. We give bounds in $L^2$ norm for these estimators and we show that they are adaptive in the minimax sense over a large class of Besov spaces. …
We consider a stochastic individual-based model in continuous time to describe a size-structured population for cell divisions. This model is motivated by the detection of cellular aging in biology. We address here the problem of nonparametric estimation of the kernel ruling the divisions based on the eigenvalue problem related to the asymptotic behavior in large population. This inverse problem involves a multiplicative deconvolution operator. Using Fourier technics we derive a nonparametric estimator whose consistency is studied. The main difficulty comes from the non-standard equations conn…
This paper deals with nonparametric maximum likelihood estimation for Gaussian locally stationary processes. Our nonparametric MLE is constructed by minimizing a frequency domain likelihood over a class of functions. The asymptotic behavior of the resulting estimator is studied. The results depend on the richness of the class of functions. Both sieve estimation and global estimation are considered. Our results apply, in particular, to estimation under shape constraints. As an example, autoregressive model fitting with a monotonic variance function is discussed in detail, including algorithmic …
In this study, we develop an asymptotic theory of nonparametric regression for locally stationary random fields (LSRFs) $\{{\bf X}_{{\bf s}, A_{n}}: {\bf s} \in R_{n} \}$ in $\mathbb{R}^{p}$ observed at irregularly spaced locations in $R_{n} =[0,A_{n}]^{d} \subset \mathbb{R}^{d}$. We first derive the uniform convergence rate of general kernel estimators, followed by the asymptotic normality of an estimator for the mean function of the model. Moreover, we consider additive models to avoid the curse of dimensionality arising from the dependence of the convergence rate of estimators on the number…
We consider the problem of estimating the context tree of a stationary ergodic process with finite alphabet without imposing additional conditions on the process. As a starting point we introduce a Hamming metric in the space of irreducible context trees and we use the properties of the weak topology in the space of ergodic stationary processes to prove that if the Hamming metric is unbounded, there exist no consistent estimators for the context tree. Even in the bounded case we show that there exist no two-sided confidence bounds. However we prove that one-sided inference is possible in this …
We consider the problem of estimating the slope parameter in functional linear regression, where scalar responses Y1,...,Yn are modeled in dependence of second order stationary random functions X1,...,Xn. An orthogonal series estimator of the functional slope parameter with additional thresholding in the Fourier domain is proposed and its performance is measured with respect to a wide range of weighted risks covering as examples the mean squared prediction error and the mean integrated squared error for derivative estimation. In this paper the minimax optimal rate of convergence of the estimat…
With regard to a three-step estimation procedure, proposed without theoretical discussion by Li and You in Journal of Applied Statistics and Management, for a nonparametric regression model with time-varying regression function, local stationary regressors and time-varying AR(p) (tvAR(p)) error process , we established all necessary asymptotic properties for each of estimator. We derive the convergence rate and asymptotic normality of the preliminary estimation of nonparametric regression function, establish the asymptotic distribution of time-varying coefficient functions in the error term, a…
We investigate the behavior of the Generalized Likelihood Ratio Test (GLRT) (Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153-193]) for time varying coefficient models where the regressors and errors are non-stationary time series and can be cross correlated. It is found that the GLRT retains the minimax rate of local alternative detection under weak dependence and non-stationarity. However, in general, the Wilks phenomenon as well as the classic residual bootstrap are sensitive to either conditional heteroscedasticity of the errors, non-stationarity or temporal dependence. An averaged test i…
In this paper, we develop nonparametric inference on spatial regression models as an extension of Lu and Tj\ostheim(2014), which develops nonparametric inference on density functions of stationary spatial processes under domain expanding and infill (DEI) asymptotics. In particular, we derive multivariate central limit theorems of mean and variance functions of nonparametric spatial regression models. Built upon those results, we propose a method to construct confidence bands for mean and variance functions.
We provide a framework for empirical process theory of locally stationary processes using the functional dependence measure. Our results extend known results for stationary Markov chains and mixing sequences by another common possibility to measure dependence and allow for additional time dependence. Our main result is a functional central limit theorem for locally stationary processes. Moreover, maximal inequalities for expectations of sums are developed. We show the applicability of our theory in some examples, for instance we provide uniform convergence rates for nonparametric regression wi…
In this paper some general theory is presented for locally stationary processes based on the stationary approximation and the stationary derivative. Laws of large numbers, central limit theorems as well as deterministic and stochastic bias expansions are proved for processes obeying an expansion in terms of the stationary approximation and derivative. In addition it is shown that this applies to some general nonlinear non-stationary Markov-models. In addition the results are applied to derive the asymptotic properties of maximum likelihood estimates of parameter curves in such models.
We consider discrete time models for asset prices with a stationary volatility process. We aim at estimating the multivariate density of this process at a set of consecutive time instants. A Fourier type deconvolution kernel density estimator based on the logarithm of the squared process is proposed to estimate the volatility density. Expansions of the bias and bounds on the variance are derived.
In this contribution we introduce weakly locally stationary time series through the local approximation of the non-stationary covariance structure by a stationary one. This allows us to define autoregression coefficients in a non-stationary context, which, in the particular case of a locally stationary Time Varying Autoregressive (TVAR) process, coincide with the generating coefficients. We provide and study an estimator of the time varying autoregression coefficients in a general setting. The proposed estimator of these coefficients enjoys an optimal minimax convergence rate under limited smo…
Many economic and causal parameters depend on nonparametric or high dimensional first steps. We give a general construction of locally robust/orthogonal moment functions for GMM, where moment conditions have zero derivative with respect to first steps. We show that orthogonal moment functions can be constructed by adding to identifying moments the nonparametric influence function for the effect of the first step on identifying moments. Orthogonal moments reduce model selection and regularization bias, as is very important in many applications, especially for machine learning first steps. We gi…
We provide new asymptotic theory for kernel density estimators, when these are applied to autoregressive processes exhibiting moderate deviations from a unit root. This fills a gap in the existing literature, which has to date considered only nearly integrated and stationary autoregressive processes. These results have applications to nonparametric predictive regression models. In particular, we show that the null rejection probability of a nonparametric t test is controlled uniformly in the degree of persistence of the regressor. This provides a rigorous justification for the validity of the …
The present paper deals with a nonparametric M-estimation for right censored regression model with stationary ergodic data. Defined as an implicit function, a kernel type estimator of a family of robust regression is considered when the covariate take its values in R^d (d >= 1) and the data are sampled from stationary ergodic process. The strong consistency (with rate) and the asymptotic distribution of the estimator are established under mild assumptions. Moreover, a usable confidence interval is provided which does not depend on any unknown quantity. Our results hold without any mixing condi…
There exists a wide literature on modelling strongly dependent time series using a longmemory parameter d, including more recent work on semiparametric wavelet estimation. As a generalization of these latter approaches, in this work we allow the long-memory parameter d to be varying over time. We embed our approach into the framework of locally stationary processes. We show weak consistency and a central limit theorem for our log-regression wavelet estimator of the time-dependent d in a Gaussian context. Both simulations and a real data example complete our work on providing a fairly general a…
In this paper we propose a new test for the hypothesis of a constant coefficient of variation in the common nonparametric regression model. The test is based on an estimate of the $L^2$-distance between the square of the regression function and variance function. We prove asymptotic normality of a standardized estimate of this distance under the null hypothesis and fixed alternatives and the finite sample properties of a corresponding bootstrap test are investigated by means of a simulation study. The results are applicable to stationary processes with the common mixing conditions and are used…
This paper outlines a new nonparametric estimation procedure for unobserved phi-mixing processes. It is assumed that the only information on the stationary hidden states (Xk) is given by the process (Yk), where Yk is a noisy observation of f(Xk). The paper introduces a maximum pseudo-likelihood procedure to estimate the function f and the distribution of the hidden states using blocks of observations of length b. The identifiability of the model is studied in the particular cases b=1 and b=2. The consistency of the estimators of f and of the distribution of the hidden states as the number of o…
In this paper, we investigate a nonparametric approach to provide a recursive estimator of the transition density of a non-stationary piecewise-deterministic Markov process, from only one observation of the path within a long time. In this framework, we do not observe a Markov chain with transition kernel of interest. Fortunately, one may write the transition density of interest as the ratio of the invariant distributions of two embedded chains of the process. Our method consists in estimating these invariant measures. We state a result of consistency and a central limit theorem under some gen…
We consider a continuous-time stochastic volatility model. The model contains a stationary volatility process, the multivariate density of the finite dimensional distributions of which we aim to estimate. We assume that we observe the process at discrete instants in time. The sampling times will be equidistant with vanishing distance. A multivariate Fourier-type deconvolution kernel density estimator based on the logarithm of the squared processes is proposed to estimate the multivariate volatility density. An expansion of the bias and a bound on the variance are derived. Key words: stochastic…
This paper aims at providing statistical guarantees for a kernel based estimation of time varying parameters driving the dynamic of local stationary processes. We extend the results of Dahlhaus et al. (2018) considering the local stationary version of the infinite memory processes of Doukhan and Wintenberger (2008). The estimators are computed as localized M-estimators of any contrast satisfying appropriate contraction conditions. We prove the uniform consistency and pointwise asymptotic normality of such kernel based estimators. We apply our result to usual contrasts such as least-square, lea…
We consider two kinds of stochastic volatility models. Both kinds of models contain a stationary volatility process, the density of which, at a fixed instant in time, we aim to estimate. We discuss discrete time models where for instance a log price process is modeled as the product of a volatility process and i.i.d. noise. We also consider samples of certain continuous time diffusion processes. The sampled time instants will be be equidistant with vanishing distance. A Fourier type deconvolution kernel density estimator based on the logarithm of the squared processes is proposed to estimate t…
In this paper, we consider a piecewise deterministic Markov process (PDMP), with known flow and deterministic transition measure, and unknown jump rate $λ$. To estimate nonparametrically the jump rate, we first construct an adaptive estimator of the stationary density, then we derive a quotient estimator $\hatλ_n$ of $λ$. We provide uniform bounds for the risk of these estimators, and prove that the estimator of the jump rate is nearly minimax (up to a $\ln^2(n)$ factor). Simulations illustrate the behavior of our estimator.
We study the nonparametric estimation of the jump density of a renewal reward process from one discretely observed sample path over [0,T]. We consider the regime when the sampling rate goes to 0. The main difficulty is that a renewal reward process is not a Levy process: the increments are non stationary and dependent. We propose an adaptive wavelet threshold density estimator and study its performance for the Lp loss over Besov spaces. We achieve minimax rates of convergence for sampling rates that vanish with T at polynomial rate. In the same spirit as Buchmann and Grübel (2003) and Duval (2…
We establish asymptotic properties of $M$-estimators, defined in terms of a contrast function and observations from a continuous-time locally stationary process. Using the stationary approximation of the sequence, $θ$-weak dependence, and hereditary properties, we give sufficient conditions on the contrast function that ensure consistency and asymptotic normality of the $M$-estimator. As an example, we obtain consistency and asymptotic normality of a localized least squares estimator for observations from a sequence of time-varying Lévy-driven Ornstein-Uhlenbeck processes. Furthermore, for a s…
In this work a method for statistical analysis of time series is proposed, which is used to obtain solutions to some classical problems of mathematical statistics under the only assumption that the process generating the data is stationary ergodic. Namely, three problems are considered: goodness-of-fit (or identity) testing, process classification, and the change point problem. For each of the problems a test is constructed that is asymptotically accurate for the case when the data is generated by stationary ergodic processes. The tests are based on empirical estimates of distributional distan…
We study the problem of nonparametric estimation of the fractional derivative of unknown spectral function of Gaussian stationary sequence (time series) and show that these problems is well posed with the classical speed of convergence when the order of derivative is less than 0.5. We prove also the asymptotical unbiaseness and normality of offered estimates with optimal speed of convergence. For the construction of the confidence region in some functional norm we establish the Central Limit Theorem in correspondent space of continuous functions for offered estimates.
The subject of this paper is the problem of estimating service time distribution of the $M/G/\infty$ queue from incomplete data on the queue. The goal is to estimate $G$ from observations of the queue--length process at the points of the regular grid on a fixed time interval. We propose an estimator and analyze its accuracy over a family of target service time distributions. The original $M/G/\infty$ problem is closely related to the problem of estimating derivatives of the covariance function of a stationary Gaussian process. We consider the latter problem and derive lower bounds on the minim…
The paper presents a systematic theory for asymptotic inference of autocovariances of stationary processes. We consider nonparametric tests for serial correlations based on the maximum (or ${\cal L}^\infty$) and the quadratic (or ${\cal L}^2$) deviations. For these two cases, with proper centering and rescaling, the asymptotic distributions of the deviations are Gumbel and Gaussian, respectively. To establish such an asymptotic theory, as byproducts, we develop a normal comparison principle and propose a sufficient condition for summability of joint cumulants of stationary processes. We adopt …
This paper provides a rigorous study of the nonparametric estimation of filaments or ridge lines of a probability density $f$. Points on the filament are considered as local extrema of the density when traversing the support of $f$ along the integral curve driven by the vector field of second eigenvectors of the Hessian of $f$. We `parametrize' points on the filaments by such integral curves, and thus both the estimation of integral curves and of filaments will be considered via a plug-in method using kernel density estimation. We establish rates of convergence and asymptotic distribution resu…
We derive an asymptotic theory of nonparametric estimation for a time series regression model $Z_t=f(X_t)+W_t$, where \ensuremath\{X_t\} and \ensuremath\{Z_t\} are observed nonstationary processes and $\{W_t\}$ is an unobserved stationary process. In econometrics, this can be interpreted as a nonlinear cointegration type relationship, but we believe that our results are of wider interest. The class of nonstationary processes allowed for $\{X_t\}$ is a subclass of the class of null recurrent Markov chains. This subclass contains random walk, unit root processes and nonlinear processes. We deriv…
In this paper, we survey some recent results on statistical inference (parametric and nonparametric statistical estimation, hypotheses testing) about the spectrum of stationary models with tapered data, as well as, a question concerning robustness of inferences, carried out on a linear stationary process contaminated by a small trend. We also discuss some question concerning tapered Toeplitz matrices and operators, central limit theorems for tapered Toeplitz type quadratic functionals, and tapered Fejér-type kernels and singular integrals. These are the main tools for obtaining the correspondi…
In this work we deal with the problem of fitting an error density to the goodness-of-fit test of the errors in nonlinear autoregressive time series models with stationary $α$-mixing error terms. The test statistic is based on the integrated squared error of the nonparametric error density estimate and the null error density. By deriving the asymptotic normality of test statistics in these models, we extend the result of Cheng and Sun (Statist. Probab. Lett. \textbf{78}, 1(2008), 50-59) in the model with i.i.d error terms to the more general case.
We consider a one-dimensional diffusion process $(X_t)$ which is observed at $n+1$ discrete times with regular sampling interval $Δ$. Assuming that $(X_t)$ is strictly stationary, we propose nonparametric estimators of the drift and diffusion coefficients obtained by a penalized least squares approach. Our estimators belong to a finite-dimensional function space whose dimension is selected by a data-driven method. We provide non-asymptotic risk bounds for the estimators. When the sampling interval tends to zero while the number of observations and the length of the observation time interval te…
We consider kernel estimation of marginal densities and regression functions of stationary processes. It is shown that for a wide class of time series, with proper centering and scaling, the maximum deviations of kernel density and regression estimates are asymptotically Gumbel. Our results substantially generalize earlier ones which were obtained under independence or beta mixing assumptions. The asymptotic results can be applied to assess patterns of marginal densities or regression functions via the construction of simultaneous confidence bands for which one can perform goodness-of-fit test…
We introduce a hierarchical nonparametric model for probability measures based on a multi-resolution transformation of probability distributions. The model allows a varying amount of shrinkage to be applied to data features of different scales and/or at different locations in the sample space, and the varying shrinkage level is locally adaptive to the empirical behavior of the data. Moreover, the model's hierarchical design---through a latent Markov tree structure---allows borrowing of information across locations and scales in setting the adaptive shrinkage level. Inference under the model pr…
Stochastic volatility modelling of financial processes has become increasingly popular. The proposed models usually contain a stationary volatility process. We will motivate and review several nonparametric methods for estimation of the density of the volatility process. Both models based on discretely sampled continuous time processes and discrete time models will be discussed. The key insight for the analysis is a transformation of the volatility density estimation problem to a deconvolution model for which standard methods exist. Three type of nonparametric density estimators are reviewed: …
We consider identification and inference about mean functionals of observed covariates and an outcome variable subject to nonignorable missingness. By leveraging a shadow variable, we establish a necessary and sufficient condition for identification of the mean functional even if the full data distribution is not identified. We further characterize a necessary condition for $\sqrt{n}$-estimability of the mean functional. This condition naturally strengthens the identifying condition, and it requires the existence of a function as a solution to a representer equation that connects the shadow va…
Two adaptive bandwidth selection methods for nonparametric estimators in locally stationary processes are proposed. We investigate a cross validation approach and a method based on contrast minimization and derive asymptotic properties of both methods. The results are applicable for different statistics under a broad setting of locally stationarity including nonlinear processes. At the same time we deepen the general framework for local stationarity based on stationary approximations. For example a general Bernstein inequality is derived for such processes. A simulation study performed on the …
This study examines a nonparametric inference on a stationary Lévy-driven Ornstein-Uhlenbeck (OU) process $X = (X_{t})_{t \geq 0}$ with a compound Poisson subordinator. We propose a new spectral estimator for the Lévy measure of the Lévy-driven OU process $X$ under macroscopic observations. We also derive, for the estimator, multivariate central limit theorems over a finite number of design points, and high-dimensional central limit theorems in the case wherein the number of design points increases with an increase in the sample size. Built on these asymptotic results, we develop methods to co…
Cascades of Poisson processes are probabilistic models for spatio-temporal phenomena in which (i) previous events may trigger subsequent events, and (ii) both the background and triggering processes are conditionally Poisson. Such phenomena are typically "data rich but knowledge poor", in the sense that large datasets are available yet a mechanistic understanding of the background and triggering processes which generate the data are unavailable. In these settings nonparametric estimation plays a central role. However existing nonparametric estimators have computational and storage complexity $…
In this paper, we study nonparametric models allowing for locally stationary regressors and a regression function that changes smoothly over time. These models are a natural extension of time series models with time-varying coefficients. We introduce a kernel-based method to estimate the time-varying regression function and provide asymptotic theory for our estimates. Moreover, we show that the main conditions of the theory are satisfied for a large class of nonlinear autoregressive processes with a time-varying regression function. Finally, we examine structured models where the regression fu…
Data can be assumed to be continuous functions defined on an infinite-dimensional space for many phenomena. However, the infinite-dimensional data might be driven by a small number of latent variables. Hence, factor models are relevant for functional data. In this paper, we study functional factor models for time-dependent functional data. We propose nonparametric estimators under stationary and nonstationary processes. We obtain estimators that consider the time-dependence property. Specifically, we use the information contained in the covariances at different lags. We show that the proposed …
A novel approach towards the spectral analysis of stationary random bivariate signals is proposed. Using the Quaternion Fourier Transform, we introduce a quaternion-valued spectral representation of random bivariate signals seen as complex-valued sequences. This makes possible the definition of a scalar quaternion-valued spectral density for bivariate signals. This spectral density can be meaningfully interpreted in terms of frequency-dependent polarization attributes. A natural decomposition of any random bivariate signal in terms of unpolarized and polarized components is introduced. Nonpara…
