A wavelet method for unfolding sphere size distributions

Antoniadis, Anestis; Fan, Jianqing; Gijbels, Irène

doi:10.2307/3316076

Cited by 9 publications

(8 citation statements)

References 29 publications

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“…Throughout this paper, as in, e.g., Hall and Smith (1988), Golubev and Levit (1998), Antoniadis, Fan and Gijbels (2001) and others, we consider squared radii of both the unobserved spheres of interest and the observed circular sections, which is more convenient mathematically and sometimes also more natural to interpret. As noted by Hall and Smith (1988, p. 411), "the practical motivation is that the squared radius is proportional to the observed cross-sectional area, which may be easier to measure than the radius."…”

Section: Introductionmentioning

confidence: 99%

Nonparametric confidence bands in Wicksell's problem

Wojdyła¹,

Szkutnik²

2018

STAT SINICA

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Although asymptotic nonparametric confidence bands have been constructed in the last decade in some inverse problems, like density deconvolution, inverse regression with a convolution operator, and regression with errors in variables, there seems to be no such construction for practically important inverse problems of stereology. Working with a kernel-type nonparametric estimator of the density of squared radii in the stereological Wicksell's problem, we partially fill this gap by constructing some corresponding asymptotic uniform confidence bands and an automatic bandwidth selection method, tuned to perform well in finite samples in terms of both area and coverage probability of the confidence bands. The performance of the new procedures is investigated in simulations and demonstrated with some astronomical data related to the M62 globular cluster.

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Section: Introductionmentioning

confidence: 99%

Nonparametric confidence bands in Wicksell's problem

Wojdyła¹,

Szkutnik²

2018

STAT SINICA

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“…It mainly considers one-sample estimation with two-dimensional observations, and many statistical and numerical procedures have been developed to overcome the ill-posed nature of the problem. For example, Hall & Smith (1988) , Van Es & Hoogendoorn (1990) and Golubev & Levit (1998) considered kernel smoothing methods, Nychka et al (1984) studied a spline-based method, Antoniadis et al (2001) proposed a wavelet method, and Groeneboom & Jongbloed (1995) considered an isotonized estimator. Many other statistical and numerical methods are surveyed in Chiu et al (2013) , who comment that no method has clear advantages.…”

Section: Introductionmentioning

confidence: 99%

Nonparametric maximum likelihood estimation for the multisample Wicksell corpuscle problem

Chan

Qin

2016

Biometrika

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We study nonparametric maximum likelihood estimation for the distribution of spherical radii using samples containing a mixture of one-dimensional, two-dimensional biased and three-dimensional unbiased observations. Since direct maximization of the likelihood function is intractable, we propose an expectation-maximization algorithm for implementing the estimator, which handles an indirect measurement problem and a sampling bias problem separately in the E- and M-steps, and circumvents the need to solve an Abel-type integral equation, which creates numerical instability in the one-sample problem. Extensions to ellipsoids are studied and connections to multiplicative censoring are discussed.

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“…In Abramovitch and Silverman [1], this method was compared with the similar vaguelette-wavelet decomposition. Other wavelet approaches, might be mentioned such as Antoniadis and Bigot [2], Antoniadis & al [3] and especially for the deconvolution problem, Penski & Vidakovic [37], Fan & Koo [17], Kalifa & Mallat [24], Neelamani & al [34].…”

Section: Projection Methodsmentioning

confidence: 99%

Estimation in inverse problems and second-generation wavelets

Kerkyacharian¹,

Picard²

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. We consider the problem of recovering a function f , when we receive a blurred (by a linear operator) and noisy version : Yε = Kf + εẆ . We will have as guides 2 famous examples of such inverse problems : the deconvolution and the Wicksell problem. The direct problem (K is the identity) isolates the denoising operation. It can't be solved unless accepting to estimate a smoothed version of f : for instance, if f has an expansion on a basis, this smoothing might correspond to stopping the expansion at some stage m. Then a crucial problem lies in finding an equilibrium for m considering the fact that for m large, the difference between f and its smoothed version is small, whereas the random effect introduces an error which is increasing with m. In the true inverse problem, in addition to denoising we have to 'inverse the operator' K, which operation not only creates the usual difficulties, but also introduces the necessity to control the additional instability due to the inversion of the random noise. Our purpose here is to emphasize the fact that in such a problem, there generally exists a basis which is fully adapted to the problem, where for instance the inversion remains very stable : this is the Singular Value Decomposition basis. On the other hand, the SVD basis might be difficult to determine and manipulate numerically, it also might not be appropriate for the accurate description of the solution with a small number of parameters. Also in many practical situations, the signal provides inhomogeneous regularity, and its local features are especially interesting to recover. In such cases, other bases (in particular localised bases such as wavelet bases) may be much more appropriate to give a good representation of the object at hand. Our approach here will be to produce estimation procedures trying to keep the advantages of localisation without loosing the stability and computability of SVD decompositions. We will especially consider two cases. In the first one (which is the case of the deconvolution example) we show that a fairly simple algorithm (WAVE-VD) using an appropriate thresholding technique performed on a standard wavelet system, enables us to estimate the object with rates which are almost optimal up to logarithm factors for any Lp loss function, and on the whole range of Besov spaces. In the second case (which is the case of the Wicksell example where the SVD bases lies in the range of Jacobi polynomials), we prove that quite a similar algorithm (NEED-VD) can be performed provided replacing the standard wavelet system by a second generation wavelet-type basis : the Needlets. We use here the construction (essentially following the work of Petrushev and co-authors) of a localised frame linked with a prescribed basis (here Jacobi polynomials) using a Littlewood Paley decomposition combined with a cubature formula. Section 5 describes the direct case (K = I). It has its own interest and will act as a guide for understanding the 'true' inverse models for a reader which is unfamiliar with nonpar...

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A wavelet method for unfolding sphere size distributions

Cited by 9 publications

References 29 publications

Nonparametric confidence bands in Wicksell's problem

Nonparametric confidence bands in Wicksell's problem

Nonparametric maximum likelihood estimation for the multisample Wicksell corpuscle problem

Estimation in inverse problems and second-generation wavelets

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