Sharp optimality for density deconvolution with dominating bias

By Cristina Butucea, Alexandre B. Tsybakov. 2004

114 citations recorded by the source corpus.

We consider estimation of the common probability density $f$ of i.i.d. random variables $X_i$ that are observed with an additive i.i.d. noise. We assume that the unknown density $f$ belongs to a class $\mathcal{A}$ of densities whose characteristic function is described by the exponent $\exp(-\alpha |u|^r)$ as $|u|\to \infty$, where $\alpha >0$, $r>0$. The noise density is supposed to be known and such that its characteristic function decays as $\exp(-\beta |u|^s)$, as $|u| \to \infty$, where $\beta >0$, $s>0$. Assuming that $r<s$, we suggest a kernel type estimator that is optimal in sharp asymptotical minimax sense on $\mathcal{A}$ simultaneously under the pointwise and the $\mathbb{L}_2$-risks. The variance of the estimators turns out to be asymptotically negligible w.r.t. its squared bias. For $r<s/2$ we construct a sharp adaptive estimator of $f$. We discuss some effects of dominating bias, such as superefficiency of minimax estimators.

Subfields: Biophysics, Radiation, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/50a9e20b781f045a8f31a753cec58043cc64e94c

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Citations in the corpus

A ridge-parameter approach to deconvolution (2007, methodology): “An approach of this type has recently been discussed by Butucea and Tsybakov [3].”
GLOBAL UNIFORM RISK BOUNDS FOR WAVELET DECONVOLUTION ESTIMATORS (2011, methodology): “It has been extensively studied: we refer to Carroll and Hall [9], Stefanski [37], Stefanski and Carroll [38], Fan [14, 15], Diggle and Hall [12], Goldenshluger [22], Pensky and Vidakovic [36], Delaigle and Gijbels [11], Hesse and Meister [24], Johnstone et al. [25], Johnstone and Raimondo [26], Bis”
Nonparametric estimation of the characteristic triplet of a discretely observed Lévy process (2008, background): “1 The proof makes use of some of the ideas found in [27,34].”
Data‐driven density estimation in the presence of additive noise with unknown distribution (2011, methodology): “Moreover, it follows from results in Butucea and Tsybakov (2007) that a loss may be unavoidable in the adaptive procedure; in that case, the rate is called adaptive optimal.”
ON DECONVOLUTION OF DISTRIBUTION FUNCTIONS (2010, background): “The outlined problem is closely related to the density deconvolution problem that has been extensively studied in the literature; see, for example, [4, 5, 13, 18, 24, 27, 28] and references therein.”
Bayesian inverse problems with non-conjugate priors (2012, background): “Non-conjugate Bayesian inverse problems 2519 In this setting, unless the prior satisfies an analytic smoothness condition, the bias of the linear estimator dominates its variance [5, 24] and consequently the minimum of the prior smoothness and the unknown true smoothness determines the rate.”
Estimation of distributions, moments and quantiles in deconvolution problems (2008, background): “…convergence rates in cases where the unknown density belongs to a class of supersmooth functions, and the error distribution is ordinary-smooth; and Butucea and Tsybakov (2008), who provided sharp optimality results in settings where the unknown density and unknown error distribution are both…”
Spectral estimation of the fractional order of a Lévy process (2010, methodology): “, Cont and Tankov (2004) or Belomestny and Reiss (2006)], we still can get an asymptotically consistent estimates for the whole triple (μ,a, ν) given a consistent estimate of φt.”
Adaptive functional linear regression (2011, background): “Cardot et al. [2003] or Crambes et al. [2009]) and the estimation not only of the slope function (see e.”
Goodness-of-fit testing and quadratic functional estimation from indirect observations (2006, methodology): “Exact minimax rates have been found; see, for example, Lepski and Tsybakov [22] for the regression model with pointwise and sup-norm distances.”