Adaptive Estimation of the Spectrum of a Stationary Gaussian Sequence

Comte, Fabienne

doi:10.2307/3318739

Cited by 26 publications

(56 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In times series analysis such covariance matrices describe among others the linear ARMA processes. The problem of adaptive estimation of the spectral density of an ARMA process has been studied by [17] (for known α) and adaptively to α via wavelet based methods by [28] and by model selection by [10]. In the case of an ARFIMA process, obtained by fractional differentiation of order d ∈ (−1/2, 1/2) of a casual invertible ARMA process, [31] gave adaptive estimators of the spectral density based on the log-periodogram regression model when the covariance matrix belongs to E(A, L).…”

Section: Introductionmentioning

confidence: 99%

Sharp minimax tests for large Toeplitz covariance matrices with repeated observations

Butucea

Zgheib

2016

Journal of Multivariate Analysis

View full text Add to dashboard Cite

We observe a sample of n independent p-dimensional Gaussian vectors with Toeplitz covariance matrix Σ = [σ |i−j| ] 1≤i,j≤p and σ 0 = 1. We consider the problem of testing the hypothesis that Σ is the identity matrix asymptotically when n → ∞ and p → ∞.We suppose that the covariances σ k decrease either polynomially ( k≥1 k 2α σ 2 k ≤ L for α > 1/4 and L > 0) or exponentially ( k≥1 e 2Ak σ 2 k ≤ L for A, L > 0). We consider a test procedure based on a weighted U-statistic of order 2, with optimal weights chosen as solution of an extremal problem. We give the asymptotic normality of the test statistic under the null hypothesis for fixed n and p → +∞ and the asymptotic behavior of the type I error probability of our test procedure. We also show that the maximal type II error probability, either tend to 0, or is bounded from above. In the latter case, the upper bound is given using the asymptotic normality of our test statistic under alternatives close to the separation boundary. Our assumptions imply mild conditions: n = o(p 2α−1/2 ) (in the polynomial case), n = o(e p ) (in the exponential case).We prove both rate optimality and sharp optimality of our results, for α > 1 in the polynomial case and for any A > 0 in the exponential case.A simulation study illustrates the good behavior of our procedure, in particular for small n, large p.

show abstract

Section: Introductionmentioning

confidence: 99%

Sharp minimax tests for large Toeplitz covariance matrices with repeated observations

Butucea

Zgheib

2016

Journal of Multivariate Analysis

View full text Add to dashboard Cite

show abstract

“…Applying a χ 2 -type inequality which initially appeared in Laurent and Massart (1998), was improved by Comte (2001) and furthermore by Gendre (2013), we derive that, for any x > 0,…”

Section: Proof Of Propositionmentioning

confidence: 92%

Laplace deconvolution with noisy observations

Abramovich¹,

Pensky²,

Rozenholc

2013

Electron. J. Statist.

View full text Add to dashboard Cite

International audienceIn the present paper we consider Laplace deconvolution on the basis of discrete noisy data observed on the interval which length may increase with a sample size. Although this problem arises in a variety of applications, to the best of our knowledge, it has not been systematically studied in statistical literature and the objective is to fill this gap. The present paper essentially provides the first comprehensive statistical treatment of Laplace deconvolution problem. The main contribution of the paper is the explicit construction of the Laplace deconvolution estimator. We show that the original Laplace deconvolution problem can be essentially reduced to estimating regression function and its derivatives on the interval of growing length T_n. Whereas the forms of the estimators essentially remain standard, the choices of the parameters and the minimax convergence rates, which are expressed in terms of T_n^2/n in this case, are affected by the asymptotic growth of the length of the interval. We derive an adaptive kernel estimator of the function of interest, and establish its asymptotic minimaxity over a range of Sobolev classes. We illustrate the theory by examples of construction of explicit expressions of Laplace deconvolution estimators. A limited simulation study shows that, in addition to providing asymptotic optimality as the number of observations turns to infinity, the proposed estimator demonstrates good performance in finite sample examples. The paper is concluded by a study of application of Laplace deconvolution to analysis of Dynamic Contrast Enhanced Computed Tomography data

show abstract

“…Moreover, it is possible to use π ∞ whereπ is some estimator of π . This method of random penalty (specifically with infinite norm) is successfully used in [7] and [12] for example, and can be applied here even if it means considering π regular enough. This is proved in Appendix A.…”

Section: Theoremmentioning

confidence: 97%

Adaptive estimation of the transition density of a Markov chain

Lacour¹

2007

Annales de l'Institut Henri Poincare (B) Probability and Statis

View full text Add to dashboard Cite

In this paper a new estimator for the transition density $\pi$ of an homogeneous Markov chain is considered. We introduce an original contrast derived from regression framework and we use a model selection method to estimate $\pi$ under mild conditions. The resulting estimate is adaptive with an optimal rate of convergence over a large range of anisotropic Besov spaces $B_{2,\infty}^{(\alpha_1,\alpha_2)}$. Some simulations are also presented

show abstract

Adaptive Estimation of the Spectrum of a Stationary Gaussian Sequence

Cited by 26 publications

References 30 publications

Sharp minimax tests for large Toeplitz covariance matrices with repeated observations

Sharp minimax tests for large Toeplitz covariance matrices with repeated observations

Laplace deconvolution with noisy observations

Adaptive estimation of the transition density of a Markov chain

Contact Info

Product

Resources

About