arXiv subjects

Theofanis Sapatinas

Publications and source records attributed to Theofanis Sapatinas.

25 records · Page 2Linked to original sources

arXiv2010

On convergence rates equivalency and sampling strategies in functional deconvolution models

Using the asymptotical minimax framework, we examine convergence rates equivalency between a continuous functional deconvolution model and its real-life discrete counterpart over a wide range of Besov balls and for the $L^2$-risk. For this purpose, all possible models are divided into three groups. For the models in the first group, which we call uniform, the convergence rates in the discrete and the continuous models coincide no matter what the sampling scheme is chosen, and hence the replacement of the discrete model by its continuous counterpart is legitimate. For the models in the second group, to which we refer as regular, one can point out the best sampling strategy in the discrete model, but not every sampling scheme leads to the same convergence rates; there are at least two sampling schemes which deliver different convergence rates in the discrete model (i.e., at least one of the discrete models leads to convergence rates that are different from the convergence rates in the continuous model). The third group consists of models for which, in general, it is impossible to devise the best sampling strategy; we call these models irregular. We formulate the conditions when each of these situations takes place. In the regular case, we not only point out the number and the selection of sampling points which deliver the fastest convergence rates in the discrete model but also investigate when, in the case of an arbitrary sampling scheme, the convergence rates in the continuous model coincide or do not coincide with the convergence rates in the discrete model. We also study what happens if one chooses a uniform, or a more general pseudo-uniform, sampling scheme which can be viewed as an intuitive replacement of the continuous model.

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arXiv2009

On Bayesian "testimation" and its application to wavelet thresholding

We consider the problem of estimating the unknown response function in the Gaussian white noise model. We first utilize the recently developed Bayesian maximum a posteriori "testimation" procedure of Abramovich et al. (2007) for recovering an unknown high-dimensional Gaussian mean vector. The existing results for its upper error bounds over various sparse $l_p$-balls are extended to more general cases. We show that, for a properly chosen prior on the number of non-zero entries of the mean vector, the corresponding adaptive estimator is simultaneously asymptotically minimax in a wide range of sparse and dense $l_p$-balls. The proposed procedure is then applied in a wavelet context to derive adaptive global and level-wise wavelet estimators of the unknown response function in the Gaussian white noise model. These estimators are then proven to be, respectively, asymptotically near-minimax and minimax in a wide range of Besov balls. These results are also extended to the estimation of derivatives of the response function. Simulated examples are conducted to illustrate the performance of the proposed level-wise wavelet estimator in finite sample situations, and to compare it with several existing

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arXiv2009

Minimax Goodness-of-Fit Testing in Multivariate Nonparametric Regression

We consider an unknown response function $f$ defined on $Δ=[0,1]^d$, $1\le d\le\infty$, taken at $n$ random uniform design points and observed with Gaussian noise of known variance. Given a positive sequence $r_n\to 0$ as $n\to\infty$ and a known function $f_0 \in L_2(Δ)$, we propose, under general conditions, a unified framework for the goodness-of-fit testing problem for testing the null hypothesis $H_0: f=f_0$ against the alternative $H_1: f\in\CF, \|f-f_0\|\ge r_n$, where $\CF$ is an ellipsoid in the Hilbert space $ L_2(Δ)$ with respect to the tensor product Fourier basis and $\|\cdot\|$ is the norm in $ L_2(Δ)$. We obtain both rate and sharp asymptotics for the error probabilities in the minimax setup. The derived tests are inherently non-adaptive. Several illustrative examples are presented. In particular, we consider functions belonging to ellipsoids arising from the well-known multidimensional Sobolev and tensor product Sobolev norms as well as from the less-known Sloan-Wo$\rm\acute{z}$niakowski norm and a norm constructed from multivariable analytic functions on the complex strip. Some extensions of the suggested minimax goodness-of-fit testing methodology, covering the cases of general design schemes with a known product probability density function, unknown variance, other basis functions and adaptivity of the suggested tests, are also briefly discussed.

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arXiv2009

Moment properties of multivariate infinitely divisible laws and criteria for self-decomposability

Ramachandran (1969, Theorem 8) has shown that for any univariate infinitely divisible distribution and any positive real number $α$, an absolute moment of order $α$ relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding L$\acute{\rm e}$vy measure. A generalized version of this result in the case of multivariate infinitely divisible distributions, involving the concept of g-moments, is given by Sato (1999, Theorem 25.3). We extend Ramachandran's theorem to the multivariate case, keeping in mind the immediate requirements under appropriate assumptions of cumulant studies of the distributions referred to; the format of Sato's theorem just referred to obviously varies from ours and seems to be having a different agenda. Also, appealing to a further criterion based on the L$\acute{\rm e}$vy measure, we identify in a certain class of multivariate infinitely divisible distributions the distributions that are self-decomposable; this throws new light on structural aspects of certain multivariate distributions such as the multivariate generalized hyperbolic distributions studied by Barndorff-Nielsen (1977) and others. Various points of relevance to the study are also addressed through specific examples.

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arXiv2009

Functional deconvolution in a periodic setting: Uniform case

We extend deconvolution in a periodic setting to deal with functional data. The resulting functional deconvolution model can be viewed as a generalization of a multitude of inverse problems in mathematical physics where one needs to recover initial or boundary conditions on the basis of observations from a noisy solution of a partial differential equation. In the case when it is observed at a finite number of distinct points, the proposed functional deconvolution model can also be viewed as a multichannel deconvolution model. We derive minimax lower bounds for the $L^2$-risk in the proposed functional deconvolution model when $f(\cdot)$ is assumed to belong to a Besov ball and the blurring function is assumed to possess some smoothness properties, including both regular-smooth and super-smooth convolutions. Furthermore, we propose an adaptive wavelet estimator of $f(\cdot)$ that is asymptotically optimal (in the minimax sense), or near-optimal within a logarithmic factor, in a wide range of Besov balls. In addition, we consider a discretization of the proposed functional deconvolution model and investigate when the availability of continuous data gives advantages over observations at the asymptotically large number of points. As an illustration, we discuss particular examples for both continuous and discrete settings.

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arXiv2008

Normalized least-squares estimation in time-varying ARCH models

We investigate the time-varying ARCH (tvARCH) process. It is shown that it can be used to describe the slow decay of the sample autocorrelations of the squared returns often observed in financial time series, which warrants the further study of parameter estimation methods for the model. Since the parameters are changing over time, a successful estimator needs to perform well for small samples. We propose a kernel normalized-least-squares (kernel-NLS) estimator which has a closed form, and thus outperforms the previously proposed kernel quasi-maximum likelihood (kernel-QML) estimator for small samples. The kernel-NLS estimator is simple, works under mild moment assumptions and avoids some of the parameter space restrictions imposed by the kernel-QML estimator. Theoretical evidence shows that the kernel-NLS estimator has the same rate of convergence as the kernel-QML estimator. Due to the kernel-NLS estimator's ease of computation, computationally intensive procedures can be used. A prediction-based cross-validation method is proposed for selecting the bandwidth of the kernel-NLS estimator. Also, we use a residual-based bootstrap scheme to bootstrap the tvARCH process. The bootstrap sample is used to obtain pointwise confidence intervals for the kernel-NLS estimator. It is shown that distributions of the estimator using the bootstrap and the ``true'' tvARCH estimator asymptotically coincide. We illustrate our estimation method on a variety of currency exchange and stock index data for which we obtain both good fits to the data and accurate forecasts.

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arXiv2005

A Functional Wavelet-Kernel Approach for Continuous-time Prediction

We consider the prediction problem of a continuous-time stochastic process on an entire time-interval in terms of its recent past. The approach we adopt is based on functional kernel nonparametric regression estimation techniques where observations are segments of the observed process considered as curves. These curves are assumed to lie within a space of possibly inhomogeneous functions, and the discretized times series dataset consists of a relatively small, compared to the number of segments, number of measurements made at regular times. We thus consider only the case where an asymptotically non-increasing number of measurements is available for each portion of the times series. We estimate conditional expectations using appropriate wavelet decompositions of the segmented sample paths. A notion of similarity, based on wavelet decompositions, is used in order to calibrate the prediction. Asymptotic properties when the number of segments grows to infinity are investigated under mild conditions, and a nonparametric resampling procedure is used to generate, in a flexible way, valid asymptotic pointwise confidence intervals for the predicted trajectories. We illustrate the usefulness of the proposed functional wavelet-kernel methodology in finite sample situations by means of three real-life datasets that were collected from different arenas.

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