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Register a new userDetecting relevant differences in the covariance operators of functional time series -- a sup-norm approach
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Holger Dette
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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In this paper we propose statistical inference tools for the covariance operators of functional time series in the two sample and change point problem. In contrast to most of the literature the focus of our approach is not testing the null hypothesis of exact equality of the covariance operators. Instead we propose to formulate the null hypotheses in them form that the distance between the operators is small, where we measure deviations by the sup-norm. We provide powerful bootstrap tests for these type of hypotheses, investigate their asymptotic properties and study their finite sample properties by means of a simulation study.
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For testing hypothesis on the covariance operator of functional time series, we suggest to use the full functional information and to avoid dimension reduction techniques. The limit distribution follows from the central limit theorem of the weak convergence of the partial sum process in general Hilbert space applied to the product space. In order to obtain critical values for tests, we generalize bootstrap results from the independent to the dependent case. This results can be applied to covariance operators, autocovariance operators and cross covariance operators. We discuss one sample and changepoint tests and give some simulation results.
In the Gaussian white noise model, we study the estimation of an unknown multidimensional function $f$ in the uniform norm by using kernel methods. The performances of procedures are measured by using the maxiset point of view: we determine the set of functions which are well estimated (at a prescribed rate) by each procedure. So, in this paper, we determine the maxisets associated to kernel estimators and to the Lepski procedure for the rate of convergence of the form $(log n/n)^{be/(2be+d)}$. We characterize the maxisets in terms of Besov and Holder spaces of regularity $beta$.
The problem of constructing a simultaneous confidence band for the mean function of a locally stationary functional time series $ { X_{i,n} (t) }_{i = 1, ldots, n}$ is challenging as these bands can not be built on classical limit theory. On the one hand, for a fixed argument $t$ of the functions $ X_{i,n}$, the maximum absolute deviation between an estimate and the time dependent regression function exhibits (after appropriate standardization) an extreme value behaviour with a Gumbel distribution in the limit. On the other hand, for stationary functional data, simultaneous confidence bands can be built on classical central theorems for Banach space valued random variables and the limit distribution of the maximum absolute deviation is given by the sup-norm of a Gaussian process. As both limit theorems have different rates of convergence, they are not compatible, and a weak convergence result, which could be used for the construction of a confidence surface in the locally stationary case, does not exist. In this paper we propose new bootstrap methodology to construct a simultaneous confidence band for the mean function of a locally stationary functional time series, which is motivated by a Gaussian approximation for the maximum absolute deviation. We prove the validity of our approach by asymptotic theory, demonstrate good finite sample properties by means of a simulation study and illustrate its applicability analyzing a data example.
Let $X$ be a centered Gaussian random variable in a separable Hilbert space ${mathbb H}$ with covariance operator $Sigma.$ We study a problem of estimation of a smooth functional of $Sigma$ based on a sample $X_1,dots ,X_n$ of $n$ independent observations of $X.$ More specifically, we are interested in functionals of the form $langle f(Sigma), Brangle,$ where $f:{mathbb R}mapsto {mathbb R}$ is a smooth function and $B$ is a nuclear operator in ${mathbb H}.$ We prove concentration and normal approximation bounds for plug-in estimator $langle f(hat Sigma),Brangle,$ $hat Sigma:=n^{-1}sum_{j=1}^n X_jotimes X_j$ being the sample covariance based on $X_1,dots, X_n.$ These bounds show that $langle f(hat Sigma),Brangle$ is an asymptotically normal estimator of its expectation ${mathbb E}_{Sigma} langle f(hat Sigma),Brangle$ (rather than of parameter of interest $langle f(Sigma),Brangle$) with a parametric convergence rate $O(n^{-1/2})$ provided that the effective rank ${bf r}(Sigma):= frac{{bf tr}(Sigma)}{|Sigma|}$ (${rm tr}(Sigma)$ being the trace and $|Sigma|$ being the operator norm of $Sigma$) satisfies the assumption ${bf r}(Sigma)=o(n).$ At the same time, we show that the bias of this estimator is typically as large as $frac{{bf r}(Sigma)}{n}$ (which is larger than $n^{-1/2}$ if ${bf r}(Sigma)geq n^{1/2}$). In the case when ${mathbb H}$ is finite-dimensional space of dimension $d=o(n),$ we develop a method of bias reduction and construct an estimator $langle h(hat Sigma),Brangle$ of $langle f(Sigma),Brangle$ that is asymptotically normal with convergence rate $O(n^{-1/2}).$ Moreover, we study asymptotic properties of the risk of this estimator and prove minimax lower bounds for arbitrary estimators showing the asymptotic efficiency of $langle h(hat Sigma),Brangle$ in a semi-parametric sense.
Prediction for high dimensional time series is a challenging task due to the curse of dimensionality problem. Classical parametric models like ARIMA or VAR require strong modeling assumptions and time stationarity and are often overparametrized. This paper offers a new flexible approach using recent ideas of manifold learning. The considered model includes linear models such as the central subspace model and ARIMA as particular cases. The proposed procedure combines manifold denoising techniques with a simple nonparametric prediction by local averaging. The resulting procedure demonstrates a very reasonable performance for real-life econometric time series. We also provide a theoretical justification of the manifold estimation procedure.
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