Nonparametric density estimation from observations with multiplicative measurement errors

Belomestny, Denis; Goldenshluger, Alexander

doi:10.1214/18-aihp954

Cited by 11 publications

(13 citation statements)

References 24 publications

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“…Let us now have a closer look at the second summand in Proposition 2.2 which bounds the variance term of the estimator. In fact, the growth of the second summand in k ∈ R + is determined by the decay of the Mellin transform of the error density g. In analogy to the usual deconvolution setting (compare [11]) and to the work of [5] we define the function class G 1.γ ⊂ G 0,1 of smooth error densities with decay γ…”

Section: Case Of Censored Observationmentioning

confidence: 99%

“…In this work, covering all those three variations of the multiplicative censoring model we consider a density estimator using the Mellin transform and a spectral cut-off regularization of its inverse, which borrows ideas from [5]. The key to the analysis of the multiplicative deconvolution problem is the convolution theorem of the Mellin transform M, which roughly states M[f Y ] = M[f ]M[g] for a density f Y = f * g. Exploiting the convolution theorem [5] introduce a kernel density estimator of f allowing more generally X and U to be real-valued. Moreover, they point out that the following widely used naive approach is a special case of their estimation strategy.…”

Section: Introductionmentioning

confidence: 99%

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Spectral cut-off regularisation for density estimation under multiplicative measurement errors

Miguel

Comte

Johannes

2021

Electron. J. Statist.

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We study the non-parametric estimation of an unknown density f with support on R + based on an i.i.d. sample with multiplicative measurement errors. The proposed fully-data driven procedure is based on the estimation of the Mellin transform of the density f , a regularisation of the inverse of the Mellin transform by a spectral cut-off and a data-driven model selection in order to deal with the upcoming bias-variance trade-off. We introduce and discuss further Mellin-Sobolev spaces which characterize the regularity of the unknown density f through the decay of its Mellin transform. Additionally, we show minimax-optimality over Mellin-Sobolev spaces of the data-driven density estimator and hence its adaptivity.

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Section: Case Of Censored Observationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Spectral cut-off regularisation for density estimation under multiplicative measurement errors

Miguel

Comte

Johannes

2021

Electron. J. Statist.

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“…We refer to the Discussion below and to Johannes [25] for general results on the quality of density estimators in a deconvolution problem when the "noise" law is unknown, generalizing results such as Fan [18] when the "noise" law is known. See also the study of Belomestny and Goldenshluger [4] in the case of a multiplicative measurement error leading to the use of Mellin transform techniques (instead of an additive error leading to the use of Fourier transform ones).…”

Section: Numerical Solution For U Bxmentioning

confidence: 99%

Estimating the division rate from indirect measurements of single cells

Doumic¹,

Olivier

2020

Discrete &Amp; Continuous Dynamical Systems - B

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Is it possible to estimate the dependence of a growing and dividing population on a given trait in the case where this trait is not directly accessible by experimental measurements, but making use of measurements of another variable? This article adresses this general question for a very recent and popular model describing bacterial growth, the so-called incremental or adder model. In this model, the division rate depends on the increment of size between birth and division, whereas the most accessible trait is the size itself. We prove that estimating the division rate from size measurements is possible, we state a reconstruction formula in a deterministic and then in a statistical setting, and solve numerically the problem on simulated and experimental data. Though this represents a severely ill-posed inverse problem, our numerical results prove to be satisfactory.

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“…and Φ O is given by (2). Note that the function αΦ O (φ/α; A i (x)) is obtained from continuous convexconcave function Φ O (·, ·) by projective transformation in the convex argument, and affine substitution in the concave argument, so that the former function is convex-concave and continuous on the domain {α > 0, φ ∈ F} × X i .…”

Section: The Estimatementioning

confidence: 99%

“…Suppose that the density f ξ is smooth with bounded second derivative, and that observations are subjected to "mixed" multiplicative censoring (see, e.g. [24,1,6,2]): the exact value of ξ k is observed with probability 0 ≤ θ ≤ 1, and with complementary probability, the available observation is η k ξ k , where η k is uniformly distributed over [0, 1].…”

Section: Illustration: Estimating Survival Ratementioning

confidence: 99%

Near-optimal recovery of linear and N-convex functions on unions of convex sets

Juditsky

Nemirovski

2019

Information and Inference: A Journal of the IMA

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In this paper we build provably near-optimal, in the minimax sense, estimates of linear forms and, more generally, "N -convex functionals" (an example being the maximum of several fractionallinear functions) of unknown "signal" from indirect noisy observations, the signal assumed to belong to the union of finitely many given convex compact sets. Our main assumption is that the observation scheme in question is good in the sense of [15], the simplest example being the Gaussian scheme where the observation is the sum of linear image of the signal and the standard Gaussian noise. The proposed estimates, same as upper bounds on their worst-case risks, stem from solutions to explicit convex optimization problems, making the estimates "computation-friendly." Immediate examples are affine-fractional functions f (x) = (a T x + a)/(b T x + b) with denominators positive on X , in particular, affine functions (N = 1), and piecewise linear functions like max[a T x + a, min[b T x + b, c T x + c]] (N = 3). A less trivial example is conditional quantile of a discrete distribution (N = 2), see Section 4.2.2 Our main results can be easily extended to the more general case of simple families -families of distributions specified in terms of upper bounds on their moment-generating functions, see [23,22] for details. Restricting the framework to the case of good observation schemes is aimed at streamlining the presentation.• Discrete o.s., where ω t are independent across t realizations of discrete random variable taking values 1, ..., m with probabilities affinely parameterized by x.The problem of (near-)optimal recovery of linear function f (x) on a convex compact set or a finite union of convex sets X has received much attention in the statistical literature (see, e.g., [20,10,12,13,14,11,7,8,9,21]). In particular, D. Donoho proved, see [11], that in the case of Gaussian observation scheme (1) and convex and compact X, the worst-case, over x ∈ X, risk of the minimax optimal affine in observations estimate is within factor 1.2 of the actual minimax risk. 3 Later, in [21], this near-optimality result was extended to other good observation schemes. In [8,9] the minimax affine estimator was used as "working horse" to build the near-optimal estimator of a linear functional over a finite union X of convex compact sets in the Gaussian observation scheme. As compared to the existing results, our contribution here is twofold. First, we pass from Gaussian o.s. to essentially more general good o.s.'s, extending in this respect the results of [8,9]. Second, we relax the requirement of affinity of the function to be recovered to N -convexity of the function. It should be stressed that the actual "common denominator" of the cited contributions and of the present work is the "operational nature" of the results, as opposed to typical results of non-parametric statistics which can be considered as descriptive. The traditional results present near-optimal estimates and their risks in a "closed analytical form," the toll being severe restrictions on...

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Nonparametric density estimation from observations with multiplicative measurement errors

Cited by 11 publications

References 24 publications

Spectral cut-off regularisation for density estimation under multiplicative measurement errors

Spectral cut-off regularisation for density estimation under multiplicative measurement errors

Estimating the division rate from indirect measurements of single cells

Near-optimal recovery of linear and N-convex functions on unions of convex sets

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