Rate of uniform convergence of statistical estimators of spectral density in spaces of differentiable functions

Bentkus, R.

doi:10.1007/bf00966738

Cited by 10 publications

(10 citation statements)

References 6 publications

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“…A prototypical assumption of this kind could, e.g., limit the magnitude of the derivative of f in a neighborhood of zero uniformly over . That is, one could require the existence of a finite constant M and a positive such that f ≤ M for all < and all f ∈ Indeed, under this type of assumption on the class , convergence to zero of the minimax risk for estimating f 0 and rates of convergence have been established; see Samarov (1977), Farrell (1979), and Bentkus (1985). Although assumptions on like the one just given seem to be indispensable if one wants to establish convergence of the minimax risk to zero, Theorem 4.1 demonstrates that such assumptions are less than innocuous as they exclude standard classes of spectral densities like AR MA ARMA , and ARMA r s r ≥ 1 s ≥ 1.…”

Section: Lower Risk Boundsmentioning

confidence: 99%

“…Similar results can be given for regression or density function estimation. We note that there is a growing number of papers abandoning the "pointwise" approach in favor of a minimax point of view, e.g., Samarov (1977), Farrell (1979), Ibragimov andKhashminskii (1980, 1981), Stone (1982), Bentkus (1985), Efroimovich and Pinsker (1986), Donoho and Johnston (1994), Donoho et al (1995), Lepski and Spokoiny (1997), to mention a few. The typical approach in these papers is to restrict the set sufficiently such as to rule out the discontinuity of the map h .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Lower Risk Bounds and Properties of Confidence Sets for Ill-Posed Estimation Problems with Applications to Spectral Density and Persistence Estimation, Unit Roots, and Estimation of Long Memory Parameters

Pötscher¹

2002

Econometrica

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Important estimation problems in econometrics like estimating the value of a spectral density at frequency zero, which appears in the econometrics literature in the guises of heteroskedasticity and autocorrelation consistent variance estimation and long run variance estimation, are shown to be "ill-posed" estimation problems. A prototypical result obtained in the paper is that the minimax risk for estimating the value of the spectral density at frequency zero is infinite regardless of sample size, and that confidence sets are close to being uninformative. In this result the maximum risk is over commonly used specifications for the set of feasible data generating processes. The consequences for inference on unit roots and cointegration are discussed. Similar results for persistence estimation and estimation of the long memory parameter are given. All these results are obtained as special cases of a more general theory developed for abstract estimation problems, which readily also allows for the treatment of other ill-posed estimation problems such as, e.g., nonparametric regression or density estimation.

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Section: Lower Risk Boundsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lower Risk Bounds and Properties of Confidence Sets for Ill-Posed Estimation Problems with Applications to Spectral Density and Persistence Estimation, Unit Roots, and Estimation of Long Memory Parameters

Pötscher¹

2002

Econometrica

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“…Remark 3.1. It is known from Bentkus (1985) that the minimax risk over some Besov ball f f aj f j â,2 < Rg cannot be smaller than C9R 2a(12â) n À2âa(2â1) , where C9 is some positive constant, and therefore our estimator is minimax (up to constants) over all such balls simultaneously.…”

Section: Adaptation To Unknown Smoothnessmentioning

confidence: 95%

“…In the nonparametric framework, Bentkus and Rudzkis (1976) proved large-deviation results for a projection spectral estimate based on a tapered periodogram. Bentkus (1985) computed optimal rates of convergence of spectral estimates in some spaces of differentiable functions. In both cases, the variables are Gaussian but the rates of convergence are asymptotic and the methods are not adaptive: this means that the de®nition of their estimator requires a priori knowledge of the smoothness of the function f .…”

Section: Introductionmentioning

confidence: 99%

Adaptive Estimation of the Spectrum of a Stationary Gaussian Sequence

Comte¹

2001

Bernoulli

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In this paper, we study the problem of nonparametric adaptive estimation of the spectral density f of a stationary Gaussian sequence. For this purpose, we consider a collection of ®nite-dimensional linear spaces (e.g. linear spaces spanned by wavelets or piecewise polynomials on possibly irregular grids or spaces of trigonometric polynomials). We estimate the spectral density by a projection estimator based on the periodogram and constructed on a data-driven choice of linear space from the collection. This data-driven choice is made via the minimization of a penalized projection contrast. The penalty function depends on k f k I , but we give results including the estimation of this bound. Moreover, we give extensions to the case of unbounded spectral densities (long-memory processes). In all cases, we state non-asymptotic risk bounds in L 2 -norm for our estimator, and we show that it is adaptive in the minimax sense over a large class of Besov balls.

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“…Essa taxa de convergência é um pouco mais lenta do que se os erros fossem i.i.d. onde, em tal caso, o melhor estimador linear de ondaletas é ótimo do ponto de vista da taxa de convergência minimax (BIERENS, 1983;BENTKUS, 1985). Note que a taxa de convergência do melhor estimador linear reside entre aqueles estimadores citados no Teorema 5.…”

Section: Melhor Estimador Linearunclassified

Regressão não-paramétrica com erros correlacionados via ondaletas.

Porto¹

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In this thesis, rates of convergence to zero are obtained for the estimation risk, for non-parametric regression using wavelets, when the errors are correlated. Four non-parametric regression methods using wavelets, with unequally spaced design are studied in the presence of correlated errors, that come from stochastic processes. Conditions on the errors and adaptrations to the procedures are presented, so that the estimators achieve quasi-minimax rates of convergence. Whenever is possible, rates of convergence are obtained for the estimators in the domain of the function, under mild conditions on the function to be estimated, on the design and on the error correlation. Through simulation studies, the behavior of some of the proposed methods is evaluated, when used on nite samples. Generally, it is suggested to use one of the studied methods, however applying thresholds by level. Since the estimation of the detail coecients can be dicult in some cases, it is also proposed a general semi-parametric iterative procedure, for wavelet methods in the presence of time-series errors.

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Rate of uniform convergence of statistical estimators of spectral density in spaces of differentiable functions

Cited by 10 publications

References 6 publications

Lower Risk Bounds and Properties of Confidence Sets for Ill-Posed Estimation Problems with Applications to Spectral Density and Persistence Estimation, Unit Roots, and Estimation of Long Memory Parameters

Lower Risk Bounds and Properties of Confidence Sets for Ill-Posed Estimation Problems with Applications to Spectral Density and Persistence Estimation, Unit Roots, and Estimation of Long Memory Parameters

Adaptive Estimation of the Spectrum of a Stationary Gaussian Sequence

Regressão não-paramétrica com erros correlacionados via ondaletas.

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